fuzzy (f-)transform of functions of two variables … · pr ace vlo zenym v datab azi dipl2. prohla...

UNIVERSITY OF OSTRAVA

FACULTY OF SCIENCE

DEPARTMENT OF MATHEMATICS

FUZZY (F-)TRANSFORM OF FUNCTIONS

OF TWO VARIABLES

AND ITS APPLICATIONS IN IMAGE

PROCESSING

Ph.D. THESIS

AUTHOR: Petra Hod’akova

SUPERVISOR: Irina Perfilieva

2014

OSTRAVSKA UNIVERZITA V OSTRAVE

PRIRODOVEDECKA FAKULTA

KATEDRA MATEMATIKY

FUZZY (F-)TRANSFORMACE FUNKCI DVOU

PROMENNYCH A JEJI APLIKACE

VE ZPRACOVANI OBRAZU

DOKTORSKA DISERTACNI PRACE

AUTOR: Petra Hod’akova

VEDOUCI PRACE: Irina Perfilieva

2014

Ja, nıze podepsana studentka, tımto cestne prohlasuji, ze text mnou odevzdane

zaverecne prace v pısemne podobe i na CD nosici je totozny s textem zaverecne

prace vlozenym v databazi DIPL2.

Prohlasuji, ze predlozena prace je mym puvodnım autorskym dılem, ktere jsem

vypracovala samostatne. Veskerou literaturu a dalsı zdroje, z nichz jsem pri zpra-

covanı cerpala, v praci radne cituji a jsou uvedeny v seznamu pouzite literatury.

Ostrava . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(podpis)

Beru na vedomı, ze tato doktorska disertacnı prace je majetkem Ostravske univerzity

(autorsky zakon C. 121/2000 Sb., §60 odst. 1), bez jejıho souhlasu nesmı byt nic

z obsahu prace publikovano.

Souhlasım s prezencnım zprıstupnenım sve prace v Univerzitnı knihovne Ostravske

univerzity.

Ostrava . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(podpis)

Acknowledgements

I want to express my gratitude to my supervisor, Prof. Irina Perfilieva, CSc.,

for her support, valuable comments and continuous encouragement, which enabled

the completion of this thesis. In addition, I would like to thank to all my colleagues

from the Institute for Research and Applications of Fuzzy Modeling for provid-

ing a friendly and creative atmosphere. In particular, I acknowledge Petr Hurtık

and Marek Vajgl for their fruitful co-operation and experimental and software sup-

port.

Ostrava, August 2014 Petra Hod’akova

5

Summary

The original fuzzy transform (F -transform, for short) was inspired by fuzzy mo-

deling. The F -transform was developed as a universal method that encompasses

classical transforms and fuzzy approximation models. Generally, the F -transform

transforms the original universe of functions into a universe of “skeleton models”

of those functions to facilitate further computation. This technique is very general

and is a powerful tool for addressing problems in a variety of fields, including image

processing, time series analysis, numerical solution of differential equations, and data

mining.

The F -transform was initially introduced for functions of one or two variables.

Subsequently, the ordinary F -transform of functions of one variable was generalized

to the F -transform of a higher degree (F s-transform, s ≥ 1) for functions of one

variable. The generalized F s-transform was proven to have better approximation

properties and to be more powerful in applications.

The goal of this thesis is to depict the general technique of F -transform of func-

tions of two variables on a background of a special Hilbert space. In our nota-

tion, the ordinary F -transform is denoted as the F 0-transform. Moreover, we ex-

tend the F s-transform, s ≥ 1, for functions of two variables, and on the basis

of this extension, we approximate partial derivatives of the original function by us-

ing the F s-transform components. We discuss and prove many interesting properties

of the F s-transform and its components. A significant portion of this thesis is de-

voted to the F 1-transform, which is further presented and used in the last part

of this thesis in an edge detection application.

The thesis is organized as follows. Chapter 1 is devoted to introducing the back-

ground and goal of the study. In Chapter 2, the basic tenets of various fuzzy parti-

tions are recalled, and a special Hilbert space is introduced. Properties of the Hilbert

space that are employed later in this thesis are also described. Chapter 3 defines

6

the F 0-transform for functions of two variables on the basis of orthogonal pro-

jection and recalls the main properties of the F 0-transform. Chapter 4 introduces

the F s-transforms, s ≥ 1, for functions of two variables. Moreover, the F 1-transform

and its properties are described in extensive detail, and the discrete F 1-transform

is ultimately introduced. In Chapter 5, the two main theorems for the approxima-

tion of the first partial derivatives and the approximation of the higher-order partial

derivatives using the F s-transform components are presented. Chapter 6 is devoted

to applications, particularly to the problem of detecting edges in an image. Two

approaches to the edge detection problem are presented. The first approach is based

on the F 0-transform, and the second is based on the F 1-transform. Finally, the re-

sults presented in this thesis are summarized in Chapter 7.

Keywords: F -transform, F 0-transform, F 1-transform, F s-transform, Approxima-

tion, Partial derivatives, Edge detection.

7

Anotace

Fuzzy transformace (kratce F -transformace) puvodne vychazı z oboru fuzzy

modelovanı. Technika F -transformace byla zavedena jako univerzalnı metoda, ktera

v sobe zahrnuje jak techniku klasicke transformace, tak i model pro fuzzy aproximaci.

Obecne, F -transformace transformuje univerzum funkcı do univerza tzv. koster

puvodnıch funkcı, aby jejich dalsı zpracovanı bylo jednodussı. Tato metoda se

ukazala byt velmi silnou ve smyslu obecneho nastroje pro resenı problemu v ruznych

oblastech, na prıklad ve zpracovanı obrazu, analyze casovych rad, numerickem resenı

diferencialnıch rovnic, zıskavanı informacı z dat, apod.

Nejprve byla F -transformace zavedena pro funkce jedne a dvou promennych.

Pozdeji byla puvodnı F -transformace funkcı jedne promenne zobecnena na F -trans-

formaci vyssıho radu (F s-transformaci, s ≥ 1) pro funkce jedne promenne. Bylo

dokazano, ze tato zobecnena F s-transformace ma lepsı aproximacnı vlastnosti, a tedy

i vetsı potencial pro aplikace.

Cılem teto disertacnı prace je popsat obecne techniku F -transformace pro funkce

dvou promennych definovanou na zaklade vlastnostı specialnıho Hilbertova prostoru.

V teto praci jsme zavedli jine oznacenı, a tedy puvodnı F -transformace je zde prezen-

tovana jako F 0-transformace. Dale jsme rozsırili techniku F s-transformace, s ≥ 1,

pro funkce dvou promennych a ukazali, jak lze pomocı komponent F s-transformace

aproximovat parcialnı derivace puvodnı funkce. Jsou zde take popsany a dokazany

dalsı zajımave vlastnosti F s-transformace a jejıch komponent. Podstatna cast prace

je venovana F 1-transformaci, ktera je pak dale prezentovana a pouzita v poslednı

casti teto prace, v aplikaci detekce hran.

Prace je strukturovana nasledovne. Kapitola 1 je venovana uvedenı do problemu

a motivaci tohoto vyzkumu. V Kapitole 2 jsou zavedeny zakladnı pojmy tykajıcı

se ruznych fuzzy rozkladu a je zde definovan specialnı Hilbertuv prostor. Uvadıme

take nektere vlastnosti Hilbertova prostoru, ktere jsou dale uzıvany v teto praci.

8

Kapitola 3 definuje F 0-transformaci pro funkce dvou promennych na zaklade orto-

gonalnı projekce a uvadı hlavnı vlastnosti F 0-transformace. Kapitola 4 definuje F s-

transformaci, s ≥ 1, pro funkce dvou promennych. Navıc, je zde podrobne rozebrana

F 1-transformace a jejı vlastnosti, vcetne disktretnı podoby teto metody. V Kapi-

tole 5 jsou uvedeny dva hlavnı teoremy tykajıcı se aproximace prvnıch parcialnıch

derivacı a aproximace parcialnıch derivacı vyssıch radu pomocı komponent F s-

transformace. Kapitola 6 je venovana aplikacım, konkretne, problemu detekce hran

v obraze. Jsou zde prezentovany dva prıstupy k resenı detekce hran. Prvnı prıstup

pouzıva k resenı F 0-transformaci, druhy prıstup pak F 1-transformaci. Shrnutı

vysledku prezentovanych v teto praci je uvedeno v Kapitole 7.

Klıcova slova: F -transformace, F 0-transformace, F 1-transformace, F s-transformace,

Aproximace, Parcialnı derivace, Detekce hran.

9

Table of Contents

Acknowledgements 5

Summary 6

Anotace 8

Table of Contents 10

List of Figures 12

1 Introduction 15

2 Preliminaries 18

2.1 Fuzzy Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.1 Fuzzy Partition With the Ruspini Condition . . . . . . . . . . 18

2.1.2 Fuzzy Partition With the Generalized Ruspini Condition . . . 19

2.1.3 Generalized Fuzzy Partition . . . . . . . . . . . . . . . . . . . 21

2.1.4 Fuzzy Partition of a Two-Dimensional Universe . . . . . . . . 23

2.2 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Space L2(Ak) . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.2 Subspace Lp2(Ak) . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.3 Space L2(Ak)× L2(Bl) . . . . . . . . . . . . . . . . . . . . . . 27

2.2.4 Subspace Ls2(Ak ×Bl) . . . . . . . . . . . . . . . . . . . . . . 28

3 F 0-transform of Functions of Two Variables 31

3.1 Direct F 0-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

10

3.1.1 F 0-transform Components Based on Various Partitions . . . . 37

3.2 Inverse F 0-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 F s-transform of Functions of Two Variables 43

4.1 Direct F s-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Inverse F s-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 F 1-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.1 Direct F 1-transform . . . . . . . . . . . . . . . . . . . . . . . 55

4.3.2 Simpler Form of F 1-transform Components . . . . . . . . . . . 58

4.3.3 Inverse F 1-transform . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.4 Discrete F 1-transform . . . . . . . . . . . . . . . . . . . . . . 63

5 Partial Derivatives 66

5.1 First Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Higher-Order Partial Derivatives . . . . . . . . . . . . . . . . . . . . . 72

6 Application to the Edge Detection Problem 80

6.1 Edge Detection Using the F 0-transform . . . . . . . . . . . . . . . . . 81

6.1.1 Main Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.1.2 F 0-transform-Based Algorithm for Edge Detection . . . . . . . 82

6.1.3 Experiments and Examples . . . . . . . . . . . . . . . . . . . 84

6.2 F 1-transform Edge Detector Inspired by Canny’s Algorithm . . . . . 88

6.2.1 Main Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2.2 Modified Canny’s algorithm . . . . . . . . . . . . . . . . . . . 90

6.2.3 Experiments and Examples . . . . . . . . . . . . . . . . . . . 92

7 Conclusion 96

Author’s Contributions 99

List of Author’s Publications 100

Bibliography 102

Index 107

11

List of Figures

2.1 The h-uniform fuzzy partition of [a, b] with the Ruspini condition. . . 20

2.2 Fuzzy 3-partition of [a, b] with the generalized Ruspini condition. . . . 21

2.3 Generalized fuzzy partition. . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Triangular shaped (0.4; 1)-uniform fuzzy partition of [1, 3] × [1, 4]

with the Ruspini condition. . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Cosine shaped (1; 1.5)-uniform fuzzy partition of [1, 3]×[1, 4] with the Rus-

pini condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Original function f(x, y) = (90/x+ x · sin(x/7))(90/y + y · sin(y/7)). 34

3.2 Illustration of the direct F 0-transform of function f(x, y) = (90/x+x·sin(x/7))(90/y+y · sin(y/7)) - the whole set of components displayed

on the original function - example 1. . . . . . . . . . . . . . . . . . . 35

3.3 Illustration of the direct F 0-transform of function f(x, y) = (90/x+x·sin(x/7))(90/y+y · sin(y/7)) - the whole set of components displayed

on the original function - example 2. . . . . . . . . . . . . . . . . . . 36

3.4 Illustration of the inverse F 0-transform of function f(x, y) = (90/x+

x · sin(x/7))(90/y + y · sin(y/7)) - example 1. . . . . . . . . . . . . . . 41



4.1 Illustration of the direct F 1-transform - original function x · y, single

F 0-transform and F 1-transform components. . . . . . . . . . . . . . . 57

12

4.2 Illustration of the direct F 1-transform of function f(x, y) = x · y -

the whole set of components - example 1. . . . . . . . . . . . . . . . . 57

4.3 Illustration of the direct F 1-transform of function f(x, y) = x · y -

the whole set of components - example 2. . . . . . . . . . . . . . . . . 58

4.4 Illustration of the direct F 1-transform of function f(x, y) = (90/x +

x · sin(x/7))(90/y + y · sin(y/7)) - the whole set of components. . . . 59





6.1 Edge detection by the F 0-transform - Original image. . . . . . . . . . 84

6.2 Edge detection by the F 0-transform - Image edge function Er (4 pixels

covered by one basic function). . . . . . . . . . . . . . . . . . . . . . 85

6.3 Edge detection by the F 0-transform - Image edge function Er (10

pixels covered by one basic function). . . . . . . . . . . . . . . . . . . 85

6.4 Edge detection by the F 0-transform - Image edge function Er (20

pixels covered by one basic function). . . . . . . . . . . . . . . . . . . 86

6.5 Partially blurred image - Original. . . . . . . . . . . . . . . . . . . . . 87

6.6 Edge detection by the F 0-transform - Partially blurred image (Er

with 10 pixels covered by one basic function). . . . . . . . . . . . . . 87

6.7 Edge detection by the F 0-transform - Partially blurred image (Er

with 100 pixels covered by one basic function). . . . . . . . . . . . . . 88

6.8 The F 1-transform edge detector: (a) original image; (b) gradient com-

ponents in the horizontal (x) direction; (c) gradient components in

the vertical (y) direction; (d) gradient magnitude; (e) gradient angles

discretization (0-degree edge is colored in yellow, 45 - in green, 90 - in

blue, and 135 - in red); (f) non-maxima suppression; (g) tracing edges

with hysteresis thresholding - the F 1-transform detector output; (h)

the Canny detector output. . . . . . . . . . . . . . . . . . . . . . . . 92

13

6.9 Lena - edge detection. . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.10 Geometrical patterns. Left: Original; Middle: Canny detector out-

put; Right: F 1-transform detector output. . . . . . . . . . . . . . . . 95

14

Chapter 1

Introduction

Integral transforms, for example, Fourier, Laplace, wavelet, etc., are used in classical

mathematics as powerful methods for solving various problems. The underlying

concept is to transform an original space of functions into a special space of functions

that facilitates further computation. In other words, the problems can be then solved

in “better” space under simpler conditions. Transformation back to the original

space produces either the original function or its approximation.

In general, an integral transform is performed using a kernel. The kernel is rep-

resented by a function of two variables and can be understood as a “collection

of local factors”. Each particular projection of the kernel characterizes some local

area of a domain of the space of the original functions.

In this work, we deal with a special transformation called fuzzy transform (F -

transform, for short). This method was developed in 2001 as a general method

for fuzzy modeling. Fuzzy modeling is regarded as a modern technique with a non-

classical background, evidenced by recent developments and applications of tech-

niques based on fuzzy modeling.

Similar to the integral transforms mentioned above, the F -transform performs

a transformation (projection) of an original universe of objects (functions) into a uni-

verse of their “skeleton models”. More specifically, the F -transform establishes a cor-

respondence between a set of continuous functions on an interval of real numbers

(space of reals) and a set of n-dimensional real vectors (matrices). The F -transform

15

can be characterized by a “discrete” kernel that generates a finite collection of fuzzy

subsets (a “fuzzy partition” of the space). Each component of the resulting skele-

ton model is a weighted local mean of the original function over an area “cove-

red” by a corresponding fuzzy subset from the fuzzy partition. The fuzzy subsets

in the partition can change, but such a change does not influence the approximation

properties of the F -transform. Therefore, the F -transform is a simplified representa-

tion of the original function and can be used instead of the original function. In this

respect, the F -transform can be as useful in applications as traditional transforms.

Moreover, the F -transform is sometimes more efficient than its counterparts.

Originally, the technique of F -transform (for functions of one or two variables)

was established as a vector of elements (fuzzy transform components) in which

the components are given explicitly by the defined expression. Generalization oc-

curred with the introduction of the fuzzy transform of a higher degree (for functions

of one variable) defined on the basis of orthogonal projection. The main goal of this

thesis is to unify the representation of the F -transform in general. We introduce

the theory of the ordinary F -transform and the F -transform of the higher degree

for functions of two variables on the basis of orthogonal projection in a special

Hilbert space.

The technique of F -transform was first introduced by I. Perfilieva in [29]. Ini-

tially, the F -transform was established for functions of one variable. The extension

of the F -transform for functions of two variables was introduced in [48]. Extended

results, developments and applications were subsequently presented in [25, 26, 27]

and deeply studied in [45].

Generalization of the ordinary F -transform to the F -transform of a higher degree

(F s-transform, s ≥ 1) for functions of one variable was introduced and developed

in [30, 31, 36]. The aim was to capture more information than that contained

in weighted local means of the original function. Components of the higher-degree

F -transform consist of polynomials in which the coefficients of the constituent mono-

mials are weighted local means of the corresponding derivatives of the original func-

tion. To retain the same notation, the original F -transform is denoted as the F 0-

transform.

16

In general, the method of F -transform is very powerful in many applications,

such as time series analysis [47, 46, 23], data analysis [37, 18], and differential equa-

tions [25, 49, 50, 38]. The F -transform of functions of two variables illustrates

the great potential in image processing, particularly image compression [21, 10],

image fusion [32, 44], and edge detection [4, 35, 34].

17

Chapter 2

Preliminaries

This chapter is devoted to the introduction of preliminary concepts. These concepts

will be used in the subsequent chapters as a background for definitions.

2.1 Fuzzy Partitions

Let us recall a conception of a fuzzy partition of a universe. We summarize various

fuzzy partitions with their special properties and present illustrative examples.

In the following definitions of fuzzy partitions, we first assume an interval [a, b]

on the real line R as a universe. We will later extend the notion of the fuzzy partition

to a two-dimensional universe [a, b]× [c, d].

2.1.1 Fuzzy Partition With the Ruspini Condition

The first fuzzy partition is called the fuzzy partition with the Ruspini condition

(or simply Ruspini partition) [27]. This fuzzy partition is specified by the Ruspini

condition, which implies normality of the respective fuzzy partition and therefore

can be denoted as the “partition-of-unity”.

Definition 1 Let x1, . . . , xn ∈ [a, b] be fixed nodes such that a = x1 < . . . < xn = b

and n ≥ 3. Fuzzy sets A1, . . . , An : [a, b] → [0, 1] identified with their member-

ship functions defined on [a, b] establish a Ruspini partition of [a, b] if they fulfill

18

the following conditions for k = 1, . . . , n:

1. Ak(xk) = 1;

2. Ak(x) = 0 if x ∈ [a, b] \ (xk−1, xk+1) and we set x0 = a, xn+1 = b;

3. Ak(x) is continuous on [xk−1, xk+1];

4. Ak(x) for k = 2, . . . , n strictly increases on [xk−1, xk] and for k = 1, . . . , n− 1

strictly decreases on [xk, xk+1];

5. for all x ∈ [a, b] holds the Ruspini condition (partition-of-unity)

n∑k=1

Ak(x) = 1. (2.1)

The membership functions A1, . . . , An are called basic functions. A point x ∈ [a, b]

is covered by the basic function Ak if Ak(x) > 0.

If the nodes x1, . . . , xn are h-equidistant, i.e., for all k = 2, . . . , n, xk = xk−1 + h,

where h = (b− a)/(n− 1), and hold two additional properties for k = 2, . . . , n− 1:

6. Ak(xk − x) = Ak(xk + x) for all x ∈ [0, h];

7. Ak(x) = Ak−1(x− h) and Ak+1(x) = Ak(x− h) for all x ∈ [xk, xk+1];

then the fuzzy partition A1, . . . , An is h-uniform.

The shape of the basic functions is not predetermined and therefore can be chosen

based on additional requirements (e.g., smoothness).

An illustration of an h-uniform fuzzy partition established by triangular shaped

basic functions is given in Figure 2.1.

2.1.2 Fuzzy Partition With the Generalized Ruspini Condi-

tion

The Ruspini condition (2.1) can be generalized by replacing “partition-of-unity”

by “partition-of-r”. The fuzzy partition with the generalized Ruspini condition (fuzzy

r-partition) was introduced in [42] by the following definition.

19

0

1

a = x1

A1

A2 A

3A

4 A5 A

6A

7A

8

x3

x2

x4

x5

b = x8

x6 x

7

Figure 2.1: The h-uniform fuzzy partition of [a, b] with the Ruspini condition.

Definition 2 Let r ≥ 1 and n ≥ 2 be fixed integers such that r ≤ n. Let a = x1 <

. . . < xn = b be nodes within [a, b], and let x1−r < . . . < x0 < a and b < xn+1 < . . . <

xn+r be nodes outside of [a, b]. A fuzzy r-partition of [a, b] is a family of n+ 2r − 2

continuous, normal, convex fuzzy sets A(r)2−r, . . . , A

(r)1 , . . . , A

(r)n , . . . , A

(r)n+r−1 such that

the following conditions are fulfilled:

1. for k = 1, . . . , n, A(r)k is a continuous function on [a, b] such that A

(r)k (xk) = 1

and A(r)k (x) = 0 for x /∈ [max(xk−r, a),min(xk+r, b)];

2. for k = 2, . . . , n− 1, A(r)k is increasing on [max(xk−r, a), xk] and decreasing

on [xk,min(xk+r, b)];

3. for k = −r + 2, . . . , 1, A(r)k is decreasing on [max(xk, a), xk+r];

4. for k = n, . . . , n+ r − 1, A(r)k is increasing on [xk−r,min(xk, b)];

5. for all x ∈ [a, b] the following generalized Ruspini condition (partition-of-r)

holdsn+r−1∑k=−r+2

A(r)k (x) = r. (2.2)

In the case of r = 1 the fuzzy r-partition (with the generalized Ruspini condition

(2.2)) becomes the partition-of-unity (with the standard Ruspini condition(2.1)).

20

0

1

a = x1

A3 A

4A

5A

6A

2A

7A1

A0

A−1

A8

A9

A10

x2

x3

x4

x5

x6

x7 b = x

8

Figure 2.2: Fuzzy 3-partition of [a, b] with the generalized Ruspini condition.

An example of the fuzzy partition with the generalized Ruspini condition is il-

lustrated in Figure 2.2.

2.1.3 Generalized Fuzzy Partition

The universal partition that will mostly be used in this thesis is the generalized fuzzy

partition [28]. The generalization in this case consists of the overlapping of basic

functions.

Definition 3 Let x0, x1, . . . , xn, xn+1 ∈ [a, b] be fixed nodes such that a = x0 ≤ x1 <

. . . < xn ≤ xn+1 = b, n ≥ 2. The fuzzy sets A1, . . . , An : [a, b] → [0, 1] constitute

a generalized fuzzy partition of [a, b] if for every k = 1, . . . , n, there exist h′

k, h′′

k ≥ 0

such that h′

k + h′′

k > 0, [xk − h′

k, xk + h′′

k] ⊆ [a, b] and the following conditions are

fulfilled:

1. (locality) – Ak(x) > 0 if x ∈ (xk − h′

k, xk + h′′

k) and Ak(x) = 0 if x ∈ [a, b] \(xk − h

′

k, xk + h′′

k);

2. (continuity) – Ak is continuous on [xk − h′

k, xk + h′′

k];

3. (covering) – for x ∈ [a, b],∑n

k=1Ak(x) > 0.

21

Figure 2.3: Generalized fuzzy partition.

By the locality and continuity conditions, it follows that∫ b

a

Ak(x)dx > 0.

If the nodes x0 = x1, x2, . . . , xn−1, xn = xn+1 are h-equidistant, i.e., for all k =

1, . . . , n+ 1, xk = xk−1 + h, where h = (b− a)/(n+ 1), h′> h/2 and two additional

properties are satisfied:

4. h′1 = h

′′n = 0, h

′′1 = h

′2 = · · · = h

′′n−1 = h

′n = h

′, and Ak(xk − x) = Ak(xk + x)

for all x ∈ [0, h′], k = 2, . . . , n− 1;

5. Ak(x) = Ak−1(x − h) and Ak+1(x) = Ak(x − h) for all x ∈ [xk, xk+1], k =

2, . . . , n− 1;

then the fuzzy partition is called (h, h′)-uniform generalized fuzzy partition.

The generalized fuzzy partition is illustrated in Figure 2.3.

An (h, h′)-uniform generalized fuzzy partition of [a, b] can also be defined using

the generating function

A0 : [−1, 1]→ [0, 1], (2.3)

which is assumed to be even, continuous and positive everywhere except on bound-

aries, where it vanishes. Then, basic functions Ak of an (h, h′)-uniform generalized

fuzzy partition are shifted copies of A0 in the sense that for all k = 1, . . . , n;

Ak(x) =

A0(x−xkh′

), x ∈ [xk − h′, xk + h

′],

0, otherwise.

As an example, we note that the function A0(x) = 1−|x| is a generating function

for all uniform triangular partitions. The difference between them is in parameters

h and h′.

22

2.1.4 Fuzzy Partition of a Two-Dimensional Universe

The original concept of the fuzzy partition of a one-dimensional universe can be fully

extended to an arbitrary finite-dimensional universe. In this section, we introduce

the fuzzy partition of a two-dimensional universe (two dimensions are sufficient

for the purposes of this thesis).

We assume a rectangle [a, b] × [c, d] as a universe and define a fuzzy partition

of this rectangle [27].

Definition 4 If fuzzy sets {Ak | k = 1, . . . , n} establish a fuzzy partition of [a, b]

and {Bl | l = 1, . . . ,m} establish a fuzzy partition of [c, d] then the Cartesian product

{Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} of these fuzzy partitions establishes a fuzzy

partition of the Cartesian product [a, b]× [c, d]. The membership function Ak ×Bl :

[a, b] × [c, d] → [0, 1] is equal to the product Ak · Bl of the respective membership

functions.

Remark 1 Analogous to the one-dimensional case, the fuzzy partition {Ak × Bl |k = 1, . . . , n, l = 1, . . . ,m} of [a, b]×[c, d] can be distinguished based on the particular

properties of the respective fuzzy partitions of [a, b] and [c, d] in the following way:

• If fuzzy sets {Ak | k = 1, . . . , n} establish an hx-uniform fuzzy partition

(with the Ruspini condition) of [a, b] and {Bl | l = 1, . . . ,m} establish an hy-

uniform fuzzy partition (with the Ruspini condition) of [c, d] then fuzzy sets

{Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} establish an (hx;hy)-uniform fuzzy par-

tition (with the Ruspini condition) of [a, b]× [c, d]. Moreover, if hx = hy = h,

we call it simply the h-uniform fuzzy partition of [a, b]× [c, d].

• If fuzzy sets {Ak | k = 1, . . . , n} establish a fuzzy r1-partition (with the gene-

ralized Ruspini condition) of [a, b] and {Bl | l = 1, . . . ,m} establish a fuzzy

r2-partition (with the generalized Ruspini condition) of [c, d] then fuzzy sets

{Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} establish a fuzzy (r1; r2)-partition

(with the generalized Ruspini condition) of [a, b] × [c, d]. When r1 = r2 = r,

we denote this partition as the fuzzy r-partition of [a, b]× [c, d].

23

Figure 2.4: Triangular shaped (0.4; 1)-uniform fuzzy partition of [1, 3] × [1, 4]with the Ruspini condition.

• If fuzzy sets {Ak | k = 1, . . . , n} establish an (hx, h′x)-uniform generalized

fuzzy partition of [a, b] and {Bl | l = 1, . . . ,m} establish an (hy, h′y)-uniform

generalized fuzzy partition of [c, d] then fuzzy sets {Ak × Bl | k = 1, . . . , n, l =

1, . . . ,m} establish an ((hx, h′x); (hy, h

′y))-uniform generalized fuzzy partition

of [a, b] × [c, d]. Analogous to the standard uniform fuzzy partition, if hx =

hy = h and h′x = h

′y = h

′, then the partition is called the (h, h

′)-uniform

generalized fuzzy partition of [a, b]× [c, d].

Illustrations of two (hx;hy)-uniform fuzzy partitions (with the Ruspini condition)

of [a, b]× [c, d] with different shapes of fuzzy sets are shown in Figures 2.4, 2.5.

2.2 Hilbert Space

In this section, we introduce a particular space of square integrable functions that

will be assumed as a Hilbert space with special properties. This Hilbert space will be

used in the following chapter as a background for the definition of the F s-transform,

24

1

2

3

4

1

1.5

2

2.5

30

0.2

0.4

0.6

0.8

1

Figure 2.5: Cosine shaped (1; 1.5)-uniform fuzzy partition of [1, 3] × [1, 4]with the Ruspini condition.

s ≥ 0.

The general theory and properties used in this section were mostly studied in [15,

5, 41, 3, 17]. Let us recall the main definitions of the Hilbert space.

Definition 5 A Hilbert space is a vector space H over R (C) together with an inner

product 〈., .〉 such that relative to the metric d(x, y) induced by the inner product, His a complete metric space.

Definition 6 Let H be a Hilbert space with the inner product 〈., .〉. The functions

f, g ∈ H are orthogonal in H if

〈f, g〉 = 0. (2.4)

If E ⊂ H then E⊥ is the set of all f ∈ H that are orthogonal to every g ∈ E.

The following theorem depicts an important aspect of the relationship between

a Hilbert space and its subspace; more specifically, it describes the best approxima-

tion in a Hilbert space.

Theorem 1 Let H be a Hilbert space with the norm ‖ · ‖, and let E be its subspace.

Then, for every element f ∈ H, there exists a unique best approximation g0 ∈ E

in the sense that g0 fulfills

‖f − g0‖ = infg∈E‖f − g‖.

25

The best approximation g0 ∈ E of f ∈ H is called an orthogonal projection of f

on E, and moreover, f − g0 ∈ E⊥.

Based on the definitions given above, in the following subsections, we will in-

troduce a particular Hilbert space of square-integrable functions. First, we present

a one-dimensional space to have a clear idea of the structure, followed by extension

to two-dimensional space.

2.2.1 Space L2(Ak)

Let us assume an interval [a, b] as the universe, and let fuzzy sets {Ak | k = 1, . . . , n}establish a fuzzy partition1 of [a, b] with nodes x0, x1, . . . , xn, xn+1. We fix one integer

k from {1, . . . , n}.

We introduce a Hilbert space L2(Ak) as a space of square-integrable functions:

f : [xk−1, xk+1]→ R,

where the inner product 〈f, g〉k of functions f, g ∈ L2(Ak) is given by

〈f, g〉k =

∫ xk+1

xk−1

f(x)g(x)Ak(x)dx. (2.5)

Let us remark that the inner product (2.5) is defined with respect to the weight

function given by Ak.

The corresponding norm ‖f‖k and the distance ρk(f, g) are as follows:

‖f‖k =√〈f, f〉k, (2.6)

ρk(f, g) = ‖f − g‖k. (2.7)

In the sequel, by L2([a, b]) we denote a set of functions f : [a, b] → R such that

for all k = 1, . . . , n,

f |[xk−1,xk+1]∈ L2(Ak),

where f |[xk−1,xk+1]is the restriction of f on [xk−1, xk+1].

1If there are not any restrictions or requirements for the fuzzy partition, then by the term “fuzzypartition”, we generally denote any type of fuzzy partition introduced in Section 2.1.

26

2.2.2 Subspace Lp2(Ak)

We still assume that k is a fixed integer from {1, . . . , n} and L2(Ak) is the Hilbert

space introduced above.

We denote by Lp2(Ak), p ≥ 0 a linear subspace of L2(Ak) with the basis given

by orthogonal polynomials

{P ik(x)}i=0,...,p, (2.8)

where p denotes a maximal degree of polynomials and orthogonality is considered

in the sense of (2.4). It is easy to see that the space Lp2(Ak), p ≥ 0 consists of all

polynomials of degree r ≤ p, and moreover, we can write the following chain:

L02(Ak) ⊂ L1

2(Ak) ⊂ · · · ⊂ Lp2(Ak) ⊂ . . .

The lemma below describes how an orthogonal projection of an arbitrary function

f ∈ L2([a, b]) on the subspace Lp2(Ak) can be characterized.

Lemma 1 Let f ∈ L2([a, b]) and let a polynomial F pk be the orthogonal projection

of f on the linear subspace Lp2(Ak) with the orthogonal basis given by (2.8). Then

F pk = c0kP

0k + c1kP

1k + · · ·+ cpkP

pk ,

where cik for i = 0, . . . , p is as follows:

cik =〈f, P i

k〉k〈P i

k, Pik〉k

=

∫ xk+1

xk−1f(x)P i

k(x)Ak(x)dx∫ xk+1

xk−1(P i

k(x))2Ak(x)dx.

The proof can be found in [31].

2.2.3 Space L2(Ak)× L2(Bl)

As a natural extension, let us now assume a rectangle [a, b]× [c, d] as the universe.

Let fuzzy sets {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} establish a fuzzy partition2

of [a, b]× [c, d] with the nodes x0, x1, . . . , xn, xn+1 ∈ [a, b] and y0, y1, . . . , ym, ym+1 ∈[c, d].


27

Moreover, we fix integers k, l from {1, . . . , n}, {1, . . . ,m}, respectively, and as-

sume L2(Ak), L2(Bl) are the Hilbert spaces defined in Section 2.2.1.

The Cartesian product of L2(Ak) and L2(Bl) is the Hilbert space L2(Ak)×L2(Bl)

of the functions of two variables:

f : [xk−1, xk+1]× [yl−1, yl+1]→ R.

The inner product 〈f, g〉kl is defined with respect to the weight functionsAk(x)Bl(y),

analogous to (2.5), i.e.,

〈f, g〉kl =

∫ xk+1

xk−1

∫ yl+1

yl−1

f(x, y)g(x, y)Ak(x)Bl(y)dxdy. (2.9)

The corresponding norm ‖f‖kl and the distance ρkl(f, g) are also given analo-

gously to (2.6) and (2.7), respectively.

Remark 2 Similar to the case of one-dimensional space L2(Ak), for the two-dimen-

sional space L2(Ak)× L2(Bl) holds the same property of orthogonality of two func-

tions, i.e., f, g ∈ L2(Ak)× L2(Bl) are orthogonal if

〈f, g〉kl = 0. (2.10)

Analogously, the important Theorem 1 holds in L2(Ak)× L2(Bl).

In the sequel, by L2([a, b]×[c, d]) we denote a set of functions f : [a, b]×[c, d]→ Rsuch that for all k = 1, . . . , n, l = 1, . . . ,m,

f |[xk−1,xk+1]×[yl−1,yl+1]∈ L2(Ak)× L2(Bl),

where f |[xk−1,xk+1]×[yl−1,yl+1]is the restriction of f on [xk−1, xk+1]× [yl−1, yl+1].

2.2.4 Subspace Ls2(Ak ×Bl)

Analogous to the extension of the space L2(Ak) to the space L2(Ak) × L2(Bl), we

introduce the extension of Lp2(Ak) to Ls2(Ak ×Bl).

We still have fixed k, l from {1, . . . , n}, {1, . . . ,m}, respectively, and assume

L2(Ak)× L2(Bl) is the Hilbert space defined in the previous section.

28

Let spaces Lp2(Ak), p ≥ 0 and Lr2(Bl), r ≥ 0 be linear subspaces of L2(Ak)

and L2(Bl), respectively, with the orthogonal bases given by polynomials:

{P ik(x)}i=0,...,p, {Qj

l (y)}j=0,...,r,

where p and r denote maximal degrees of polynomials and orthogonality is consid-

ered in the sense of (2.4).

Then, we can introduce the space Ls2(Ak × Bl), s ≥ 0 as a linear subspace

of L2(Ak) × L2(Bl) with the basis given by the following orthogonal polynomials

(orthogonality is now in the sense of (2.10)):

{Sijkl(x, y)}i=0,...,p; j=0,...,r; i+j≤s = {P ik(x) ·Qj

l (y)}i=0,...,p; j=0,...,r; i+j≤s. (2.11)

Remark 3 Let us remark that the space Ls2(Ak ×Bl) is not the same as the Carte-

sian product Lp2(Ak)×Lr2(Bl); the difference is the use of fewer of the possible com-

binations of orthogonal basis polynomials. Therefore, s ≤ p + r. In the case where

s = p+ r, the space Ls2(Ak ×Bl) coincides with Lp2(Ak)× Lr2(Bl).

In fact, s is the maximal degree of products P ik(x)Qj

l (y) such that i + j ≤ s.

For example, the basis of the space L12(Ak ×Bl) is established by the following poly-

nomials:

P 0k (x)Q0

l (y)︸︷︷︸S00kl (x,y)

, P 1k (x)Q0

l (y)︸︷︷︸S10kl (x,y)

, P 0k (x)Q1

l (y)︸︷︷︸S01kl (x,y)

. (2.12)

Remark 4 As it was introduced above, the space Ls2(Ak ×Bl), s ≥ 0 consists of all

polynomials of degree t ≤ s; therefore, we can write the following chain:

L02(Ak ×Bl) ⊂ L1

2(Ak ×Bl) ⊂ · · · ⊂ Ls2(Ak ×Bl) ⊂ . . .

Analogous to Lemma 1, the following lemma characterizes the orthogonal pro-

jection of a function f ∈ L2([a, b]× [c, d]) or the best approximation of f in the space

Ls2(Ak ×Bl).

Lemma 2 Let f ∈ L2([a, b] × [c, d]) and let Ls2(Ak × Bl) be a linear subspace

of L2(Ak) × L2(Bl), as specified above. Then, the orthogonal projection F skl of f

on Ls2(Ak ×Bl), s ≥ 0 is equal to

F skl =

∑0≤i+j≤s

cijklSijkl, (2.13)

29

where

cijkl =〈f, Sijkl〉kl〈Sijkl, S

ijkl〉kl

=

∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)Sijkl(x, y)Ak(x)Bl(y)dxdy∫ yl+1

yl−1

∫ xk+1

xk−1(Sijkl(x, y))2Ak(x)Bl(y)dxdy

. (2.14)

The proof is constructed analogously to the case of functions of one variable

(Lemma 1 [31]).

30

Chapter 3

F 0-transform of Functions of Two

Variables

In this chapter, we provide an overview of the main ideas and principles of the F 0-

transform of functions of two variables. We specify the form of components based

on the various fuzzy partitions introduced in Section 2.1.

Let us remark that by the term F 0-transform, we denote the ordinary F -transform,

which was first introduced in [29] and then extended for functions of two variables

in [48, 27]. Originally, the technique of F -transform was established as a vector (ma-

trix) of components in which the components are defined by the explicit expressions.

In this thesis, we define the F -transform as the orthogonal projection of a linear

space to its special subspace in which the expressions for the F -transform compo-

nents are derived from the properties of the orthogonal projection.

We assume that the reader is familiar with the theory of the F -transform; there-

fore, we do not describe all details in this chapter. We present the main ideas

of the F -transform from our perspective. On the basis of this new representation

of the F -transform and to maintain the same notation throughout the thesis, we

denote it as the F 0-transform.

31

3.1 Direct F 0-transform

Let [a, b] × [c, d] be the universe, and let fuzzy sets {Ak × Bl | k = 1, . . . , n, l =

1, . . . ,m} establish a fuzzy partition1 of [a, b]× [c, d].

Moreover, let L2(Ak) × L2(Bl), k = 1, . . . , n, l = 1, . . . ,m, be the Hilbert

space, and let f ∈ L2([a, b] × [c, d]), i.e., for all k = 1, . . . , n, l = 1, . . . ,m,

f |[xk−1,xk+1]×[yl−1,yl+1]∈ L2(Ak)× L2(Bl) (see Section 2.2.3).

To introduce the F 0-transform of f ∈ L2([a, b] × [c, d]), we assume L02(Ak) ⊆

L2(Ak), k = 1, . . . , n and L02(Bl) ⊆ L2(Bl), l = 1, . . . ,m are the linear spans

of the sets consisting of zero-degree polynomials, i.e., the orthogonal bases of L02(Ak)

and L02(Bl) are the following constant functions of variables x and y, respectively,

P 0k (x) = 1, Q0

l (y) = 1.

Then, consequently, let L02(Ak × Bl) ⊆ L2(Ak) × L2(Bl) be the linear span

of the set consisting of the following constant function of two variables x, y

S00kl (x, y) = 1. (3.1)

The following definition introduces the direct F 0-transform of f ∈ L2([a, b] ×[c, d]) based on the orthogonal projection of f on L0

2(Ak × Bl), k = 1, . . . , n, l =

1, . . . ,m (see Lemma 2).

Definition 7 Let f ∈ L2([a, b]× [c, d]) and let {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m}be a fuzzy partition of [a, b] × [c, d]. Moreover, let F 0

kl be the orthogonal projection

(2.13) of f on subspace L02(Ak×Bl), k = 1, . . . , n, l = 1, . . . ,m, with the basis given

by (3.1).

A (n×m) matrix F0nm[f ] is the direct F 0-transform of f with respect to {Ak×Bl |

k = 1, . . . , n, l = 1, . . . ,m} where

F0nm[f ] =

F 011 . . . F 0

1m

......

...

F 0n1 . . . F 0

nm

.


32

F 0kl, k = 1, . . . , n, l = 1, . . . ,m is called the F 0-transform component.

By Lemma 2, for each k = 1, . . . , n, l = 1, . . . ,m, the F 0-transform component

has the following representation:

F 0kl = c00kl · 1,

where 1 denotes the constant function from the basis (3.1) of the space L02(Ak×Bl).

More precisely, for each k = 1, . . . , n, l = 1, . . . ,m,

F 0kl =

∫ xk+1

xk−1

∫ yl+1

yl−1f(x, y)Ak(x)Bl(y)dxdy∫ xk+1

xk−1

∫ yl+1

yl−1Ak(x)Bl(y)dxdy

. (3.2)

Remark 5 Originally, the F 0-transform components were explicitly defined on [a, b]×[c, d] for each k = 1, . . . , n, l = 1, . . . ,m as follows:

F 0kl =

∫ ba

∫ dcf(x, y)Ak(x)Bl(y)dxdy∫ b

a

∫ dcAk(x)Bl(y)dxdy

.

In our definition, the F 0-transform components rely on the polynomials S00kl (x, y)

(3.1) defined on [xk−1, xk+1] × [yl−1, yl+1], k = 1, . . . , n, l = 1, . . . ,m. To en-

sure the correspondence between both approaches, the polynomials S00kl (x, y) can be

formally extended to [a, b] × [c, d] by setting S00kl (x, y) = 0 outside the rectangle

[xk−1, xk+1]× [yl−1, yl+1], k = 1, . . . , n, l = 1, . . . ,m.

Graphical examples of two direct F 0-transforms of a function f (original in Fig-

ure 3.1) as sets of different numbers of constant components (points) are shown

in Figures 3.2, 3.3.

In the following, we recall the main properties of the (direct) F 0-transform [28].

Let f ∈ L2([a, b]× [c, d]), and let {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m} be the fixed

fuzzy partition of [a, b] × [c, d] and F0nm[f ] = (F 0

kl), k = 1, . . . , n, l = 1, . . . ,m, be

the F 0-transform of f with respect to {Ak ×Bl | k = 1, . . . , n, l = 1, . . . ,m}.

(A) F0nm[f ] is an image of a linear mapping from L2([a, b]× [c, d]) to L0

2(A1×B1)×· · · × L0

2(An ×Bm) where for all functions f, g, h ∈ L2([a, b]× [c, d]) such that

f = αg + βh where α, β ∈ R the following holds

F0nm[f ] = αF0

nm[g] + βF0nm[h].

33

Figure 3.1: Original function f(x, y) = (90/x+ x · sin(x/7))(90/y + y · sin(y/7)).

(B) The kl-th component of F0nm[f ] = (F 0

kl), k = 1, . . . , n, l = 1, . . . ,m gives

minimum to the function

Φkl(t) =

∫ b

a

∫ d

c

(f(x, y)− t)2Ak(x)Bl(y)dxdy, (3.3)

defined on L02(Ak × Bl), k = 1, . . . , n, l = 1, . . . ,m. Therefore, F 0

kl is the best

approximation of f in L02(Ak ×Bl), k = 1, . . . , n, l = 1, . . . ,m.

(C) If for all (x, y) ∈ [a, b]×[c, d] the function f(x, y) = C, then for all k = 1, . . . , n,

l = 1, . . . ,m the components F 0kl = C.

Let us present a lemma confirming approximation properties of the F 0-transform

components. It was proven that the components of the F 0-transform of f are

equal to precise values of f at the respective nodes of the uniform fuzzy parti-

tion (with the Ruspini condition) up to a certain accuracy ([45]). In the follow-

ing lemma, we generalize this result for F 0-transform components of the function

f ∈ L2([a, b]× [c, d]) with the generalized fuzzy partition.

Lemma 3 Let f ∈ L2([a, b]×[c, d]) and let {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m} be

an (h, h′)-uniform generalized fuzzy partition of [a, b]× [c, d]. Moreover, let functions

f , Ak, Bl be twice continuously differentiable on [a, b] × [c, d]. Then, for every

k = 1, . . . , n, l = 1, . . . ,m the following holds:

c00kl = f(xk, yl) +O(h′).

34

Figure 3.2: Illustration of the direct F 0-transform of function f(x, y) = (90/x+ x ·sin(x/7))(90/y+y ·sin(y/7)) - the whole set of components displayed on the originalfunction - example 1.

proof: We recall that h = xk − xk−1 = yl − yl−1; k = 1, . . . , n + 1; l =

1, . . . ,m+1 and h′> h/2. We will give the proof for fixed values of k, l from {1, . . . , n}

and {1, . . . ,m}, respectively. Let

c00kl =

∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)Ak(x)Bl(y)dxdy

(∫ xk+1

xk−1Ak(x)dx)(

∫ yl+1

yl−1Bl(y)dy)

.

We denote the particular integrals from the formula given above as follows:

I1(y) =

∫ xk+1

xk−1

f(x, y)Ak(x)dx =

∫ xk+h′

xk−h′f(x, y)Ak(x)dx,

I2 =

∫ yl+1

yl−1

I1(y)Bl(y)dy =

∫ yl+h′

yl−h′I1(y)Bl(y)dy,

I3 =

∫ xk+1

xk−1

Ak(x)dx =

∫ xk+h′

xk−h′Ak(x)dx,

I4 =

∫ yl+1

yl−1

Bl(y)dy =

∫ yl+h′

yl−h′Bl(y)dy.

Then, we apply the trapezoidal rule with nodes xk−h′, xk, xk+h

′to the integrals

I1, I3 and the same rule with nodes yl − h′, yl, yl + h

′to the integrals I2, I4. We use

the properties:

35

Figure 3.3: Illustration of the direct F 0-transform of function f(x, y) = (90/x+ x ·sin(x/7))(90/y+y ·sin(y/7)) - the whole set of components displayed on the originalfunction - example 2.

Ak(xk − h′) = Ak(xk + h

′) = 0,

Bl(yl − h′) = Bl(yl + h

′) = 0.

I1(y) =

∫ xk+h′

xk−h′f(x, y)Ak(x)dx =

h′[1/2f(xk − h

′, y)Ak(xk − h

′) + f(xk, y)Ak(xk)+

1/2f(xk + h′, y)Ak(xk + h

′)] +R1(y) =

h′[f(xk, y)Ak(xk)] +R1(y),

I2 =

∫ yl+h′

yl−h′I1(y)Bl(y)dy =

∫ yl+h′

yl−h′[h′f(xk, y)Ak(xk) +R1(y)]Bl(y)dy =

h′[(h′f(xk, yl)Ak(xk) +R1(yl))Bl(yl)] +R2 =

h′2f(xk, yl)Ak(xk)Bl(yl) + h

′Bl(yl)R1(yl) +R2,

I3 =

∫ xk+h′

xk−h′Ak(x)dx = h

′Ak(xk) +R3,

36

I4 =

∫ yl+h′

yl−h′Bl(y)dy = h

′Bl(yl) +R4.

By the trapezoidal rule, the estimations of errors R1, R2, R3, R4 are as follows:

Ri = − [(xk + h′)− (xk − h

′)]h′2Mi

12= −h

′3Mi

6; i = 1, 3,

Rj = − [(yl + h′)− (yl − h

′)]h′2Mj

12= −h

′3Mj

6; j = 2, 4,

where

M1 =∂2[fAk]

∂x2

∣∣∣∣(ξ,y)

M2 = [f(xk, ·)Bl(·)]′′(η),

M3 = A′′

k(ϑ),

M4 = B′′

l (ζ).

and ξ, ϑ ∈ (xk − h′, xk + h

′), η, ζ ∈ (yl − h

′, yl + h

′).

Then, the final estimation of the coefficient c00kl is the following:

c00kl =h′2f(xk, yl)Ak(xk)Bl(yl) + h

′Bl(yl)R1(yl) +R2

h′2Ak(xk)Bl(yl) + h′Ak(xk)R4 + h′Bl(yl)R3 +R3R4

=

f(xk, yl) +O(h′2) +O(h

′)

1 +O(h′2) +O(h′2) +O(h′4)= f(xk, yl) +O(h

′).

2

3.1.1 F 0-transform Components Based on Various Parti-

tions

The F 0-transform of a function f is defined with respect to the chosen fuzzy partition

of the domain of f . In particular applications, we can use various fuzzy partitions.

In this subsection, we show how the F 0-transform components (3.2) can be simplified

according to the chosen fuzzy partition (see Section 2.1).

37

Let f ∈ L2([a, b] × [c, d]) and F0nm[f ] = (F 0

kl), k = 1, . . . , n, l = 1, . . . ,m be

the F 0-transform of f with respect to the chosen fuzzy partition.

• Let fuzzy sets {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m} establish an ((hx, h′x); (hy, h

′y))-

uniform generalized fuzzy partition of [a, b]×[c, d]. Moreover, let the particular

fuzzy partitions of [a, b] and [c, d] be defined by the generating functions A0

and B0 (2.3), respectively. We denote

a0 =

∫ 1

−1A0dx, b0 =

∫ 1

−1B0dy.

Then, for every k = 1, . . . , n, l = 1, . . . ,m, the F 0-transform components (3.2)

can be simplified as follows:

F 0kl =

∫ xk+1

xk−1

∫ yl+1

yl−1f(x, y)Ak(x)Bl(y)dxdy

h′xh′ya0b0

, (3.4)

where we use the fact that∫ xk+1

xk−1

Ak(x)dx = h′

x

∫ 1

−1A0dx = h

′

xa0,

and analogously ∫ yl+1

yl−1

Bl(y)dy = h′

y

∫ 1

−1B0dy = h

′

yb0.

• Let fuzzy sets {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} establish an (hx;hy)-

uniform fuzzy partition (with the Ruspini condition) of [a, b] × [c, d]. Then,

the F 0-transform components (3.2) are in the following form (proof in [45]):

F 011 =

4

hxhy

∫ xk+1

xk−1

∫ yl+1

yl−1

f(x, y)A1(x)B1(y)dxdy,

F 01m =

4

hxhy

∫ xk+1

xk−1

∫ yl+1

yl−1

f(x, y)A1(x)Bm(y)dxdy,

F 0n1 =

4

hxhy

∫ xk+1

xk−1

∫ yl+1

yl−1

f(x, y)An(x)B1(y)dxdy,

F 0nm =

4

hxhy

∫ xk+1

xk−1

∫ yl+1

yl−1

f(x, y)An(x)Bm(y)dxdy,

38

and for all k = 2, . . . , n− 1, l = 2, . . . ,m− 1

F 0k1 =

2

hxhy

∫ xk+1

xk−1

∫ yl+1

yl−1

f(x, y)Ak(x)B1(y)dxdy,

F 0km =

2

hxhy

∫ xk+1

xk−1

∫ yl+1

yl−1

f(x, y)Ak(x)Bm(y)dxdy,

F 01l =

2

hxhy

∫ xk+1

xk−1

∫ yl+1

yl−1

f(x, y)A1(x)Bl(y)dxdy,

F 0nl =

2

hxhy

∫ xk+1

xk−1

∫ yl+1

yl−1

f(x, y)An(x)Bl(y)dxdy,

F 0kl =

1

hxhy

∫ xk+1

xk−1

∫ yl+1

yl−1

f(x, y)Ak(x)Bl(y)dxdy.

3.2 Inverse F 0-transform

In this section, we present an inversion formula that shows how to “reconstruct”

a function f using the F 0-transform. By applying the direct F 0-transform to the func-

tion f , we obtain F0nm[f ] = (F 0

kl), k = 1, . . . , n, l = 1, . . . ,m and we lose some in-

formation about the original function f . Therefore, we cannot precisely reconstruct

the original function f but obtain the approximation of the function f .

Definition 8 Let f ∈ L2([a, b]× [c, d]), and let the matrix of components F0nm[f ] =

(F 0kl), k = 1, . . . , n, l = 1, . . . ,m be the F 0-transform of f with respect to the chosen

fuzzy partition {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m}. Then, the inverse F 0-

transform of f with respect to the same fuzzy partition is a function f 0 : [a, b] ×[c, d]→ R defined by the following formula:

f 0(x, y) =

∑nk=1

∑ml=1 F

0klAk(x)Bl(y)∑n

k=1

∑ml=1Ak(x)Bl(y)

. (3.5)

Analogous to the direct F 0-transform of f , the formula of the inverse F 0-transform

of f can be simplified according to the chosen fuzzy partition (see Section 2.1) [28].

• Let fuzzy sets {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} establish an (hx;hy)-

uniform fuzzy partition (with the Ruspini condition) of [a, b] × [c, d], then

39

the inversion formula (3.5) has the following form:

f 0(x, y) =n∑k=1

m∑l=1

F 0klAk(x)Bl(y). (3.6)

• Let fuzzy sets {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m} establish a fuzzy (r1; r2)-

partition (with the generalized Ruspini condition) of [a, b]× [c, d], then the in-

version formula (3.5) has the following form:

f 0(x, y) =

∑nk=1

∑ml=1 F

0klAk(x)Bl(y)

r1r2. (3.7)

• Let us remark that for generalized fuzzy partition of [a, b]×[c, d] holds the origi-

nal inversion formula (3.5).

It was proven that for arbitrary precision, there exists the inverse F 0-transform

of f with respect to the chosen fuzzy partition that approximates the original func-

tion f ([45]). The following theorem recalls that fact.

Theorem 2 Let f ∈ L2([a, b] × [c, d]). Then, for any ε > 0, there exist nε,mε

and fuzzy partition {Ak × Bl | k = 1, . . . , nε, l = 1, . . . ,mε} of [a, b] × [c, d], such

that for all (x, y) ∈ [a, b]× [c, d]

|f(x, y)− f 0(x, y)| ≤ ε,

where f 0(x, y) is the inverse F 0-transform of f with respect to {Ak × Bl | k =

1, . . . , nε, l = 1, . . . ,mε}.

Illustrations of two inverse F 0-transforms of the function f (original in Figure 3.1)

and their error functions with different numbers of basic functions in the fuzzy

partitions are shown in Figures 3.4, 3.5.

40

(a) The inverse F 0-transform of f with less basic functions.

(b) Error function - difference between the original function and its inverse F 0-transform.

Figure 3.4: Illustration of the inverse F 0-transform of function f(x, y) = (90/x+ x ·sin(x/7))(90/y + y · sin(y/7)) - example 1.

41

(a) The inverse F 0-transform of f with more basic functions.



42

Chapter 4

F s-transform of Functions of Two

Variables

In this chapter, we present a fuzzy transform of a higher degree (F s-transform,

s ≥ 1) of functions of two variables. More specifically, we extend the F 0-transform

of f where the components were characterized by polynomials of the zeroth degree

(constants) to the F s-transform of f where the components are described generally

by polynomials of the s-th degree. We follow the main ideas of the F 0-transform

and generalize its properties for the F s-transform.

First, the technique of the F s-transform, s ≥ 1, was introduced for functions

of one variable [30, 31]. We extend it for functions of two variables.

4.1 Direct F s-transform

Let [a, b] × [c, d] be the universe, and let fuzzy sets {Ak × Bl | k = 1, . . . , n, l =

1, . . . ,m} establish a fuzzy partition1 of [a, b]× [c, d].

Let L2(Ak) × L2(Bl), k = 1, . . . , n, l = 1, . . . ,m, be the Hilbert space, and let

f ∈ L2([a, b] × [c, d]), i.e., for all k = 1, . . . , n, l = 1, . . . ,m, f |[xk−1,xk+1]×[yl−1,yl+1]∈

L2(Ak)×L2(Bl) (see Section 2.2.3). Moreover, let Ls2(Ak×Bl), s ≥ 1, k = 1, . . . , n,


43

l = 1, . . . ,m, be the linear subspace of L2(Ak)×L2(Bl), with the basis given by or-

thogonal polynomials:

{Sijkl(x, y)}i=0,...,s; j=0,...,s; i+j≤s = {P ik(x) ·Qj

l (y)}i=0,...,s; j=0,...,s; i+j≤s. (4.1)

Let us recall that s denotes the maximal degree of polynomials Sijkl(x, y) and the or-

thogonality is in the sense of (2.10) (see Section 2.2.4).

Moreover, we recall that by F skl, k = 1, . . . , n, l = 1, . . . ,m, we denote the or-

thogonal projection of f ∈ L2([a, b] × [c, d]) on Ls2(Ak × Bl), s ≥ 1, k = 1, . . . , n,

l = 1, . . . ,m (see Lemma 2).

In the following, we define the direct F s-transform, s ≥ 1, of function f ∈L2([a, b]× [c, d]).

Definition 9 Let f ∈ L2([a, b]× [c, d]) and let {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m}be a fuzzy partition of [a, b] × [c, d]. Moreover, let F s


of f on subspace Ls2(Ak×Bl), s ≥ 1, k = 1, . . . , n, l = 1, . . . ,m, with the basis given

by (4.1).

A (n × m) matrix Fsnm[f ] is the direct F s-transform of f , s ≥ 1, with respect

to {Ak ×Bl | k = 1, . . . , n, l = 1, . . . ,m} where

Fsnm[f ] =

F s11 . . . F s

1m

......

...

F sn1 . . . F s

nm

and F s

kl, k = 1, . . . , n, l = 1, . . . ,m is called the F s-transform component.

By Lemma 2, for each k = 1, . . . , n, l = 1, . . . ,m, the F s-transform components

have the following representation:

F skl =

∑0≤i+j≤s

cijklSijkl, (4.2)

where

cijkl =〈f, Sijkl〉kl〈Sijkl, S

ijkl〉kl

=

∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)Sijkl(x, y)Ak(x)Bl(y)dxdy∫ yl+1

yl−1

∫ xk+1


. (4.3)

44

Let us present the main properties of the direct F s-transform of f , s ≥ 1,

in a manner similar to that for the F 0-transform (see Section 3.1). We do not

present the proofs; the following properties are the generalized properties of the F 0-

transform, and the proofs can be analogously derived.

Let f ∈ L2([a, b] × [c, d]), and let {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} be

the fixed fuzzy partition of [a, b] × [c, d] and Fsnm[f ] = (F s

kl), s ≥ 1, k = 1, . . . , n,

l = 1, . . . ,m be the F s-transform of f with respect to {Ak × Bl | k = 1, . . . , n, l =

1, . . . ,m}.

(A) Fsnm[f ] is an image of a linear mapping from L2([a, b] × [c, d]) to (Ls2(A1 ×

B1)×· · ·×Ls2(An×Bm)) where for all functions f, g, h ∈ L2([a, b]× [c, d]) such

that f = αg + βh where α, β ∈ R the following holds:

Fsnm[f ] = αFs

nm[g] + βFsnm[h]. (4.4)

(B) The kl-th component of Fsnm[f ] = (F s

kl), s ≥ 1, k = 1, . . . , n, l = 1, . . . ,m

gives the minimum to the function:

c00kl , . . . , cijkl =

∫ yl+1

yl−1

∫ xk+1

xk−1

(f(x, y)−∑

0≤i+j≤s

cijklSijkl)

2Ak(x)Bl(y)dxdy.

Therefore, F skl is the best approximation of f in Ls2(Ak×Bl), k = 1, . . . , n, l =

1, . . . ,m.

(C) Let f be a polynomial of degree t ≤ s. Then, any F s-transform component

F skl of f , s ≥ 1, k = 1, . . . , n, l = 1, . . . ,m coincides with f on [xk−1, xk+1] ×

[yl−1, yl+1].

(D) Every F s-transform component F skl, s ≥ 1, k = 1, . . . , n, l = 1, . . . ,m fulfills

the following recurrent equation:

F skl = F s−1

kl +∑i+j=s

cijklSijkl. (4.5)

We mentioned previously that the F s-transform, s ≥ 1, is an extension of the F 0-

transform. The following lemma describes the relationship between the F 0-transform

and F s-transform components (the proof was inspired by the analogous result for func-

tions of one variable [31]).

45

Lemma 4 Let Fsnm[f ] = (F s

kl), s ≥ 1, k = 1, . . . , n, l = 1, . . . ,m be the F s-

transform of f with respect to {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} where F skl =∑

0≤i+j≤s cijklS

ijkl. Then, (c00kl ), k = 1, . . . , n, l = 1, . . . ,m is the F 0-transform of f

with respect to the same fuzzy partition {Ak ×Bl | k = 1, . . . , n, l = 1, . . . ,m}.

proof: The proof comes directly from the Gram-Schmidt orthogonalization

process [17], in which we create the orthogonal polynomials {P ik(x)}i=0,...,s,

{Qjl (y)}j=0,...,s, from the sets {1, x, x2, . . . , xs} and {1, y, y2, . . . , ys}, respectively.

Then, for all k = 1, . . . , n, l = 1, . . . ,m,

S00kl (x, y) = P 0

k (x)Q0l (y) = 122. (4.6)

By using the polynomials (4.6) and the formula (4.3), then for every k = 1, . . . , n,

l = 1, . . . ,m, we obtain

c00kl =〈f, S00

kl 〉kl〈S00

kl , S00kl 〉kl

=

∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)Ak(x)Bl(y)dxdy∫ yl+1

yl−1

∫ xk+1

xk−1Ak(x)Bl(y)dxdy

= F 0kl.

2

Any F s-transform component F 0kl, F

1kl, . . . , F

skl, s ≥ 1, k = 1, . . . , n, l = 1, . . . ,m

defined by the orthogonal projection can approximate the original function f ∈L2([a, b]×[c, d]) restricted to [xk−1, xk+1]×[yl−1, yl+1] (see Theorem 1). The following

lemma demonstrates that the quality of the approximation increases with the degree

of the polynomial (the proof was inspired by the analogous result for functions of one

variable [31]).

Lemma 5 Let the polynomials F skl, F

s+1kl , s ≥ 1, k = 1, . . . , n, l = 1, . . . ,m be

the orthogonal projections of f ∈ L2([a, b] × [c, d]) on the subspaces Ls2(Ak × Bl)

and Ls+12 (Ak ×Bl), respectively. Then, for all k = 1, . . . , n, l = 1, . . . ,m,

‖ f − F s+1kl ‖kl≤‖ f − F

skl ‖kl .

proof: The function f is from the space L2([a, b] × [c, d]), i.e., for all k =

1, . . . , n, l = 1, . . . ,m, f |[xk−1,xk+1]×[yl−1,yl+1]∈ L2(Ak)×L2(Bl) where L2(Ak)×L2(Bl)

22P 0k (x) = 1, Q0

l (y) = 1 and S00kl (x, y) = 1 are constant functions of variable x, y and two

variables x, y, respectively.

46

is the Hilbert space. By Theorem 1, for every element f ′ ∈ L2(Ak)× L2(Bl), there

exists a unique best approximation F skl ∈ Ls2(Ak ×Bl) in the sense that

‖ f ′ − F skl ‖kl= inf

g∈Ls2(Ak×Bl)

‖f ′ − g‖.

By Remark 4, it holds that Ls2(Ak ×Bl) ⊂ Ls+12 (Ak ×Bl). Therefore

‖ f ′ − F s+1kl ‖kl= inf

g′∈Ls+12 (Ak×Bl)

‖f ′ − g′‖ ≤ infg∈Ls

2(Ak×Bl)‖f ′ − g‖ =‖ f ′ − F s

kl ‖kl .

2

The following lemma estimates the quality of the approximation of the original

function f by the F s-transform component. The proof was inspired by the Theo-

rem about approximation of derivatives using the F s-transform of functions of one

variable [36] and is based on differential and integral calculus [11, 15, 12, 13].

Lemma 6 Let n,m ≥ 2 and let functions f , Ak, Bl, k = 1, . . . , n, l = 1, . . . ,m, be

(s + 1)-times continuously differentiable on [a, b] × [c, d]. Moreover, let {Ak × Bl |k = 1, . . . , n, l = 1, . . . ,m} establish an (h, h

′)-uniform generalized fuzzy partition

of [a, b] × [c, d] and let Fsnm[f ] = (F s

kl), s ≥ 1, k = 1, . . . , n, l = 1, . . . ,m, be

the F s-transform of f with respect to the given partition where

F skl =

∑0≤i+j≤s

cijklSijkl.

Then for every (x, y) ∈ [a, b]× [c, d] and for every s ≥ 1 there exist k, l such that the

following holds true

|F skl(x, y)− f(x, y)| ≤ Cs · (h′)s+1,

where Cs → 0 for s→∞.

proof: Let the assumptions of the lemma be fulfilled. Moreover, based on the dif-

ferentiability of f it follows that f ∈ L2([a, b] × [c, d]), i.e., for all k = 1, . . . , n,

l = 1, . . . ,m, f |[xk−1,xk+1]×[yl−1,yl+1]∈ L2(Ak) × L2(Bl) where L2(Ak) × L2(Bl),

k = 1, . . . , n, l = 1, . . . ,m, is the Hilbert space (see Section 2.2.3).

Now, let us fix k, l from {1, . . . , n}, {1, . . . ,m}, respectively, and restrict our

interest on f ∈ L2(Ak)× L2(Bl).

47

Based on the assumptions, fuzzy sets {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m}establish an (h, h

′)-uniform generalized fuzzy partition of [a, b]× [c, d]. In addition,

we assume that the chosen n,m ≥ 2 and h = xk+1 − xk = yl+1 − yl, h′> h/2, are

large enough in the sense that the rectangle [xk−1, xk+1] × [yl−1, yl+1] ⊆ O(xk, yl)

where O(xk, yl) is a neighborhood of (xk, yl) such that we can apply the Taylor’s

theorem to the function f and for (x, y) ∈ [xk−1, xk+1] × [yl−1, yl+1] ⊆ O(xk, yl)

obtain

f(x, y) = tsf (x, y) + r(x, y), (4.7)

where tsf (x, y) is the s-th order Taylor’s polynomial of f near (xk, yl) and the re-

mainder term r(x, y) can be expressed by

r(x, y) =1

(s+ 1)!

s+1∑i=0

(s+ 1

i

)∂s+1f

∂xs+1−i∂yi

∣∣∣∣(ξ,η)

(x− xk)s+1−i(y − yl)i, (4.8)

where (ξ, η) is an inner point at the line segment between (x, y) and (xk, yl).

Let us apply the direct F s-transform to f represented by (4.7). Based on the prop-

erty (4.4), it holds that the F s-transform of f is given as follows

F skl(x, y) = T skl(x, y) +Rs

kl(x, y),

(x, y) ∈ [xk−1, xk+1]× [yl−1, yl+1],

where T skl(x, y) and Rskl(x, y) are the F s-transform components of tsf (x, y) and r(x, y),

respectively.

Moreover, by the property C (in Section 4.1), the F s-transform of an s-th degree

polynomial coincides with the polynomial itself. Therefore,

F skl(x, y) = tsf (x, y) +Rs

kl(x, y). (4.9)

By using (4.7) and (4.9), we obtain the following representation of F skl(x, y):

F skl(x, y) = f(x, y)− r(x, y) +Rs

kl(x, y). (4.10)

Based on (4.10), to estimate the difference |F skl(x, y)− f(x, y)| for all (x, y) ∈

[xk−1, xk+1]× [yl−1, yl+1] such that

|F skl(x, y)− f(x, y)| = |Rs

kl(x, y)− r(x, y)| , (4.11)

48

we need to estimate r(x, y) specified above by (4.8) and Rskl(x, y) which is the F s-

transform component of r(x, y).

• The reminder r(x, y) defined by (4.8) can be bounded for all (x, y) ∈ [xk−1, xk+1]×[yl−1, yl+1] as follows

|r(x, y)| =∣∣∣∣∣ 1

(s+ 1)!

s+1∑i=0

(s+ 1

i

)∂s+1f

∂xs+1−i∂yi

∣∣∣∣(ξ,η)

(x− xk)s+1−i(y − yl)i∣∣∣∣∣ ≤

1

(s+ 1)!

s+1∑i=0

(s+ 1

i

)·

∣∣∣∣∣ ∂s+1f

∂xs+1−i∂yi

∣∣∣∣(ξ,η)

∣∣∣∣∣ · |x− xk|s+1−i · |y − yl|i .

Let us denote

Kkl = maxi=0,...,s+1

∣∣∣∣∣ ∂s+1f

∂xs+1−i∂yi

∣∣∣∣(ξ,η)

∣∣∣∣∣ . (4.12)

Then,

|r(x, y)| ≤ 1

(s+ 1)!

s+1∑i=0

(s+ 1

i

)·Kkl · |x− xk|s+1−i · |y − yl|i . (4.13)

We consider the values (x, y) ∈ [xk−1, xk+1] × [yl−1, yl+1]. More specifically,

for the (h, h′)-uniform generalized fuzzy partition, we consider all x ∈ [xk −

h′, xk + h′] and all y ∈ [yl − h′, yl + h′], and thus

|x− xk| ≤ h′, |y − yl| ≤ h′. (4.14)

Substituting (4.14) in the inequality (4.13), we obtain

|r(x, y)| ≤ Kkl

(s+ 1)!

s+1∑i=0

(s+ 1

i

)(h′)s+1−i(h′)i, (4.15)

and moreover, by using the binomial theorem, we can rewrite the inequality

(4.15) as follows

|r(x, y)| ≤ Kkl

(s+ 1)!· (2h′)s+1 =

2s+1 ·Kkl

(s+ 1)!· (h′)s+1, (4.16)

where Kkl is specified above by (4.12).

49

• Let us estimate Rskl(x, y) for all (x, y) ∈ [xk−1, xk+1] × [yl−1, yl+1]. According

to the Definition 9, the component Rskl(x, y) is defined by

Rskl(x, y) =

∑0≤i+j≤s

cijklSijkl(x, y), (4.17)

where

cijkl =

∫ yl+1

yl−1

∫ xk+1

xk−1r(x, y)Sijkl(x, y)Ak(x)Bl(y)dxdy∫ yl+1

yl−1

∫ xk+1


, (4.18)

and {Sijkl(x, y)}0≤i+j≤s is the orthogonal basis of polynomials of the space

L2(Ak)× L2(Bl).

To estimate Rskl(x, y) represented by (4.17), we will estimate first the polyno-

mials Sijkl(x, y) and then the coefficients cijkl.

Let us denote by Mkl the maximal possible value of all absolute values of poly-

nomials Sijkl(x, y) for all (x, y) ∈ [xk−1, xk+1]× [yl−1, yl+1], i.e.,

Mkl = max0≤i+j≤s

∣∣Sijkl(x, y)∣∣ . (4.19)

To estimate the coefficients cijkl represented by (4.18), let us bound the nomi-

nator and the denominator separately.

By using the expressions (4.16) and (4.19), the nominator of cijkl can be bounded

as follows:∣∣∣∣∣∫ yl+1

yl−1

∫ xk+1

xk−1

r(x, y)Sijkl(x, y)Ak(x)Bl(y)dxdy

∣∣∣∣∣ ≤∫ yl+1

yl−1

∫ xk+1

xk−1

2s+1 ·Kkl

(s+ 1)!· (h′)s+1 ·Mkl · Ak(x)Bl(y)dxdy =

2s+1 ·Kkl ·Mkl

(s+ 1)!· (h′)s+1 ·

∫ yl+1

yl−1

∫ xk+1

xk−1

Ak(x)Bl(y)dxdy. (4.20)

To bound the denominator of cijkl, we use the First mean value theorem for inte-

gration. Since the function (Sijkl(x, y))2 is continuous on [xk−1, xk+1]×[yl−1, yl+1]

and (Sijkl(x, y))2 ≥ 0 for all (x, y) ∈ [xk−1, xk+1] × [yl−1, yl+1], then there exist

50

a point (ζij, µij) ∈ [xk−1, xk+1]× [yl−1, yl+1], such that∫ yl+1

yl−1

∫ xk+1

xk−1

(Sijkl(x, y))2Ak(x)Bl(y)dxdy =

(Sijkl(ζij, µij))2

∫ yl+1

yl−1

∫ xk+1

xk−1


Since∫ yl+1

yl−1

∫ xk+1

xk−1(Sijkl(x, y))2Ak(x)Bl(y)dxdy 6= 0, then (Sijkl(ζij, µij))

2 6= 0.

Let us denote bymkl the minimal possible value of all mean values (Sijkl(ζij, µij))2,

i.e.,

mkl = min0≤i+j≤s

(Sijkl(ζij, µij))2. (4.22)

Therefore, the denominator of cijkl can be bounded as follows:∫ yl+1

yl−1

∫ xk+1

xk−1

(Sijkl(x, y))2Ak(x)Bl(y)dxdy ≥

mkl ·∫ yl+1

yl−1

∫ xk+1

xk−1


Based on (4.20) and (4.23), the coefficients cijkl represented by (4.18) can be

bounded for all i, j; 0 ≤ i+ j ≤ s, as follows:

∣∣cijkl∣∣ =

∣∣∣∣∣∫ yl+1

yl−1

∫ xk+1

xk−1r(x, y)Sijkl(x, y)Ak(x)Bl(y)dxdy∫ yl+1

yl−1

∫ xk+1


∣∣∣∣∣ ≤2s+1·Kkl·Mkl

(s+1)!· (h′)s+1 ·

∫ yl+1

yl−1

∫ xk+1


mkl ·∫ yl+1

yl−1

∫ xk+1


=

2s+1 ·Kkl ·Mkl

(s+ 1)! ·mkl

· (h′)s+1. (4.24)

Moreover, based on the estimations (4.19) and (4.24) of Sijkl and cijkl, respec-

tively, the final estimation of the F s-transform component Rskl(x, y), (x, y) ∈

[xk−1, xk+1]× [yl−1, yl+1], represented by (4.17) is following

|Rskl(x, y)| =

∣∣∣∣∣ ∑0≤i+j≤s

cijklSijkl(x, y)

∣∣∣∣∣ ≤ ∑0≤i+j≤s

∣∣cijkl∣∣ · ∣∣Sijkl(x, y)∣∣ ≤

∑0≤i+j≤s

2s+1 ·Kkl ·Mkl

(s+ 1)! ·mkl

· (h′)s+1 ·Mkl =

∑0≤i+j≤s

2s+1 ·Kkl ·M2kl

(s+ 1)! ·mkl

· (h′)s+1. (4.25)

51

Based on the arithmetic series {ij} such that 0 ≤ i + j ≤ s, the number

of summands in the sum (4.25) is equal to

(s+ 1) + s+ (s− 1) + · · ·+ 1 =(s+ 1)(s+ 2)

2. (4.26)

Therefore, we can write

|Rskl(x, y)| ≤ (s+ 1)(s+ 2)

2· 2s+1 ·Kkl ·M2

kl

(s+ 1)! ·mkl

· (h′)s+1 =

(s+ 2) · 2s ·Kkl ·M2kl

s! ·mkl

· (h′)s+1. (4.27)

Let us go to the beginning. The aim was to estimate the following difference

|F skl(x, y)− f(x, y)| = |Rs

kl(x, y)− r(x, y)| ,

(x, y) ∈ [xk−1, xk+1]× [yl−1, yl+1].

Based on (4.16) and (4.27), we obtain

|F skl(x, y)− f(x, y)| = |Rs

kl(x, y)− r(x, y)| ≤ |Rskl(x, y)|+ |r(x, y)| ≤

(s+ 2) · 2s ·Kkl ·M2kl

s! ·mkl

· (h′)s+1 +2s+1 ·Kkl

(s+ 1)!· (h′)s+1 =(

(s+ 2) · 2s ·M2kl

s! ·mkl

+2s+1

(s+ 1)!

)·Kkl · (h′)s+1, (4.28)

where the constants Kkl,Mkl,mkl are specified above by (4.12), (4.19) and (4.22),

respectively. These constants Kkl,Mkl,mkl in the expression (4.28) are dependent

on k, l. Let us finally denote the following

K = maxk=1,...,n; l=1,...,m

Kkl,

M = maxk=1,...,n; l=1,...,m

Mkl,

Z = mink=1,...,n; l=1,...,m

mkl.

Then, we can conclude that for all k = 1, . . . , n, l = 1, . . . ,m and all (x, y) ∈[a, b]× [c, d], the difference |F s

kl(x, y)− f(x, y)| is estimated by the following

|F skl(x, y)− f(x, y)| ≤

((s+ 2) · 2s ·M2

s! · Z+

2s+1

(s+ 1)!

)·K · (h′)s+1 = Cs · (h′)s+1,

(4.29)

where Cs = K(

(s+2)·2s·M2

s!·Z + 2s+1

(s+1)!

)is dependent on s such that Cs → 0 for s→∞.

2

52

4.2 Inverse F s-transform

Similar to the inverse F 0-transform, the inverse F s-transform, s ≥ 1, of the original

function f is defined as a linear combination of the basic functions and the F s-

transform components.

Definition 10 Let f ∈ L2([a, b]×[c, d]), and let the matrix of components Fsnm[f ] =

(F skl), s ≥ 1, k = 1, . . . , n, l = 1, . . . ,m be the F s-transform of f with respect

to the chosen fuzzy partition {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m}. The function

f s : [a, b]× [c, d]→ R represented by

f s(x, y) =

∑nk=1

∑ml=1 F

sklAk(x)Bl(y)∑n

k=1

∑ml=1Ak(x)Bl(y)

(4.30)

is the inverse F s-transform of f , s ≥ 1, with respect to the same fuzzy partition

{Ak ×Bl | k = 1, . . . , n, l = 1, . . . ,m}.

Remark 6 The inversion formula of F s-transform of f can be simplified according

to the chosen partition analogous to the expressions (3.6) and (3.7) given for the in-

verse F 0-transform (see Section 3.2).

Lemma 7 Let the assumptions of the Definition 10 hold; then the inverse F s-

transform of f , s ≥ 1, fulfills the following recurrent formula:

f s(x, y) = ˆf s−1(x, y) +

∑nk=1

∑ml=1

∑i+j=s c

ijklS

ijklAk(x)Bl(y)∑n

k=1

∑ml=1Ak(x)Bl(y)

.

proof: Indeed, from the formula (4.30) of the inverse F s-transform of f

and from the property (4.5) of the direct F s-transform of f , s ≥ 1, we obtain

f s(x, y) =

∑nk=1

∑ml=1 F

sklAk(x)Bl(y)∑n

k=1

∑ml=1Ak(x)Bl(y)

=∑nk=1

∑ml=1

(F s−1kl +

∑i+j=s c

ijklS

ijkl

)Ak(x)Bl(y)∑n

k=1

∑ml=1Ak(x)Bl(y)

=

ˆf s−1(x, y) +

∑nk=1

∑ml=1

∑i+j=s c

ijklS

ijklAk(x)Bl(y)∑n

k=1

∑ml=1Ak(x)Bl(y)

.

2

53

In the following, we demonstrate that the inverse F s-transform of f , s ≥ 1, ap-

proximates an original function f , and we estimate the quality of the approximation.

Based on Lemma 6, based on the increasing value of s, the quality of the approxi-

mation is improving.

Theorem 3 Let n,m ≥ 2 and let functions f , Ak, Bl, k = 1, . . . , n, l = 1, . . . ,m,

be (s+ 1)-times continuously differentiable on [a, b]× [c, d]. Moreover, let {Ak×Bl |k = 1, . . . , n, l = 1, . . . ,m} establish an (h, h


of [a, b] × [c, d] and let f s, s ≥ 1 be the inverse F s-transform of f with respect

to the chosen fuzzy partition. Then, for every (x, y) ∈ [a, b] × [c, d] and for every

s ≥ 1, the following holds

|f(x, y)− f s(x, y)| ≤ Cs · (h′)s+1,


proof: Let the assumptions of the theorem be fulfilled. Moreover, based

on the differentiability of f , it follows that f ∈ L2([a, b]× [c, d]). According to Defi-

nition 10, we can write

∣∣∣f(x, y)− f s(x, y)∣∣∣ =

∣∣∣∣f(x, y)−∑n

k=1

∑ml=1 F

sklAk(x)Bl(y)∑n

k=1

∑ml=1Ak(x)Bl(y)

∣∣∣∣ =∣∣∣∣f(x, y)∑n

k=1

∑ml=1Ak(x)Bl(y)−

∑nk=1

∑ml=1 F

sklAk(x)Bl(y)∑n

k=1

∑ml=1Ak(x)Bl(y)

∣∣∣∣ ≤∑nk=1

∑ml=1Ak(x)Bl(y) |f(x, y)− F s

kl|∑nk=1

∑ml=1Ak(x)Bl(y)

.

Now, based on Lemma 6, we know that for all (x, y) ∈ [a, b] × [c, d] and all

k = 1, . . . , n, l = 1, . . . ,m, the following holds

|f(x, y)− F skl(x, y)| ≤ Cs · (h′)s+1,

where Cs is specified by (4.29) and it holds that Cs → 0 for s→∞.

54

Therefore, we obtain∣∣∣f(x, y)− f s(x, y)∣∣∣ ≤ ∑n

k=1

∑ml=1Ak(x)Bl(y) |f(x, y)− F s

kl|∑nk=1

∑ml=1Ak(x)Bl(y)

≤∑nk=1

∑ml=1Ak(x)Bl(y) · Cs · (h′)s+1∑nk=1

∑ml=1Ak(x)Bl(y)

= Cs · (h′)s+1,


2

4.3 F 1-transform

In the previous section, we defined the direct and inverse F s-transform of f , generally

for s ≥ 1. In this section, we restrict to the case of s = 1, thus the F 1-transform.

We focus on the F 1-transform because this technique is later used in the applica-

tion in Chapter 6. We specify the particular forms of the F 1-transform components

according to the particular fuzzy partitions. Moreover, we introduce a so-called

discrete F 1-transform, in which the original function is defined at discrete points.

Generally, the discrete case of the F s-transform, s ≥ 0, is principally used for ex-

ample in image processing.

4.3.1 Direct F 1-transform

Let [a, b] × [c, d] be the universe and let fuzzy sets {Ak × Bl | k = 1, . . . , n, l =

1, . . . ,m} establish a fuzzy partition2 of [a, b] × [c, d]. We assume Ak and Bl to be

symmetrical with respect to xk and yl, k = 1, . . . , n, l = 1, . . . ,m, respectively.

Moreover, let L2(Ak) × L2(Bl), k = 1, . . . , n, l = 1, . . . ,m, be the Hilbert space

and let f ∈ L2([a, b]× [c, d]), i.e., for all k = 1, . . . , n, l = 1, . . . ,m,

f |[xk−1,xk+1]×[yl−1,yl+1]∈ L2(Ak)× L2(Bl) (see Section 2.2.3).

To introduce the F 1-transform of f ∈ L2([a, b] × [c, d]), we assume L12(Ak) ⊆

L2(Ak), k = 1, . . . , n and L12(Bl) ⊆ L2(Bl), l = 1, . . . ,m are the linear spans


55

of the sets consisting of orthogonal zero- and first-degree polynomials, i.e.,

P 0k (x) = 133, P 1

k (x) = x− xk (Q0l (y) = 133, Q1

l (y) = y − yl).

Analogously, let L12(Ak × Bl) ⊆ L2(Ak) × L2(Bl) be the linear span of the set

consisting of three orthogonal polynomials

S00kl (x, y) = 133, S10

kl (x, y) = x− xk, S01kl (x, y) = y − yl. (4.31)

Analogous to the general definition of the F s-transform, s ≥ 1, in the following,

we introduce the F 1-transform of f ∈ L2([a, b] × [c, d]) where the components are

in the form of linear polynomials.

Definition 11 Let f ∈ L2([a, b]×[c, d]) and let {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m}be a fuzzy partition of [a, b] × [c, d]. Moreover, let F 1


(2.13) of f on subspace L12(Ak×Bl), k = 1, . . . , n, l = 1, . . . ,m, with the basis given

by (4.31).

A (n×m) matrix F1nm[f ] = (F 1

kl), k = 1, . . . , n, l = 1, . . . ,m, is the F 1-transform

of f with respect to {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m}, and F 1kl is the correspond-

ing F 1-transform component.

By Lemma 2, for every k = 1, . . . , n, l = 1, . . . ,m, the F 1-transform component

of f is in the form

F 1kl(x, y) = c00kl + c10kl (x− xk) + c01kl (y − yl), (4.32)

where the coefficients are given by

c00kl =

∫ yl+1

yl−1

∫ xk+1


(∫ xk+1

xk−1Ak(x)dx)(

∫ yl+1

yl−1Bl(y)dy)

, (4.33)

c10kl =

∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)(x− xk)Ak(x)Bl(y)dxdy

(∫ xk+1

xk−1(x− xk)2Ak(x)dx)(

∫ yl+1

yl−1Bl(y)dy)

, (4.34)

33P 0k (x) = 1, Q0

l (y) = 1 and S00kl (x, y) = 1 are constant functions of variable x, y and two

variables x, y, respectively.

56

c01kl =

∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)(y − yl)Ak(x)Bl(y)dxdy

(∫ xk+1

xk−1Ak(x)dx)(

∫ yl+1

yl−1(y − yl)2Bl(y)dy)

. (4.35)

An illustration of a particular F 0-transform component as a single point and the cor-

responding F 1-transform component in the form of a hyperplane is shown in Fig-

ure 4.1. Graphical representations of the direct F 1-transforms of two different func-

tions as sets of components (hyperplanes) are shown in Figures 4.2, 4.3, 4.4.

Figure 4.1: Illustration of the direct F 1-transform - original function x · y, singleF 0-transform and F 1-transform components.

Figure 4.2: Illustration of the direct F 1-transform of function f(x, y) = x · y -the whole set of components - example 1.

Remark 7 The coefficients c00kl (4.33) are equal to the components F 0kl (3.2), k =

1, . . . , n, l = 1, . . . ,m of the F 0-transform of the given function f . Therefore, we

57

Figure 4.3: Illustration of the direct F 1-transform of function f(x, y) = x · y -the whole set of components - example 2.

can write for each k = 1, . . . , n, l = 1, . . . ,m:

F 1kl = F 0

kl + c10kl (x− xk) + c01kl (y − yl),

where the coefficients c10kl , c01kl are given as above (4.34), (4.35), respectively.

4.3.2 Simpler Form of F 1-transform Components

Similarly to Section 3.1.1, representations (4.33) - (4.35) of the coefficients c00kl , c10kl , c

01kl

of the F 1-transform components can be simplified according to the chosen partition.

In this section, we present the case of uniform fuzzy partitions.

Lemma 8 Let f ∈ L2([a, b]× [c, d]) and let {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m} be

an ((hx, h′x); (hy, h

′y))-uniform generalized fuzzy partition of [a, b] × [c, d]. The par-

ticular fuzzy partitions of [a, b] and [c, d] are defined by the generating functions A0

and B0 (2.3), respectively. Let us denote

a0 =

∫ 1

−1A0(x)dx, b0 =

∫ 1

−1B0(y)dy.

Then, coefficients c00kl , c10kl , c

01kl , k = 1, . . . , n, l = 1, . . . ,m, in the above given

representations (4.33) - (4.35) can be rewritten as follows:

c00kl =

∫ 1

−1

∫ 1

−1 f(h′xx+ xk, h

′yy + yl)A0(x)B0(y)dxdy

a0 b0,

58

Figure 4.4: Illustration of the direct F 1-transform of function f(x, y) = (90/x+ x ·sin(x/7))(90/y + y · sin(y/7)) - the whole set of components.

c10kl =

∫ 1

−1

∫ 1


′yy + yl) x A0(x)B0(y)dxdy

b0 h′x

∫ 1

−1 x2A0(x)dx

,

c01kl =

∫ 1

−1

∫ 1


′yy + yl) y A0(x)B0(y)dxdy

a0 h′y

∫ 1

−1 y2B0(y)dy

.

proof: We use the fact that

Ak(x) = A0

(x− xkh′x

), x ∈ [xk − h

′

x, xk + h′

x],

Bl(y) = B0

(y − ylh′y

), y ∈ [yl − h

′

y, yl + h′

y],

and substitutions

t =x− xkh′x

, u =y − ylh′y

.

Then, it follows that∫ yl+1

yl−1

∫ xk+1

xk−1

f(x, y)Ak(x)Bl(y)dxdy =

h′

xh′

y

∫ 1

−1

∫ 1

−1f(h

′

xx+ xk, h′

yy + yl)A0(x)B0(y)dxdy,

59

∫ yl+1

yl−1

∫ xk+1

xk−1

f(x, y)(x− xk)Ak(x)Bl(y)dxdy =

h′2x h′

y

∫ 1

−1

∫ 1

−1f(h

′

xx+ xk, h′

yy + yl)xA0(x)B0(y)dxdy,

∫ xk+1

xk−1

(x− xk)2Ak(x)dx = h′2x

∫ 1

−1x2A0(x)dx,

∫ xk+1

xk−1

Ak(x)dx = h′

x

∫ 1

−1A0(x)dx = h

′

xa0.

Analogously for the variable y and its corresponding integrals. 2

In the following lemma, we assume generating functions of a triangular shape,

and therefore, we simplify the above given expressions further. This fact is very

important for applications because it significantly reduces computational complexity.

Lemma 9 Let f ∈ L2([a, b]× [c, d]) and let {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m} be

an (h, h′)-uniform generalized fuzzy partition of [a, b]× [c, d]. Moreover, let the fuzzy

partitions of [a, b] and [c, d] be defined by the generating functions A0 and B0 of the tri-

angular shape, respectively, i.e.,

A0(x) = 1− |x|, x ∈ [−1, 1],

and similarly for B0. Then, for every k = 1, . . . , n, l = 1, . . . ,m, the coefficients

c00kl , c10kl , c

01kl in the above given representations (4.33) - (4.35) are as follows:

c00kl =

∫ yl+1

yl−1

∫ xk+1


h′2,

c10kl =6∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)(x− xk)Ak(x)Bl(y)dxdy

h′4,

c01kl =6∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)(y − yl)Ak(x)Bl(y)dxdy

h′4.

proof: We will prove that∫ xk+1

xk−1

Ak(x)dx = h′

60

and ∫ xk+1

xk−1

(x− xk)2Ak(x)dx =h′3

6

for the triangular-shaped basic functions represented by

Ak(x) = 1− |x− xk|h′

, k = 1, . . . , n.

The same holds for∫ yl+1

yl−1Bl(y)dy and

∫ yl+1

yl−1(y − yl)2Bl(y)dy, l = 1, . . . ,m.

∫ xk+1

xk−1

Ak(x)dx =

∫ xk+h′

xk−h′

(1− |x− xk|

h′

)dx = 2

∫ xk+h′

xk

(1− x− xk

h′

)dx = h

′,

∫ xk+1

xk−1

(x− xk)2Ak(x)dx =

∫ xk+h′

xk−h′(x− xk)2

(1− |x− xk|

h′

)dx =

2

∫ xk+h′

xk

((x− xk)2 −

(x− xk)3

h′

)dx = 2

(h′3

3− h

′3

4

)=h′3

6.

2

4.3.3 Inverse F 1-transform

In Section 4.2, we defined the inverse F s-transform of f generally for s ≥ 1 as a linear

combination of basic functions Ak × Bl with coefficients given by F skl, k = 1, . . . , n,

l = 1, . . . ,m. The inversion formula for the F 1-transform of f has the same repre-

sentation where coefficients F 1kl, k = 1, . . . , n, l = 1, . . . ,m are in the form of linear

polynomials (4.32). Therefore, we do not repeat the whole definition of the inverse

F 1-transform of f .

Remark 8 Let {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} be an (h, h′)-uniform gener-

alized fuzzy partition of [a, b] × [c, d] and let f 1 be the inverse F 1-transform of f ∈L2([a, b]×[c, d]) with respect to the chosen partition. Based on the Theorem 3, we can

estimate the difference between the original function f and its inverse F 1-transform

for all (x, y) ∈ [a, b]× [c, d] as follows∣∣∣f(x, y)− f 1(x, y)∣∣∣ ≤ C1 · (h′)2,

where C1 is a constant specified by (4.29) for s = 1.

61

Illustrations of two inverse F 1-transforms of the function f (original in Figure 3.1)

and their error functions with different numbers of basic functions in the fuzzy par-

titions are shown in Figures 4.5, 4.6. For comparison, the chosen numbers of ba-

sic functions are the same as they were in the case of the inverse F 0-transform

of the same original function f in Figures 3.4, 3.5.

(a) The inverse F 1-transform of f with less basic functions.



62

(a) The inverse F 1-transform of f with more basic functions.



4.3.4 Discrete F 1-transform

Let us consider the discrete case when the original function is known only at some

discrete points. In Chapter 6, we will examine image function, which is generally

represented by an array of discrete points. We will apply the discrete form of the F 0-

transform and F 1-transform to that image function. Therefore, in this subsection,

we define the discrete F 1-transform. Let us remark that the F 0-transform is a part

of the F 1-transform (see Remark 7).

Generally, the discrete form can also be considered for the F s-transform, s > 1,

and similar properties introduced in Section 4.1 adapted to the discrete case can be

proven. For the purpose of this thesis, the discrete F 0-transform and F 1-transform

63

are sufficient.

Let [a, b] × [c, d] be a universe and let {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} be

a fuzzy partition3 of [a, b]×[c, d], where x0, x1, . . . , xn+1 ∈ [a, b] and y0, y1, . . . , ym+1 ∈[c, d] are fixed nodes. Let us assume a function f : [a, b] × [c, d] → R known only

at discrete points P = {(pi, qj) | i = 1, . . . , N ; j = 1, . . . ,M}. Moreover, we assume

that the set P of points is sufficiently dense with respect to the fixed partition, i.e.,

∀k, l ∃i, j; Ak(pi)Bl(qj) > 0.

Let us now introduce the discrete F 1-transform of the the function f given

at discrete points.

Definition 12 Let a function f : [a, b] × [c, d] → R be defined at points P =

{(pi, qj) | i = 1, . . . , N ; j = 1, . . . ,M} and let {Ak ×Bl | k = 1, . . . , n, l = 1, . . . ,m},n ≤ N,m ≤ M , be a fuzzy partition of [a, b] × [c, d]. Suppose that the set P is

sufficiently dense with respect to the chosen partition.

A (n×m) matrix F1nm[f ] = (F 1

kl), k = 1, . . . , n, l = 1, . . . ,m, is the discrete F 1-

transform of f with respect to the chosen partition if for all k = 1, . . . , n, l = 1, . . . ,m

and (pi, qj) ∈ P

F 1kl(pi, qj) = c00kl + c10kl (pi − xk) + c01kl (qj − yl),

where xk and yl, k = 1, . . . , n, l = 1, . . . ,m are fixed points from the fuzzy partition

and the coefficients are given by

c00kl =

∑Ni=1

∑Mj=1 f(pi, qj)Ak(pi)Bl(qj)∑N

i=1

∑Mj=1Ak(pi)Bl(qj)

,

c10kl =

∑Ni=1

∑Mj=1 f(pi, qj)(pi − xk)Ak(pi)Bl(qj)∑N

i=1

∑Mj=1(pi − xk)2Ak(pi)Bl(qj)

,

c01kl =

∑Ni=1

∑Mj=1 f(pi, qj)(qj − yl)Ak(pi)Bl(qj)∑N

i=1

∑Mj=1(qj − yl)2Ak(pi)Bl(qj)

.

3If there are not any restrictions or requirements for fuzzy partition; then, by the term “fuzzypartition”, we generally denote any type of fuzzy partition introduced in Section 2.1.

64

In the discrete case, we define the inverse F 1-transform of f on the same set P

on which the original function is defined.

Definition 13 Let a function f : [a, b]×[c, d]→ R be defined on a set P = {(pi, qj) |i = 1, . . . , N ; j = 1, . . . ,M} that is sufficiently dense with respect to the fuzzy par-

tition {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m}, n ≤ N,m ≤ M . Moreover, let

F1nm[f ] = (F 1

kl), k = 1, . . . , n, l = 1, . . . ,m be the discrete F 1-transform of f with

respect to the chosen fuzzy partition. Then, the function f 1 : P → R represented by

f 1(pi, qj) =

∑nk=1

∑ml=1 F

1kl(pi, qj)Ak(pi)Bl(qj)∑n

k=1

∑ml=1Ak(pi)Bl(qj)

(4.36)

is the inverse F 1-transform of f with respect to the chosen fuzzy partition.

65

Chapter 5

Partial Derivatives

The goal of this chapter is to provide approximations of partial derivatives of the given

function f of two variables, by using the technique of F s-transform, s ≥ 1, and more-

over, to estimate the qualities of these approximations.

The similar result was already proven for functions of one variable. More specif-

ically, it was proven that the F s-transform, s ≥ 1, of functions of one variable can

approximate not only the original function but also its corresponding derivatives

[36]. But in the literature, there is no tool for the estimation of values of partial

derivatives at a point of a function.

The general theory and properties used for definitions and proofs in this chapter

were mostly studied in [11, 12, 13, 16, 17].

5.1 First Partial Derivatives

In this section, we show how the coefficients of the F 1-transform components of f

characterize the behavior of an original function f . More specifically, we show how

these coefficients approximate first partial derivatives of the given function f in areas

that are closed to the nodes of the associated fuzzy partition.

First, we recall the definition of first partial derivatives of a given function f

at the exact point.

66

Definition 14 Let f : [a, b]× [c, d]→ R and (x, y), (x0, y0) ∈ [a, b]× [c, d].

• We fix the value of y0 and obtain a function f(x, y0) of one variable x. Then,

if there exists a limit

limx→x0

f(x, y0)− f(x0, y0)

x− x0,

we say that f has the first partial derivative with respect to x at the point

(x0, y0). We denote it by ∂f∂x

∣∣(x0,y0)

.

• Analogously, we can fix the value of x0 and obtain a function f(x0, y) of one

variable y. If there exists a limit

limy→y0

f(x0, y)− f(x0, y0)

y − y0,

we say that f has the first partial derivative with respect to y at the point

(x0, y0). We denote it by ∂f∂y

∣∣∣(x0,y0)

.

Now, let f ∈ L2([a, b]×[c, d]) and fuzzy sets {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m}establish a generalized fuzzy partition of [a, b]×[c, d]. The following theorem presents

approximations of the first partial derivatives of f using the coefficients of the F 1-

transform components of f and estimates the qualities of the approximations.

Theorem 4 Let n,m ≥ 2 and let functions f , Ak, Bl, k = 1, . . . , n, l = 1, . . . ,m, be

four times continuously differentiable on [a, b]× [c, d]. Moreover, let {Ak ×Bl | k =

1, . . . , n, l = 1, . . . ,m} be an (h, h′)-uniform generalized fuzzy partition of [a, b]×[c, d]

and let F1nm[f ] = ((F 1

kl) where

F 1kl = c00kl + c10kl (x− xk) + c01kl (y − yk),

k = 1, . . . , n, l = 1, . . . ,m, be the F 1-transform of f with respect to the given

partition. Then, for every k = 1, . . . , n, l = 1, . . . ,m, the following holds:

∂F 1kl

∂x

∣∣∣∣(xk,yl)

= c10kl =∂f

∂x

∣∣∣∣(xk,yl)

+O(h′), (5.1)

∂F 1kl

∂y

∣∣∣∣(xk,yl)

= c01kl =∂f

∂y

∣∣∣∣(xk,yl)

+O(h′). (5.2)

67


on the differentiability of f it follows that f ∈ L2([a, b]× [c, d]).

Let us recall that h = xk − xk−1 = yl − yl−1; k = 1, . . . , n + 1; l = 1, . . . ,m + 1

and h′> h/2; moreover, let us denote h1 = h

′/2.

Now, let us fix k, l from {1, . . . , n} and {1, . . . ,m}, respectively.

We will give the proof for (5.1) only because the proof of (5.2) can be obtained

analogously. We have

c10kl =

∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)(x− xk)Ak(x)Bl(y)dx dy

(∫ xk+1


∫ yl+1

yl−1Bl(y)dy)

and assume the following integrals:

I1(x) =

∫ yl+1

yl−1

f(x, y)Bl(y)dy =

∫ yl+h′

yl−h′f(x, y)Bl(y)dy,

I2 =

∫ xk+1

xk−1

I1(x)[x− xk]Ak(x)dx =

∫ xk+h′

xk−h′I1(x)[x− xk]Ak(x)dx,

I3 =

∫ xk+1

xk−1

[x− xk]2Ak(x)dx =

∫ xk+h′

xk−h′[x− xk]2Ak(x)dx,

I4 =

∫ yl+1

yl−1

Bl(y)dy =

∫ yl+h′

yl−h′Bl(y)dy.

Then, we apply the trapezoidal rule with three nodes yl−h′, yl, yl +h

′to the in-

tegrals I1, I4 and use the following property of basic functions:

Bl(yl − h′) = Bl(yl + h

′) = 0.

I1(x) =

∫ yl+h′

yl−h′f(x, y)Bl(y)dy =

h′[1/2f(x, yl−h

′)Bl(yl−h

′) + f(x, yl)Bl(yl) + 1/2f(x, yl +h

′)Bl(yl +h

′)] +R1(x) =

h′f(x, yl)Bl(yl) +R1(x) = 2h1f(x, yl)Bl(yl) +R1(x),

I4 =

∫ yl+h′

yl−h′Bl(y)dy = h

′Bl(yl) +R4 = 2h1Bl(yl) +R4.

68

We apply the Simpson’s rule with five nodes xk − h′, xk − h1, xk, xk + h1, xk + h

′

to the integrals I2, I3 and use the properties:

Ak(xk − h′) = Ak(xk + h

′) = 0,

Ak(xk − h1) = Ak(xk + h1).

I2 =

∫ xk+h′

xk−h′I1(x)[x−xk]Ak(x)dx =

∫ xk+h′

xk−h′[2h1f(x, yl)Bl(yl)+R1(x)][x−xk]Ak(x)dx =

h1/3{2h1f(xk−h′, yl)Bl(yl)[xk−h

′−xk]Ak(xk−h′)+R1(xk−h

′)[xk−h

′−xk]Ak(xk−h′)+

4[2h1f(xk−h1, yl)Bl(yl)[xk−h1−xk]Ak(xk−h1)+R1(xk−h1)[xk−h1−xk]Ak(xk−h1)]+

2[2h1f(xk, yl)Bl(yl)[xk − xk]Ak(xk) +R1(xk)[xk − xk]Ak(xk]+

4[2h1f(xk+h1, yl)Bl(yl)[xk+h1−xk]Ak(xk+h1)+R1(xk+h1)[xk+h1−xk]Ak(xk+h1)]+

2h1f(xk+h′, yl)Bl(yl)[xk+h

′−xk]Ak(xk+h′)+R1(xk+h

′)[xk+h

′−xk]Ak(xk+h′)}+R2 =

8h31Ak(xk + h1)Bl(yl)

3[f(xk + h1, yl)− f(xk − h1, yl)]+

4h21Ak(xk + h1)

3[R1(xk + h1)−R1(xk − h1)] +R2,

I3 =

∫ xk+h′

xk−h′[x− xk]2Ak(x)dx =

h1/3[4[xk − h1 − xk]2Ak(xk − h1) + 4[xk + h1 − xk]2Ak(xk + h1)] +R3 =

8h31Ak(xk + h1)

3+R3.

The estimations of the errorsR1, R4 with respect to the trapezoidal rule andR2, R3

with respect to the Simpson’s rule are the following:

R1 = − [(yl + h′)− (yl − h

′)]h′2M1

12= −h

′3M1

6= −4h31M1

3,

R2 = − [(xk + h′)− (xk − h

′)]h41M2

180= −h

51M2

45,

R3 = − [(xk + h′)− (xk − h

′)]h41M3

180= −h

51M3

45,

R4 = − [(yl + h′)− (yl − h

′)]h′2M4

12= −4h31M4

3,

69

where M1,M2,M3,M4 are given by

M1 =∂2[fBl]

∂y2

∣∣∣∣(x,ξ)

,

M2 = [f(·, yl)Ak(·)](4)(η),

M3 = [(· − xk)2Ak(·))](4)(ζ),

M4 = B′′

l (ϑ)

and ξ, ϑ ∈ (yl−1, yl+1); η, ζ ∈ (xk−1, xk+1).

Based on the integrals I1, I2, I3, I4, errors R1, R2, R3, R4 and the fact that Ak(xk+

h1)Bl(yl) > 0, we obtain the final estimation:

c10kl =8h31Ak(xk + h1)Bl(yl)[f(xk + h1, yl)− f(xk − h1, yl)]

16h41Ak(xk + h1)Bl(yl) + 8h31Ak(xk + h1)R4 + 6h1Bl(yl)R3 + 3R3R4

+

4h21Ak(xk + h1)[R1(xk + h1)−R1(xk − h1)] + 3R2

16h41Ak(xk + h1)Bl(yl) + 8h31Ak(xk + h1)R4 + 6h1Bl(yl)R3 + 3R3R4

=

[f(xk + h1, yl)− f(xk − h1, yl)]/2h1 +O(h1) +O(h1)

1 +O(h21) +O(h21) +O(h41)=

f(xk + h1, yl)− f(xk − h1, yl)2h1

+O(h1).

And therefore

c10kl =∂f

∂x

∣∣∣∣(xk,yl)

+O(h1).

According to the assumptions, O(h1) and O(h′) are counted to be same. 2

In classical mathematics, the following property of partial derivatives holds:

Lemma 10 Let f, g : [a, b]× [c, d]→ R, and let f, g have the first partial derivatives

with respect to x at the point (x0, y0) ∈ [a, b] × [c, d]. Then, a function f + g has

the first partial derivative with respect to x at the same point (x0, y0) ∈ [a, b]× [c, d]

and the following holds:

∂(f + g)

∂x

∣∣∣∣(x0,y0)

=∂f

∂x

∣∣∣∣(x0,y0)

+∂g

∂x

∣∣∣∣(x0,y0)

.

Analogously for the variable y.

70

The next corollary of Theorem 4 presents an analogous property of the partial

derivatives represented by the coefficients of the F 1-transform components.

Corollary 1 Let h, f, g ∈ L2([a, b]× [c, d]) such that h = f + g. Let {Ak ×Bl | k =

1, . . . , n, l = 1, . . . ,m} be an (h, h′)-uniform generalized fuzzy partition of [a, b]×[c, d]

and let F1nm[h] = (H1

kl), F1nm[f ] = (F 1

kl) and F1nm[g] = (G1

kl), k = 1, . . . , n,

l = 1, . . . ,m, be the F 1-transforms of h, f, g, respectively, with respect to the same

partition, where the components H1kl, F

1kl, G

1kl, k = 1, . . . , n, l = 1, . . . ,m, are given

by (4.32). Moreover, let functions h, f , g, Ak, Bl be four times continuously differ-

entiable on [a, b] × [c, d]. Then, for every k = 1, . . . , n, l = 1, . . . ,m, the following

holds:∂H1

kl

∂x

∣∣∣∣(xk,yl)

=∂F 1

kl

∂x

∣∣∣∣(xk,yl)

+∂G1

kl

∂x

∣∣∣∣(xk,yl)

. (5.3)

The same result holds for variable y.

proof: We fix one k, l from {1, . . . , n} and {1, . . . ,m}, respectively, and as-

sume that the components H1kl, F

1kl, G

1kl of the F 1-transforms of the functions h, f ,

g, respectively, have the following representations:

H1kl = c00kl,h + c10kl,h(x− xk) + c01kl,h(y − yl),

F 1kl = c00kl,f + c10kl,f (x− xk) + c01kl,f (y − yl),

G1kl = c00kl,g + c10kl,g(x− xk) + c01kl,g(y − yl).

Then, it follows that

∂H1kl

∂x

∣∣∣∣(xk,yl)

= c10kl,g =

∫ yl+1

yl−1

∫ xk+1

xk−1h(x, y)(x− xk)Ak(x)Bl(y)dx dy

(∫ xk+1


∫ yl+1

yl−1Bl(y)dy)

=∫ yl+1

yl−1

∫ xk+1

xk−1(f + g)(x, y)(x− xk)Ak(x)Bl(y)dx dy

(∫ xk+1


∫ yl+1

yl−1Bl(y)dy)

=∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)(x− xk)Ak(x)Bl(y)dx dy

(∫ xk+1


∫ yl+1

yl−1Bl(y)dy)

+∫ yl+1

yl−1

∫ xk+1

xk−1g(x, y)(x− xk)Ak(x)Bl(y)dx dy

(∫ xk+1


∫ yl+1

yl−1Bl(y)dy)

=

c10kl,f + c10kl,g =∂F 1

kl

∂x

∣∣∣∣(xk,yl)

+∂G1

kl

∂x

∣∣∣∣(xk,yl)

.

71

2

5.2 Higher-Order Partial Derivatives

In the previous section, we discussed the first partial derivatives of the given function

f . Those partial derivatives are functions of two variables x, y and thus they can

be differentiated again with respect to any of their variables. Then, we speak about

higher-order partial derivatives.

The goal of this section is to approximate the higher-order partial derivatives

of the given function f using the F s-transform of f , s ≥ 1. First, we recall the def-

inition of the higher-order partial derivatives of f .

Definition 15 Let (x0, y0) ∈ [a, b] × [c, d]. Let f, ∂f∂x, ∂f

∂y: [a, b] × [c, d] → R be

functions of two variables x, y. If there exists a partial derivative of the function ∂f∂x

with respect to x at the point (x0, y0), f has the second-order partial derivative with

respect to x at the point (x0, y0). We denote it by

∂

∂x

(∂f

∂x

)∣∣∣∣(x0,y0)

or∂2f

∂x2

∣∣∣∣(x0,y0)

.

Analogously, if there exists a partial derivative of the function ∂f∂x

with respect

to y at the point (x0, y0), f has the second-order partial derivative with respect to x

and with respect to y at the point (x0, y0) denoted by

∂

∂y

(∂f

∂x

)∣∣∣∣(x0,y0)

or∂2f

∂x∂y

∣∣∣∣(x0,y0)

.

Analogously to the function ∂f∂y

with respect to both variables.

Third-, fourth-, and higher-order partial derivatives can be obtained similarly

by repeating the differentiations.

Remark 9 For a function of two variables, there will be four possible second-order

derivatives:∂

∂x

(∂f

∂x

)=∂2f

∂x2,

∂

∂y

(∂f

∂y

)=∂2f

∂y2,

72

∂

∂y

(∂f

∂x

)=

∂2f

∂x∂y,

∂

∂x

(∂f

∂y

)=

∂2f

∂y∂x.

The partial derivatives ∂2f∂y∂x

and ∂2f∂x∂y

are called mixed partial derivatives.

Now, let L2(Ak)×L2(Bl), k = 1, . . . , n, l = 1, . . . ,m, be a Hilbert space, and let

f ∈ L2([a, b] × [c, d]), i.e., for all k = 1, . . . , n, l = 1, . . . ,m, f |[xk−1,xk+1]×[yl−1,yl+1]∈

L2(Ak) × L2(Bl) (see Section 2.2.3). Moreover, let Lp2(Ak) ⊆ L2(Ak), p ≥ 0, k =

1, . . . , n and Lr2(Bl) ⊆ L2(Bl), r ≥ 0, l = 1, . . . ,m be the linear spans consisting

of orthogonal polynomials:

{P ik(x)}i=0,...,p, {Qj

l (y)}j=0,...,r, respectively. (5.4)

Consequently, let Ls2(Ak × Bl) ⊆ L2(Ak) × L2(Bl) be the linear span consisting

of orthogonal polynomials

{Sijkl(x, y)}i=0,...,s; j=0,...,s; i+j≤s = {P ik(x) ·Qj

l (y)}i=0,...,s; j=0,...,s; i+j≤s. (5.5)

Let us remark that p, r, s denote the degrees of polynomials, and orthogonality

is considered in the sense of (2.10).

The following theorem presents approximations of the higher-order partial deriva-

tives of f using the coefficients of the F s-transform components of f , s ≥ 1.

The proof is based on Lemma 6 (see Section 4.1) and was inspired by the The-

orem about approximation of derivatives using the F s-transform of functions of one

variable [36]. In the proof are used facts from differential calculus [11, 15].

Theorem 5 Let n,m ≥ 2 and let functions f , Ak, Bl, k = 1, . . . , n, l = 1, . . . ,m,

be (s+ 1)-times continuously differentiable on [a, b]× [c, d]. Moreover, let {Ak×Bl |k = 1, . . . , n, l = 1, . . . ,m} establish an (h, h


of [a, b] × [c, d] and let Fsnm[f ] = (F s

kl), s ≥ 1, k = 1, . . . , n, l = 1, . . . ,m, be

the F s-transform of f with respect to the given partition, where

F skl =

∑0≤i+j≤s

cijklSijkl.

Then for every (xk, yl), k = 1, . . . , n, l = 1, . . . ,m, and for every t, d such that

0 ≤ t ≤ d and 1 ≤ d ≤ s, the following holds true∣∣∣∣∣ ∂dF skl

∂xt∂yd−t

∣∣∣∣(xk,yl)

− ∂df

∂xt∂yd−t

∣∣∣∣(xk,yl)

∣∣∣∣∣ ≤ Cd,s · (h′)s+1,

73

where Cd,s → 0 for s→∞.


on the differentiability of f it follows that f ∈ L2([a, b] × [c, d]), i.e., for all k =

1, . . . , n, l = 1, . . . ,m, f |[xk−1,xk+1]×[yl−1,yl+1]∈ L2(Ak) × L2(Bl) where L2(Ak) ×

L2(Bl), k = 1, . . . , n, l = 1, . . . ,m, is the Hilbert space (see Section 2.2.3).

Now, let us fix k, l from {1, . . . , n}, {1, . . . ,m}, respectively, and restrict our

interest on f ∈ L2(Ak) × L2(Bl). Moreover, let us fix t, d such that 0 ≤ t ≤ d

and 1 ≤ d ≤ s.

Based on the proof of Lemma 6 (expression (4.10)), the F s-transform component

F skl of the function f is equal to

F skl(x, y) = f(x, y) +Rs

kl(x, y)− r(x, y), (5.6)

(x, y) ∈ [xk−1, xk+1]× [yl−1, yl+1],

where Rskl(x, y) is the F s-transform component of the reminder r(x, y) which is rep-

resented as follows:

r(x, y) =1

(s+ 1)!

s+1∑i=0

(s+ 1

i

)∂s+1f

∂xs+1−i∂yi

∣∣∣∣(ξ,η)

(x− xk)s+1−i(y − yl)i,

where (ξ, η) is an inner point at the line segment between (x, y) and (xk, yl).

Let us consider d-th order partial derivatives of both sides of the formula (5.6)

t-times with respect to x and (d − t)-times with respect to y at the point (xk, yl).

We obtain

∂dF skl

∂xt∂yd−t

∣∣∣∣(xk,yl)

=∂df

∂xt∂yd−t

∣∣∣∣(xk,yl)

+∂dRs

kl

∂xt∂yd−t

∣∣∣∣(xk,yl)

− ∂dr

∂xt∂yd−t

∣∣∣∣(xk,yl)

.

Since ∂dr∂xt∂yd−t

∣∣∣(xk,yl)

= 0, we obtain

∂dF skl

∂xt∂yd−t

∣∣∣∣(xk,yl)

=∂df

∂xt∂yd−t

∣∣∣∣(xk,yl)

+∂dRs

kl

∂xt∂yd−t

∣∣∣∣(xk,yl)

.

To estimate the following difference∣∣∣∣∣ ∂dF skl

∂xt∂yd−t

∣∣∣∣(xk,yl)

− ∂df

∂xt∂yd−t

∣∣∣∣(xk,yl)

∣∣∣∣∣ =

∣∣∣∣∣ ∂dRskl

∂xt∂yd−t

∣∣∣∣(xk,yl)

∣∣∣∣∣ , (5.7)

74

we need to estimate the partial derivative∂dRs

kl

∂xt∂yd−t

∣∣∣(xk,yl)

of the F s-transform com-

ponent Rskl(x, y).

According to the Definition 9 (Section 4.1), the component Rskl(x, y) is defined

for all (x, y) ∈ [xk−1, xk+1]× [yl−1, yl+1] as follows

Rskl(x, y) =

∑0≤i+j≤s

cijrklSijkl(x, y),

where cijrkl are coefficients of the F s-transform component and {Sijkl(x, y)}0≤i+j≤s is

the orthogonal basis of polynomials of the space L2(Ak)×L2(Bl). The polynomials

{Sijkl(x, y)}0≤i+j≤s are constructed as follows

{Sijkl(x, y)}0≤i+j≤s = {P ik(x) ·Qj

l (y)}i=0,...,s; j=0,...,s; i+j≤s,

where i, j, s denote the degrees of polynomials (see Section 2.2.4). Therefore, we can

write

Rskl(x, y) =

∑0≤i+j≤s

cijrklPik(x)Qj

l (y). (5.8)

Let us consider the partial derivative of Rskl(x, y) represented by (5.8). We obtain

∂dRskl

∂xt∂yd−t

∣∣∣∣(xk,yl)

=∑

0≤i+j≤s

cijrkl ·∂dP i

kQjl

∂xt∂yd−t

∣∣∣∣∣(xk,yl)

. (5.9)

For t > i or d − t > j, we have∂dP i

kQjl

∂xt∂yd−t (x, y) = 0. Therefore, we can specify

the expression (5.9) as follows

∂dRskl

∂xt∂yd−t

∣∣∣∣(xk,yl)

=

i+j≤s∑i≥t, j≥d−t

cijrkl ·∂dP i

kQjl

∂xt∂yd−t

∣∣∣∣∣(xk,yl)

. (5.10)

To estimate the expression (5.10), let us bound the individual factors of it.

• First, let us determine the number of summands in the sum (5.10). Based

on the arithmetic series {ij} for i = t, t+ 1, . . . , s and j = d− t, d− t+ 1, . . . , s

such that i+ j ≤ s, the number of summands is equal to

(s− d+ 1) + (s− d) + (s− d− 1) + · · ·+ 1 =(s− d+ 2)(s− d+ 1)

2. (5.11)

75

• By Lemma 6 (expression (4.24)), the coefficients cijrkl in (5.10) can be bounded

for all i, j such that 0 ≤ i+ j ≤ s as follows:∣∣cijrkl∣∣ ≤ 2s+1 ·Kkl ·Mkl

(s+ 1)! ·mkl

· (h′)s+1, (5.12)

where Kkl,Mkl,mkl are constants specified by (4.12), (4.19) and (4.22), respec-

tively.

• Let us denote

Lijkl =∂dP i

kQjl

∂xt∂yd−t

∣∣∣∣∣(xk,yl)

and Lkl = maxi≥t, j≥d−t, i+j≤s

|Lijkl|. (5.13)

Then, based on (5.11), (5.12) and (5.13), the estimation of∂dRs

kl

∂xt∂yd−t

∣∣∣(xk,yl)

repre-

sented by (5.10) is following:∣∣∣∣∣ ∂dRskl

∂xt∂yd−t

∣∣∣∣(xk,yl)

∣∣∣∣∣ ≤i+j≤s∑

i≥t, j≥d−t

∣∣cijrkl∣∣ ·∣∣∣∣∣∣ ∂

dP ikQ

jl

∂xt∂yd−t

∣∣∣∣∣(xk,yl)

∣∣∣∣∣∣ ≤(s− d+ 2)(s− d+ 1)

2· 2s+1 ·Kkl ·Mkl

(s+ 1)! ·mkl

· (h′)s+1 · Lkl =

(s− d+ 2)(s− d+ 1) · 2s

(s+ 1)!· Kkl ·Mkl · Lkl

mkl

· (h′)s+1. (5.14)

Let us go to the beginning. The aim was to estimate the following difference∣∣∣∣∣ ∂dF skl

∂xt∂yd−t

∣∣∣∣(xk,yl)

− ∂df

∂xt∂yd−t

∣∣∣∣(xk,yl)

∣∣∣∣∣ =

∣∣∣∣∣ ∂dRskl

∂xt∂yd−t

∣∣∣∣(xk,yl)

∣∣∣∣∣ .Based on (5.14), we obtain∣∣∣∣∣ ∂dF s

kl

∂xt∂yd−t

∣∣∣∣(xk,yl)

− ∂df

∂xt∂yd−t

∣∣∣∣(xk,yl)

∣∣∣∣∣ ≤(s− d+ 2)(s− d+ 1) · 2s

(s+ 1)!· Kkl ·Mkl · Lkl

mkl

· (h′)s+1, (5.15)

where the constants Kkl,Mkl,mkl, Lkl are specified by (4.12), (4.19), (4.22) (see

the proof of Lemma 6) and (5.13), respectively. These constants Kkl,Mkl,mkl, Lkl

in the expression (5.15) are dependent on k, l. Let us finally denote the following

K = maxk=1,...,n; l=1,...,m

Kkl,

76

M = maxk=1,...,n; l=1,...,m

Mkl,

L = maxk=1,...,n; l=1,...,m

Lkl,

Z = mink=1,...,n; l=1,...,m

mkl.

Moreover, let us denote

C =K ·M · L

Z. (5.16)

Then, we can conclude that for all (xk, yl), k = 1, . . . , n, l = 1, . . . ,m, the difference∣∣∣∣ ∂dF skl

∂xt∂yd−t

∣∣∣(xk,yl)

− ∂df∂xt∂yd−t

∣∣∣(xk,yl)

∣∣∣∣ is estimated by the following

∣∣∣∣∣ ∂dF skl

∂xt∂yd−t

∣∣∣∣(xk,yl)

− ∂df

∂xt∂yd−t

∣∣∣∣(xk,yl)

∣∣∣∣∣ ≤(s− d+ 2)(s− d+ 1) · 2s

(s+ 1)!· C · (h′)s+1 = Cd,s · (h′)s+1,

where Cd,s = (s−d+2)(s−d+1)·2s(s+1)!

· C is dependent on d and s such that Cd,s → 0

for s→∞.

2

Corollary 2 Let the assumptions of Theorem 5 hold and let d = s. Then for every

(xk, yl), k = 1, . . . , n, l = 1, . . . ,m, and for every 0 ≤ t ≤ s, the following holds

∂sf

∂xt∂ys−t

∣∣∣∣(xk,yl)

= t! · (s− t)! · ct(s−t)kl +K(h′, s),

where ct(s−t)kl is the coefficient of the F s-transform component and |K(h′, s)| ≤ Cs ·

(h′)s+1, such that Cs → 0 for s→∞.

proof: Let us fix k, l from {1, . . . , n}, {1, . . . ,m}, respectively, and t such

that 0 ≤ t ≤ s.

Based on Theorem 5, we have∣∣∣∣∣ ∂sf

∂xt∂ys−t

∣∣∣∣(xk,yl)

− ∂sF skl

∂xt∂ys−t

∣∣∣∣(xk,yl)

∣∣∣∣∣ ≤ Cs · (h′)s+1,

where Cs = 2s+1

(s+1)!· C depends on s such that Cs → 0 for s → ∞ and C is a real

constant specified by (5.16).

77

Let us denote

∂sf

∂xt∂ys−t

∣∣∣∣(xk,yl)

− ∂sF skl

∂xt∂ys−t

∣∣∣∣(xk,yl)

= K(h′, s),

where |K(h′, s)| ≤ Cs · (h′)s+1. Therefore, we can write

∂sf

∂xt∂ys−t

∣∣∣∣(xk,yl)

=∂sF s

kl

∂xt∂ys−t

∣∣∣∣(xk,yl)

+K(h′, s). (5.17)

Let us focus on F skl and its partial derivative

∂sF skl

∂xt∂ys−t

∣∣∣(xk,yl)

. Similarly to the proof

of Theorem 5, we can represent F skl by the following

F skl(x, y) =

∑0≤i+j≤s

cijklPik(x)Qj

l (y), (5.18)

where cijkl are coefficients of the F s-transform component and P ik(x), Qj

l (y) are or-

thogonal polynomials created from the sets {1, x, x2, . . . , xs} and {1, y, y2, . . . , ys},respectively, by the Gram-Schmidt orthogonalization process [17].

Let us partially differentiate the expression (5.18) t-times with respect to x

and (s− t)-times with respect to y at the point (xk, yl), we obtain

∂sF skl

∂xt∂yd−t

∣∣∣∣(xk,yl)

=∑

0≤i+j≤s

cijkl ·∂sP i

kQjl

∂xt∂yd−t

∣∣∣∣∣(xk,yl)

.

Since i, j denote the degrees of polynomials P ik(x) and Qj

l (y), then, for all i < t

or j < s− t, we have∂sP i

kQjl

∂xt∂ys−t (x, y) = 0. Therefore, we obtain

∂sF skl

∂xt∂ys−t

∣∣∣∣(xk,yl)

= ct(s−t)kl · ∂

sP tkQ

(s−t)l

∂xt∂ys−t

∣∣∣∣∣(xk,yl)

= t! · (s− t)! · ct(s−t)kl . (5.19)

Based on (5.17) and (5.19), we can finally conclude that

∂sf

∂xt∂ys−t

∣∣∣∣(xk,yl)

= t! · (s− t)! · ct(s−t)kl +K(h′, s),

where |K(h′, s)| ≤ Cs · (h′)s+1, such that Cs → 0 for s→∞.

2

78

Remark 10 Let the assumptions of Corollary 2 hold. Let us fix k, l from {1, . . . , n},{1, . . . ,m}, respectively and 0 ≤ t ≤ s. Generally, to approximate the following

partial derivative∂sf

∂xt∂ys−t

∣∣∣∣(xk,yl)

,

we need to compute

t! · (s− t)! · ct(s−t)kl .

According to Lemma 2 (see Section 2.2.4), the coefficient ct(s−t)kl is given by

ct(s−t)kl =

〈f, St(s−t)kl 〉kl〈St(s−t)kl , S

t(s−t)kl 〉kl

=〈f, P t

kQs−tl 〉kl

〈P tkQ

s−tl , P t

kQs−tl 〉kl

,

where 〈., .〉 is the inner product given by (2.9) and P tk(x), Qs−t

l (y) and St(s−t)kl (x, y)

are polynomials (5.4) and (5.5).

Therefore, to compute the coefficient ct(s−t)kl , we do not need to find all orthogonal

polynomials up to the s-th degree (from the bases (5.4) and (5.5)). It is sufficient

to find only the following polynomials:

• P 0k (x), P 1

k (x), . . . , P tk(x) ∈ Lp2(Ak), x ∈ [xk−1, xk+1], 0 ≤ t ≤ p,

• Q0l (y), Q1

l (y), . . . , Qs−tl (y) ∈ Lr2(Bl), y ∈ [yl−1, yl+1], 0 ≤ s− t ≤ r.

For example, let us assume the F 5-transform of f and the following partial

derivative (for fixed k, l)∂5f

∂x3∂y2

∣∣∣∣(xk,yl)

.

To approximate this partial derivative, it is sufficient to find the following polyno-

mials:

• P 0k (x), P 1

k (x), P 2k (x), P 3

k (x),

• Q0l (y), Q1

l (y), Q2l (y),

and then to compute 12 c32kl where

c32kl =

∫ yl+1

yl−1

∫ xk+1

xk−1f(x, y)P 3

k (x)Q2l (y)Ak(x)Bl(y)dxdy∫ yl+1

yl−1

∫ xk+1

xk−1(P 3

k (x)Q2l (y))2Ak(x)Bl(y)dxdy

.

79

Chapter 6

Application to the Edge Detection

Problem

The edge detection problem is an essential problem of many image processing appli-

cations due to the wide spectrum of methods that use edge detection as a preprocess-

ing technique. For example, edge detection is an important tool in feature detection,

image segmentation, and image registration, etc. In addition, many real applications

need to solve the edge detection problem, for example, microscopy, medicine, safety

management, etc. Therefore, interest in the development of this field has increased

in recent years, particularly with respect to the following techniques: wavelet trans-

form, neural networks, soft computing techniques (see, e.g., [24, 14, 43, 19, 22, 1,

20, 51]). These techniques have been used in combination with well-recommended

approaches published in [2, 8, 7].

Generally, the edge detection problem is easy to describe but difficult to formalize

because there is no explicit definition of the term “edge”. Therefore, one of the most

fundamental components of edge detection is the characterization of the edge. The ex-

isting methods and algorithms for edge detection use different properties of edges

and thus appeal to different mathematical theories. For example, the edge can be

defined as an area with a presence of high frequencies in a frequency spectrum;

in this case, the appropriate detection is based on the use of high-pass filters. An-

other approach specifies an edge as a significant change in image intensity; in this

80

case, the detection problem is based on searching for local maxima of the gradient

magnitude.

In this chapter, we present two approaches to the edge detection problem and pro-

pose their solutions on the basis of the theory described in this thesis. The first ap-

proach is focused on the term edge as the area with a presence of high frequencies,

for which the edge detection problem is solved by the F 0-transform (see Chapter 3).

The second approach is inspired by the Canny edge detector, in which the edge is

specified as a significant change in image intensity. The Canny algorithm is modified

by the F 1-transform (see Section 4.3).

6.1 Edge Detection Using the F 0-transform

In this section, we present our first approach to the edge detection problem. The aim

is to show one of the possible methods for edge detection in images. We do not

elaborate technical details but instead describe and explain the foundation of our

approach and demonstrate it using examples. In this section, we do not focus

on the comparison of our method with other existing methods.

6.1.1 Main Idea

In this approach, the edge is assumed to be an area with a presence of high frequen-

cies in a frequency spectrum, and as we mentioned above, the detection is based

on the use of high-pass filters. The main idea of our approach is to transform an orig-

inal image function into an “edge” function. Then, the edge function is considered

as a fuzzy set whose membership function represents the degree of belonging of each

element of the function to an edge (for more about fuzzy sets, see [51]). This trans-

formation is provided by the F 0-transform of the given image function. A detailed

description of this approach is provided below.

Let an original image function be described as a discrete real function u : [1, N ]×[1,M ] → R defined on the N ×M array of pixels P = {(pi, qj) | i = 1, . . . , N, j =

1, . . . ,M}.

81

Let generally x0, x1, . . . , xn, xn+1 ∈ [1, N ] and y0, y1, . . . , ym, ym+1 ∈ [1,M ] be

fixed nodes, and let fuzzy sets {Ak × Bl | k = 1, . . . , n, l = 1, . . . ,m} establish

a fuzzy partition1 of [1, N ] × [1,M ]. Moreover, let F0nm[u] = (F 0

kl), k = 1, . . . , n,

l = 1, . . . ,m, be the direct F 0-transform of u and u0(pi, qj), where (pi, qj) ∈ P , be

the inverse F 0-transform of u (see Chapter 3).

Now, we recall two important properties of the F 0-transform. The main idea

of the edge detection in the image u solved by the F 0-transform is based on these two

properties. The properties are useful for verifying that the proposed edge detection

technique works correctly.

• The first property is the property (B) (3.3) introduced in Section 3.1 (proved

in [39]). It says that each component F 0kl, k = 1, . . . , n, l = 1, . . . ,m of the di-

rect F 0-transform of u is a local mean value of u over a support set of the re-

spective fuzzy set (Ak ×Bl), k = 1, . . . , n, l = 1, . . . ,m.

• The second property (proven in [33]) states that each component F 0kl, k =

1, . . . , n, l = 1, . . . ,m of the direct F 0-transform of u works as a low-pass

filter, i.e., by applying the direct F 0-transform to the original function u, we

remove high frequencies from the function u.

Based on these properties and the representation of the edge by areas of high

frequencies in the image function, we come to the following conclusion. The dif-

ference between an original image function u and its inverse F 0-transform u0, i.e.,

E = |u− u0|, works as a high-pass filter of the former. Therefore, this difference E

can be used as a membership function of the fuzzy set “edge”.

6.1.2 F 0-transform-Based Algorithm for Edge Detection

Now let us characterize an edge by a fuzzy set with the membership function that

assigns a degree of belonging to every image pixel.

Let n,m be numbers of fuzzy sets in a fuzzy partition of the domain P =

1In this application, we use the h-uniform fuzzy partition with the Ruspini condition.

82

[1, N ] × [1,M ] of an image u. Parameters n,m determine a robustness of the re-

spective partition, and they are important parameters in our approach. They relate

to a size of a neighborhood of chosen pixels and therefore influence the thickness

of the respective edges according to the following rule: the larger the values n,m,

the “thinner” the corresponding edges. This rule is a core of the edge detection

algorithm proposed below.

The membership function of an edge E can be represented by the formal expres-

sion

E = {(pi, qj) ∈ P | (∃k, l)(pi ∈ Ak) and (qj ∈ Bl) and |u(pi, qj)− F 0kl|}

with the following interpretation: ∃ relates to the addition; “and” to the product;

(pi ∈ Ak) is interpreted as Ak(pi) (similarly, qj ∈ Bl); and finally |u(pi, qj) − F 0kl|

is rescaled to [0, 1] value. Therefore, the membership degree E(pi, qj) is equal

to |∑n

k=1

∑ml=1Ak(pi)Bl(qj) (u(pi, qj) − F 0

kl)|. Thus, we obtain the following rep-

resentation, which will be used in the edge detection algorithm. For all (pi, qj) ∈ P

E(pi, qj) = |u(pi, qj)− u0(pi, qj)| = |n∑k=1

m∑l=1

Ak(pi)Bl(qj) (u(pi, qj)− F 0kl)| ≤

n∑k=1

m∑l=1

Ak(pi)Bl(qj) |(u(pi, qj)− F 0kl)|.

Let us describe the main steps of the edge detection algorithm (FTransform-

EDA), which is based on the above proposed characterization and uses the F 0-

transform of the image function u:

Input: Image function u, the numbers n,m of fuzzy sets in the chosen fuzzy

partition.

1. Compute F0nm[u] = (F 0

kl), k = 1, . . . , n, l = 1, . . . ,m – the direct F 0-transform

of u (see Definition 7);

2. Compute u0(pi, qj); (pi, qj) ∈ P – the inverse F 0-transform of u (see Defini-

tion 8);

3. Compute for all (pi, qj) ∈ P the residuum function

E(pi, qj) = |u(pi, qj)− u0(pi, qj)|.

83

Figure 6.1: Edge detection by the F 0-transform - Original image.

4. Rescale and round the values of E from [0,max(pi,qj)∈P E(pi, qj)] to the integers

in [0, 255], which results in the new image Er.

Output: Plot Er.

6.1.3 Experiments and Examples

As mentioned in the previous section, the input arguments n,m (the numbers

of fuzzy sets in the chosen fuzzy partition) of the algorithm FTransform-EDA

determine the “thickness” of the edges. According to the above formulated rule,

the larger the values n,m (equivalently, the fewer pixels covered by every basic

function (Ak × Bl), k = 1, . . . , n, l = 1, . . . ,m) the “thinner” the corresponding

edges. The illustrations of the image edge functions Er (residuals) with different

settings of parameters n,m are given in Figures 6.2 – 6.4, together with the original

image in Figure 6.1.

Naturally, blurred images with a various depth of the focus or objects in (partial)

shadow are problematic. Let us present here the second case and show how the set-

ting of the edge thickness (i.e., the setting of n,m) affects the result. Figure 6.5

84

Figure 6.2: Edge detection by the F 0-transform - Image edge function Er (4 pixelscovered by one basic function).


85


presents an exemplary image in which the object of interest is blurred. Obviously,

the edge elements of this object do not correspond with the characterization of edge

elements for an object in focus, i.e., the intensity change in a small neighborhood

of the edge element is not significant. Hence, the edges of the blurred objects cannot

be captured using small values of n,m (see Figure 6.6). A different situation occurs

at higher values of n,m (see Figure 6.7): in this case, lower intensity changes are

captured by the components of the F 0-transform, edges are thicker and the edge

of the blurred object is specified in a negative way, which means its edge elements

have degrees of memberships close to zero.

Remark 11 Let us remark that all “edge images” (results from the FTransform-

EDA algorithm) are gray-scale images, i.e., they are described by the membership

functions that represent the degree of belonging of each pixel to an edge (values within

[0, 1]).

Various techniques can be applied over these membership functions of edges to in-

crease their sharpness or exclude points with weak intensity, such as the technique

86

Figure 6.5: Partially blurred image - Original.

Figure 6.6: Edge detection by the F 0-transform - Partially blurred image (Erwith 10 pixels covered by one basic function).

87

Figure 6.7: Edge detection by the F 0-transform - Partially blurred image (Erwith 100 pixels covered by one basic function).

of thresholding, which assigns two values, 0 or 1, to the resulting image. If an an-

alyzed value is less than a chosen threshold, the value is changed to 0; otherwise,

the value is changed to 1.

Such processed images can then be used for comparison with other edge detection

methods. However, as mentioned at the beginning, we do not focus on comparing

our results with others in this section.

6.2 F 1-transform Edge Detector Inspired

by Canny’s Algorithm

In this section, we present our second approach to the edge detection problem.

The approach is inspired by the Canny detector, which is well known and widely

used in computer vision. We compare our approach based on the F 1-transform

with the Canny approach. We demonstrate applications of our detector using specific

images and compare the results with the Canny results.

88

6.2.1 Main Idea

In this approach, the edge is specified as a significant change in the image intensity,

and such changes are always modeled by a first derivative. In Chapter 5, we showed

how first (partial) derivatives can be approximated using the F 1-transform compo-

nents. Therefore, the main idea of our approach is to modify Canny’s algorithm

using the technique of the F 1-transform.

We focused on the Canny detector [2], denoted as an “optimal detector” that was

developed to ensure three basic criteria: good detection, good localization and mini-

mal multiple responses to a single edge. The important point was to capture the in-

tuitive criteria given above in a mathematical form that is easily solvable.

The Canny algorithm was created as a multi-step procedure for detecting edges

at local maxima of the gradient magnitude. There are many algorithmic imple-

mentations of the Canny edge detector. Below, we give the implementation that is

regarded as a typical approximation of this detector [6].

1. Step: Smooth the image and filter out any noise using the Gaussian filter.

2. Step: Approximate the first derivative of the image function by computing a gradient

of the image function to find local maxima of the gradient magnitude and its

direction. This step is realized by a convolution of the original image with a fil-

ter given by directional masks (edge detection operators such as Roberts, Pre-

witt, Sobel, etc.). This convolution returns a value for the first derivative

in the horizontal direction and the vertical direction. The gradient magni-

tude and direction are as follows:

| grad f | =

√(∂f

∂x

)2

+

(∂f

∂y

)2

, Θ = arctan∂f

∂y/∂f

∂x.

3. Step: Non-maximum suppression ([40]). The point is considered on the edge if its

gradient magnitude assumes a local maximum in the gradient direction.

4. Step: Tracing edges through the image and hysteresis thresholding. By tracing edges,

we imply a procedure that preserves continuity. A candidate edge pixel is fi-

nally selected if its intensity is greater than the high threshold or if its intensity

89

is greater than the low threshold and moreover is connected to a pixel with

intensity greater than the high threshold.

The output is a binary image in which each pixel is marked as either an edge

pixel or a non-edge pixel.

Now, let us explain how the F 1-transform can be combined with the Canny

detector and how it is advantageous with respect to the pure Canny method.

The main idea is to replace the first two steps of the above algorithmic im-

plementation by a single step in which the F 1-transform is applied to the image

function. As a result, we obtain approximate values of the first partial derivatives

at nodes of the F 1-transform. These values will then be interpolated by the inverse

F 0-transform to obtain approximate values of the first partial derivatives at every

pixel.

The advantage of the proposed method is that we can solve these two problems

in one step:

• Blur the image and filter out any noise: the presence of noise is related to high-

frequency areas of the image, and the F -transform was generally proven to re-

move high frequencies from the image (for filtering abilities of the F -transform,

see [39, 33, 9]).

• Compute the gradient magnitude: by Theorem 4 (see Section 5.1) coeffi-

cients c10kl and c01kl of the F 1-transform components approximate the first partial

derivatives of the given function.

6.2.2 Modified Canny’s algorithm

Let an original image function be described as a discrete real function u : [1, N ] ×[1,M ] → R defined on the N ×M array of pixels P = {(pi, qj) | i = 1, . . . , N, j =

1, . . . ,M}. Let generally x0, x1, . . . , xn, xn+1 ∈ [1, N ] and y0, y1, . . . , ym, ym+1 ∈[1,M ] be fixed nodes, and let fuzzy sets {Ak×Bl | k = 1, . . . , n, l = 1, . . . ,m} estab-

lish a fuzzy partition of [1, N ]× [1,M ]. Moreover, let F1nm[u] = (F 1

kl), k = 1, . . . , n,

l = 1, . . . ,m, be the (direct) F 1-transform of u.

90

In this application we use the (h, h′)-uniform generalized fuzzy partition with

the triangular-shaped basic functions (see Subsection 2.1.2), which do not satisfy

the Ruspini condition and enable a smoother image to be obtained after an appli-

cation of the F 1-transform.

The first step is to apply the direct F 1-transform to the image function u

and to obtain the coefficients c10kl , c01kl , k = 1, . . . , n, l = 1, . . . ,m (see Lemma 9

in Section 4.3.2):

c10kl =6∑N

i=1

∑Mj=1 u(pi, qj)(pi − xk)Ak(pi)Bl(qj)

h′4,

c01kl =6∑N

i=1

∑Mj=1 u(pi, qj)(qj − yl)Ak(pi)Bl(qj)

h′4.

These coefficients approximate the first partial derivatives with respect to x and y

at nodes (xk, yl), k = 1, . . . , n, l = 1, . . . ,m (see Theorem 4 in Section 5.1):

c10kl ≈∂u

∂x

∣∣∣∣(xk,yl)

, c01kl ≈∂u

∂y

∣∣∣∣(xk,yl)

.

The next step is to extend the approximation of the first partial derivatives from

nodes to the whole domain of the image function u using the inverse F 0-transform.

For all (pi, qj) ∈ [1, N ]× [1,M ], we obtain in the horizontal direction:

∂u

∂x

∣∣∣∣(pi,qj)

≈n∑k=1

m∑l=1

c10klAk(pi)Bl(qj)

and similarly in the vertical direction:

∂u

∂y

∣∣∣∣(pi,qj)

≈n∑k=1

m∑l=1

c01klAk(pi)Bl(qj).

All other steps of the Canny edge detector, namely finding a local maxima

of the gradient magnitude and its direction, non-maximum suppression, tracing

edges through the image and hysteresis thresholding, are the same as in the original

procedure.

91

6.2.3 Experiments and Examples

In this section, we illustrate our proposed modified Canny detector using specific im-

ages. The first example (image “star” in Figure 6.8) demonstrates the principal steps

of the proposed algorithm: original image (the input), gradient components in x

and y directions, gradient magnitude, gradient angles discretization, non-maxima

suppression and, finally, tracing edges with hysteresis thresholding (the output).

For comparison, the Canny detector output of “star” is illustrated in Figure 6.8

as well.

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 6.8: The F 1-transform edge detector: (a) original image; (b) gradient com-ponents in the horizontal (x) direction; (c) gradient components in the vertical (y)direction; (d) gradient magnitude; (e) gradient angles discretization (0-degree edgeis colored in yellow, 45 - in green, 90 - in blue, and 135 - in red); (f) non-maxima sup-pression; (g) tracing edges with hysteresis thresholding - the F 1-transform detectoroutput; (h) the Canny detector output.

The second example (see Figure 6.9) shows the original image and the outputs

of the proposed F 1-transform detector and Canny detector. Both detectors are

demonstrated in the grayscale image “Lena”. This image is often used as an im-

portant benchmark due to its blurred background, texture on the hat, perfect main

92

contours, noise. The original image (Figure 6.9(a)) and the Canny detector out-

put (Figure 6.9(b)) were downloaded from the web page [52]. Two other images

(Figures 6.9(c) and 6.9(d)) are the results of the proposed F 1-transform detector

with different threshold selections. The latter specifies the number of detected

edges.

(a) Original (b) Canny detector

(c) F 1-transform detector - more edges (d) F 1-transform detector - less edges

Figure 6.9: Lena - edge detection.

The last example with geometrical patterns was created by us as a benchmark

(see Figure 6.10 (Left)). We wanted to confirm that by using the F 1-transform

detector the following properties are maintained:

93

• smoothness of circular lines,

• dashed circles,

• concentric circles,

• smoothness of sharp connections.

The images for comparison that were processed by the Canny detector and the F 1-

transform detector are presented in Figure 6.10 (Middle, Right).

Remark 12 In this thesis, we demonstrated our approaches based on the theory in-

troduced here. We do not focus on qualitative comparisons of our results with the re-

sults of other detectors. Moreover, due to vague characterizations of edges and the edge

detection problem, an objective discussion is not possible. There are some accepted

benchmarks that contain “ideal” solutions. In [34], we used the Berkley Segmen-

tation Dataset and Benchmark BSDS3002 to compute qualitative measures of our

results and make a comparison with other methods.

2https://www.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/

94

Figure 6.10: Geometrical patterns. Left: Original; Middle: Canny detector output;Right: F 1-transform detector output.

95

Chapter 7

Conclusion

The main goal of the thesis was to investigate further details of the technique of fuzzy

transform (F -transform) and to extend this technique in order to estimate quali-

ties of approximations of partial derivatives. The F -transform was presented from

the theoretical perspective as well as from the point of view of possible practical

application to image processing.

We focused on functions of two variables and investigated the F 0-transform

and the F s-transform, s ≥ 1, for them. The F 0-transform is de facto the ordi-

nary F -transform, which was originally developed by I. Perfilieva as a tool for fuzzy

modeling. In our denotation, it is presented as the F 0-transform, where 0 repre-

sents the zeroth degree of polynomials (constants) of the respective F 0-transform

components. From this point of view, we extended the fuzzy transform of the ze-

roth degree (F 0-transform) to the fuzzy transform of a higher degree (F s-transform,

s ≥ 1), where the respective components are in the forms of the s-th degree poly-

nomials.

Originally, the technique of fuzzy transform (for functions of one or two vari-

ables) was established as a vector (matrix) of elements (fuzzy transform components)

in which the components are given explicitly by the defined expression. The general-

ization came with the introduction of the fuzzy transform of a higher degree (for func-

tions of one variable) defined on the basis of orthogonal projection. The main idea

of the thesis was to unify the representation of fuzzy transform in general. In other

96

words, we introduced the F 0-transform as well as the F s-transform, s ≥ 1, for func-

tions of two variables on the basis of orthogonal projection in a special Hilbert

space.

In Chapter 3, we rewrote the original theory of the F 0-transform on the ba-

sis of orthogonal projection as mentioned above. We presented the direct as well

as the inverse F 0-transform together with their main properties and approxima-

tion skills. We illustrated the graphical representation of the direct F 0-transform

in the form of constant components and demonstrated the inverse F 0-transform

using specific examples.

Analogous to the F 0-transform, we investigated the F s-transform, s ≥ 1; see

Chapter 4. We defined the direct and inverse F s-transform, s ≥ 1, of functions of two

variables and generalized and proved the main properties. Moreover, in this chapter,

we deeply studied the fuzzy transform of the first degree (F 1-transform). We focused

on the F 1-transform components and simplified them according to chosen fuzzy

partitions. We demonstrated the graphical representation of the direct F 1-transform

of particular functions in the form of a set of hyperplanes. We also illustrated

the inverse F 1-transform of a particular function as the approximation of the original

function. Finally, we presented the discrete form of the F 1-transform for subsequent

use in the application part.

The investigation of the F 1-transform components and the F s-transform compo-

nents, s ≥ 1, in general indicated that using these components enables an approx-

imation of not only the original function but also its partial derivatives. In Chap-

ter 5, we proved the main Theorems that describe the approximation of the first-

and higher-order partial derivatives of the given function and estimate the qual-

ities of these approximations. The approximations of the first partial derivatives

using the F 1-transform components was later used in the application part, in one

of the proposed approaches to edge detection.

Finally, we demonstrated the use of the presented theory in a possible practi-

cal application to the edge detection problem; see Chapter 6. We presented two

different approaches to the problem of edge detection in image processing and pro-

posed their solutions. The first approach was based on the term edge as the area

97

with a presence of high frequencies. The proposed solution was the use of the F 0-

transform, which was proven to work as a low-pass filter. The second approach was

inspired by the Canny edge detector, in which the edge is specified as a significant

change in the image intensity. The Canny algorithm is a multi-step procedure

for detecting edges at local maxima of the gradient magnitude, i.e., one of the main

steps is to investigate the derivatives of the image function. This explains why we

decided to use the technique of the F 1-transform for this application. In the pro-

posed solution, the Canny algorithm was modified by the F 1-transform technique.

Both approaches were discussed according to different settings of input parameters

and demonstrated using specific images.

98

Author’s Contributions

My contributions to the thesis are summarized in this chapter:

• Chapter 3: Rewriting the known theory of the F 0-transform based on the back-

ground of the Hilbert space and the orthogonal projection.

• Chapter 4: The extension of the F s-transform, s ≥ 1, for functions of two

variables. All definitions, properties and results that were proven in this chap-

ter. Elaboration of the F 1-transform including all properties, simplifications

of components and the discrete form.

• Chapter 5: The two main theorems, proofs and all properties concerning

the approximation of the partial derivatives using the F 1-transform and F s-

transform, s ≥ 1, generally.

• Chapter 6: In Section 6.2, the theoretical background of the edge detec-

tion method based on the F 1-transform. The comparison of our approach

and the Canny approach from a theoretical perspective.

99

List of Author’s Publications

• PERFILIEVA, I., HODAKOVA, P., HURTIK, P., Differentiation by the F-

transform and Application to Edge Detection. Fuzzy Sets and Systems, sub-

mitted.

• HODAKOVA, P., PERFILIEVA, I., HURTIK, P., F-transform and its Ex-

tension as Tool for Big Data Processing. Proc. of the 15th International

Conference on Information Processing and Management of Uncertainty in

Knowledge-Based Systems (IPMU 2014). France, 2014.

• HODAKOVA, P., PERFILJEVA, I., F 1-transform of Functions of Two Vari-

ables. Proc. of the EUSFLAT 2013, Advances in Intelligent Systems Research.

Atlantis Press, 2013, pp. 547–553, ISBN 978-90786-77-78-9.

• PERFILJEVA, I., HODAKOVA, P., VAJGL, M., DANKOVA, M., Classifica-

tion of Damages on Jewelry Stones: Preprocessing. Proc. of the 2013 Joint

IFSA World Congress, NAFIPS Annual Meeting (IFSA/NAFIPS). Edmon-

ton, Canada: IEEE Catalog No: CFP13750-USB, 2013, pp. 783–788, ISBN

978-147990347-4.

• VAJGL, M., PERFILJEVA, I., HODAKOVA, P., Advanced F-Transform-

Based Image Fusion. Advances in Fuzzy Systems. 2012, pp. 1–9.

• PERFILJEVA, I., HODAKOVA, P., HURTIK, P., F 1-transform Edge De-

tector Inspired by Canny’s algorithm. Communications in Computer and

Information Science. Advances on Computational Intelligence, 297. Berlin

Heidelberg: Springer, 2012, pp. 320–329, ISBN 978-3-642-31708-8.

100

• PERFILIEVA, I., PRADE, H., DUBOIS, D., GODO, L., ESTEVA, F., HODA-

KOVA, P., Interpolation of Fuzzy Data. Analytical Approach and Overview.

Fuzzy Sets and Systems. 2012, VOL. 192, pp. 134–158.

• PERFILJEVA, I., HODAKOVA, P., Fuzzy and Fourier Transforms. Proc. of

the EUSFLAT - LFA 2011. Atlantis Press, 2011, pp. 452–456, ISBN 978-90-

78677-00-0.

• PERFILJEVA, I., DANKOVA, M., HODAKOVA, P., VAJGL, M., F-Transform

Based Image Fusion. Rijeka, Croatia: InTech, 2011, pp. 3-22, ISBN 978-953-

307-679-9.

• DANKOVA, M., HODAKOVA, P., PERFILJEVA, I., VAJGL, M., Edge de-

tection using F-transform. Proc. of the 2011 11th International Conference

on Intelligent Systems Design and Applications. 2011, pp. 672–677. ISBN

978-1-4577-1675-1.

• PERFILJEVA, I., WRUBLOVA, M., HODAKOVA, P., Fuzzy Interpolation

according to Fuzzy and Classical Conditions. Acta Polytechnica Hungarica.

2010, VOL. 7, pp. 39–55.

• PERFILJEVA, I., DANKOVA, M., HODAKOVA, P., VAJGL, M., The Use

of F-Transform for Image Fusion Algorithms. Proc. of the 2010 International

Conference of Soft Computing and Pattern Recognition. IEEE, 2010, pp. 472–

477. ISBN 978-1-4244-7895-8.

• HODAKOVA, P., WRUBLOVA, M., PERFILJEVA, I., Fuzzy Rule Base In-

terpolation and its Graphical Representation. Proc. of the East West Fuzzy

Colloquium 2010 / 17th Zittau Fuzzy Colloquim. Zittau/Gorlitz, 2010, pp.

132–137. ISBN 978-3-9812655-4-5.

101

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Index

F 0-transform

component, 33, 37

direct, 32

inverse, 39

F 1-transform

component, 56, 58

direct, 55

discrete, 63

inverse, 61

F s-transform

component, 44

direct, 43

inverse, 53

approximation

of function

by the F 0-transform, 34, 40

by the F s-transform, 54

of partial derivative

first, 67

higher order, 73

edge detection, 80, 81, 88

algorithm

based on the F 0-transform, 82

based on the F 1-transform, 90

Canny, 89

fuzzy partition

generalized fuzzy partition, 21, 24

with the generalized Ruspini con-

dition, 20, 23

with the Ruspini condition, 18, 23

Hilbert space, 24

inner product, 26, 28

orthogonal projection, 25, 27, 29

orthogonality, 25, 28

partial derivative

first, 66

approximation, 67

higher order

approximation, 73

higher-order, 72

107

fuzzy (f-)transform of functions of two variables … · pr ace vlo zenym v datab azi dipl2. prohla...

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