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Unified approaches to time series and shape analysis and some fuzzy extensions

VASILE GEORGESCU

University of Craiova, Faculty of Economics and Business Administration

13 A.I.Cuza, Craiova, Romania

e-mail: v_geo@yahoo.com, vgeo@central.ucv.ro

Abstract. Despite their differences in nature, time series analysis and statistical shape analysis have much in common because they can share a unified methodological framework. Basically, it is based on

transforming a shape closed contour to a time series. Statistical Shape Analysis involves methods for the

geometrical study of random objects where location, rotation and scale information can be removed.

Time series analysis is a widely spread technique that takes into consideration the temporal nature of

data. However, it has been proved to be well adapted to represent or describe two-dimensional closed

contours of shapes. The aim of this paper is to extend the methods of transforming a shape closed contour

to a time series in a fuzzy context, based on recent advances in fuzzy digital geometry. The main

ingredient in this generalization is to consider geodesic distances between two points in a fuzzy object

(for example, between its centroid and its fuzzy boundaries). A geodesic distance is taken along a

geodesic path (i.e., a path of minimum length). Unlike in crisp (binary) convex objects, the shortest path

(when it exists) between to points in a fuzzy convex object is not necessarily a straight line segment. The

notions of fuzzy geodesic distance and fuzzy distance transform are foundational for developing a unitary

framework allowing to deal with both fuzzy shapes and fuzzy time series as equivalent objects.

1. TRANSFORMING A CRISP SHAPE CLOSED CONTOUR TO A TIME SERIES

The contour of a shape can be described by a function. There are two basic ideas to introduce such a

function:

i) the contour of a figure can be symmetric with respect to a line; than the orthogonal distance of the

contour point from the symmetry line as a function of position on the symmetry line can be considered

as a contour function;

ii) the contour function may be periodic, the contour itself can be considered as a periodic function;

assuming that the contour has some desirable properties such as star-shapedness (i.e., for each point

Ay ∈ , the line segment connecting y with the centroid is contained in A ) or convexity, a relatively

simple contour functions, such as the radius-vector or support functions, can be introduced, otherwise

more complex contour functions, such as the tangent-angle function, can be considered as contour

functions.

The radius-vector function rX (ϕ) is the distance from the reference point O (usually the center of

gravity) to the contour in the direction of the ϕ-ray where πϕ 20 ≤≤ (Fig. 1 a). In the general case,

however, description by the radius-vector function is not suitable for non-star-shaped figures (Fig. 1

b).

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ϕ

Xr X

(ϕ)

O

ϕ

Xr X

(ϕ)

O

a b

Figure 1. a) Radius-vector function; b) Problems with the radius-vector function occur if the figure is

not star-shaped.

An example of a star-shaped figure and its radius-vector function is given in Fig. 2.

ϕ

O

Xr X

(ϕ)

a

Figure 2. a) A start-shaped figure X ; b) Radius-vector function rX (ϕ) of the figure X . The gravity center of the figure was used as the origin to generate the radius-vector function.

When the radius-vector function rX (ϕ) of a star-shaped figure X is available some geometrical figure

parameters can be obtained. Integrating of rX (ϕ) yields the perimeter P(X), area A(X) and mean

radius-vector length Xr :

∫∫∫ ==′+=

πππ

ϕϕπ

ϕϕϕϕϕ2

0

2

0

2

2

0

22 )(2

1,)(

2

1)(,)()()( drrdrXAdrrXP XXXXX

The tangent angle at different points of the contour can also be used for the description of a shape

contour. It is assumed that the contour of the considered figure is piecewise-smooth so that a tangent

may not exist at a finite number of points. Let the perimeter of the figure X be L . Every point pl of the

contour of X can thus be identified with a number l, with 0≤ l≤ L, run through anti-clockwise. A

pointer is places at p0 so that its zero position coincides with the tangent direction at p0 . If the pointer

moves on the contour then it changes its direction in such a way that it is always in the direction of the

tangent, where its orientation is given by the direction of the movement. The angle given by the

pointer direction at pl is denoted )(lXφ where )0(Xφ = 0 and )(LXφ = π2 (Fig. 3). The function )(lXφ

is called the tangent-angle function.

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Figure 3. a, b) Definition of the tangent-angle function )(lXφ of figure X .

All these functions can be interpreted as signatures of a shape. Furthermore, they can be regarded as

parameterized functions of time, thus acting as converters from shape contours to time series.

2. REPRESENTATION OF CRISP AND FUZZY SHAPES IN A CONTINUOUS SPACE

2.1. CRISP SHAPES

Crisp shapes represent objects with crisp borders. Furthermore, if a texture is associated with the

object, it has to be uniformly represented (e.g. a digitized image, where all pixels are classified as

object pixels, or as background pixels).

The contour functions introduced in the previous section are good candidates for a shape descriptor. It

designates a signature of the shape and is based on a one-dimensional functional representation of the

two-dimensional shape boundary. The simplest way to generate a signature is to use the radial function

(also called centroid distance function), which express the radial distance from the centroid to the

boundary, as a function of the angle. Thus, for crisp objects, the shape signature function corresponds

to the Euclidean distance between each boundary point ( ))(),()( tytxtA = and the centroid

( )ccc yxA ,= of the shape:

( ) ( )22)()()( cc ytyxtxtCD −+−=

2.2. CONTINUOUS FUZZY SHAPES

In the case of a fuzzy object, boundary points are not strictly defined; there is a progressive transition

of the membership values from the support outline to the core outline. Fuzziness of an image

representation can arise from various reasons, such as limited acquisition conditions (scanning

resolution in digital images), but also as intrinsic property of the image, which may have imprecise

borders. In such cases, pixels close to the border of the object have assigned to them a fuzzy

membership value according to the extent of their belongingness to the object.

Continuous fuzzy shapes can be described as fuzzy geometric objects. A continuous fuzzy geometric

object A in pℜ is defined as a set of pairs ( ) pA xxx ℜ∈|)(, µ where [ ]1,0: →ℜ pAµ is the

membership function of A in nℜ . For any value θ ∈ [0, 1], θ-support of S, denoted by Θθ(S), is the

hard subset x | nx ℜ∈ and )(xSµ ≥ θ of nℜ . In other words, the θ-support of S is the set of all

points in nℜ with membership values greater than or equal to θ. 0-support will often be referred to as

support and be denoted by Θ(S). A fuzzy subset with a bounded support is called bounded. S is said to

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be convex if, for every three collinear points x, y, and z in nℜ such that y lies between x and z, µS(y) ≥

min[µS(x), µS(z)]. A fuzzy subset is called smooth if it is differentiable at every location nx ℜ∈ .

An alternative representation of fuzzy geometric objects is given by a set of −α cuts:

[ ] 1,0|)( ∈= ααAAC , where αµα ≥ℜ∈= )(| xxA Ap is a crisp object, whose −α level contour is

obtained for αµ =)(xA .

In (Chaudhuri, 1991), basic fuzzy geometric shapes, like point, line, circle, ellipse, and polygon, are

defined on continuous 2D support space. It is assumed that the fuzzy set has a bounded support, is

piecewise constant, and has a finite number n of distinct membership values.

A fuzzy geometric point is a fuzzy set with nonzero membership only at one point of the support

space.

A fuzzy geometric straight (curved) line is a fuzzy set for which any −α cut, ∈α (0, 1], is either

empty, or a straight (curved) line in a support space. A fuzzy line is a connected fuzzy set. A fuzzy

straight line is a convex fuzzy set.

A fuzzy geometric circle is a fuzzy set whose −α cuts, for ∈α (0, 1], are all concentric circles (figure

5(a)).

A fuzzy subset S is a ring if )(xSµ = )(~ rµ , where =r ||x −x0|| for some nx ℜ∈0 and ]1,0[:~ →ℜµ is a

membership function. A convex ring is called a fuzzy disk.

As opposite to a fuzzy geometric circle, the membership function of a fuzzy geometric disk is non-

increasing away from the interior of the object.

Figure 4. Fuzzy point, fuzzy curved line and fuzzy region

For example, in figure 5(b) is shown a fuzzy disk. Its core is a crisp disk defined by

( ) 122

21

21 | rxxxA ≤+ℜ∈= and its contour is the circle defined by ( ) 1

22

21

21 | rxxxA C =+ℜ∈= , where

1r is the length of the corresponding radix at the level 1=α . In general, for any ]1,0[∈α , the −α cut

(or equivalently, −α support) is defined by ( ) αα rxxxA ≤+ℜ∈= 22

21

2| , and the −α level contour is

defined by ( ) αα rxxxA C =+ℜ∈= 22

21

2 | .

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(a) (b)

Figure 5: (a) A fuzzy circle; (b) A fuzzy disk: centroid, core, support, α-level contours, radial distance

Two possible ways of generalizing the shape signature for a continuous fuzzy shape was proposed by

Chanussot et al. (2005):

• as s radial integral of the membership function:

( )∫=−

)(

1 )(),()(

tA

A

Afuzzy

c

dyxtCD ρρρµ

where )(tρρ = is a parameterization of the straight path between a boundary point and the centroid.

• as an average signature obtained from the −α cuts:

∫=−

1

0

2 )()( αα dtCDtCD fuzzy

where fuzzy star-shaped objects are considered, with all the boundaries of there −α cuts jointly

indexed by the same parameter t .

3. REPRESENTATION OF CRISP AND FUZZY SHAPES IN A DIGITAL SPACE

3.1. DIGITAL GEOMETRY

Discrete objects can arise from the digitization of scanned images, which involves sampling the

picture and quantizing the sampled values. A digital picture consists of a finite number of pixels (short

for “picture elements”) or voxels (short for “volume elements”), each of which is defined by a location

and a value at that location.

Digital geometry is the study of geometric or topologic properties of sets of pixels or voxels. It often

attempts to obtain quantitative information and topologic characterizations of pictures or to transform

pictures into “simpler” topologically equivalent pictures.

In the context of digital geometry, pixels and voxels refer to the elements of the medium on which

pictures reside, which is usually defined by a regular orthogonal grid. Thus, they are locations defined

by grid coordinates, and have values defined by a picture.

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In general, a digital grid is a set of points in nℜ . However, under a proper coordinate system, these

points represent the points in nZ where Z is the set of all integers. Moreover, most imaging systems

acquire images in regular orthogonal grids.

Objects in digital geometry often do not behave like Euclidean objects. For example, we can define

adjacency relations between pixels, whereas distinct points cannot be “adjacent” in Euclidean

geometry. Grid points are isolated points in the (real) plane, but, in the grid, adjacency relations

between grid points can be defined.

A digital space D is an ordered pair (G, α) where G is the underlying digital grid and α is a binary

relation on G that indicates the adjacency relationship between every two points in G. The notion of

adjacency is useful to define a path in a digital space and the boundary of a digital object. Only hard

adjacencies will be considered, because the interpretation of fuzziness of adjacencies in the context of

a path is not clear. In other words, α : Zn × Z

n →0, 1. Two points p, q ∈ Zn

are called adjacent if and

only if α(p, q) = 1. We can chose α to be standard 4- or 8-adjacency in 2D, 6-, 18-, or 26-adjacency in

3D, and their higher dimensional analogs.

Two adjacent points are often referred to as neighboring points to each other. For p = (x, y) ∈ Z2, we

define the neighborhoods

N4(p) = (x,y), (x+1,y), (x−1,y), (x,y+1), (x,y−1)

and

N8(p) = N4(p) ∪(x+1,y+1), (x+1,y−1), (x−1,y+1), (x−1,y−1) .

Two grid points p,q ∈ Z2 are called 4-adjacent or proper 4-neighbors (8-adjacent or proper 8-

neighbors) iff p ≠ q and p ∈ N4(q) (p ∈ N8(q)).

A set M (e.g., of grid points) is called connected if for all p,q ∈M there exists a sequence npp ,,0 … of

elements of M such that p0 = p, pn = q, and pi is adjacent to pi−1 (1 ≤ i ≤ n); such a sequence is called a

path and is said to join p and q in M. Maximal connected subsets of M are called (connected)

components of M.

(a) (b) (c) (d)

Figure 6. (a) Cell 1-adjacency and pixel 4-adjacency (left); (b) Neighborhoods (right)

(c) Cell 0-adjacency and pixel 8-adjacency (left); (d) Neighborhoods (right)

Figure 7. Cell- and voxel-adjacency for 3D digital objects

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3.2. FUZZY DIGITAL GEOMETRY

When an object in a scene is represented by a set of pixels in a picture, it may not always be obvious

which pixels belong to the object and which pixels belong to the background or another object. This

suggests that an object can be viewed as a fuzzy set which is specified by some membership function

defined on all picture points.

The concept of fuzzy digital geometry has been introduced by Rosenfeld (1984) and plays a key role

in many image processing applications. The application areas of fuzzy geometry are image

representation, enhancement and segmentation. The process of converting the input image into a fuzzy

set by indicating, for each pixel, the degree of membership to the object, is referred to as “fuzzy

segmentation”. The most straightforward way to perform fuzzy segmentation is to scale grey-levels of

an image to be between 0 and 1. Such grey levels reflect the area coverage of a pixel by the object, and

can be naturally used as membership values. However, in most cases, more advanced segmentation

methods are required, especially since it is rarely sufficient to use only the brightness of pixels to

calculate fuzzy membership values. For example, fully segmented image can be generated by

combining the optimum automatic thresholding procedure with edge detection to produce

continuously connected object border.

The object of interest is represented as a discrete spatial fuzzy subset of a grid. It should be noted,

however, that the discrete fuzzy objects obtained from the digitization of scanned images (say, using a

grey-level scale) are affected by multiple distortions, due to limited representation resolution.

Consequently, their properties are significantly different with respect to those of corresponding

continuous fuzzy objects. Figure 8 shows a discrete fuzzy disk (a) versus a discrete fuzzy hole (c). The

crisp counterpart of a fuzzy disk (b) can be obtained by thresholding a fuzzy (gray-level) image: pixels

with a membership degree below the threshold are lost.

a) (b) (c)

Figure 8: (a) discrete fuzzy disk; (b) crisp (binary) disk; (c) discrete fuzzy hole

For any picture P with pixel values in the range 0, . . . ,Gmax, if we divide the pixel values by Gmax,

we obtain a picture P′ in which the pixel values are in the range [0,1], so that they can be regarded as membership values of the pixels in a fuzzy subset µ of P.

3.3. Fuzzy distances and distance transforms

There are two main approaches in measuring distances when considering fuzzy objects: the first one

basically compares only the membership functions representing the concerned fuzzy object, while the

other one combines spatial distance between objects and membership functions, thus taking into

account both spatial information and information related to the imprecision attached to the image

object.

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Distances between two points in a fuzzy set are typically addressed in order to find the best path in the

geodesic sense in a spatial fuzzy set.

Distances from a point to a set are used when computing distance from a point to a complement of a

fuzzy set, i.e., performing distance transform.

The distances between sets are used in shape matching.

3.3.1. Fuzzy geodesic distance

A geodesic distance between points in a fuzzy set was introduced by Bloch (2000), being defined

conditionally to a reference set X. Thus, a geodesic distance dX(x, y) from x to y is the length of a

shortest path from x to y, completely included in X. Let µ be a fuzzy set on the space S . The

definition of the geodesic distance relies on the degree of connectivity in µ between two points x and

y of S, as defined by Rosenfeld (1984),

=

∈)(minmax),(

),(),(tyxc

yxLtyxLµµ

where L(x, y) denotes a path from x to y, consisting of a sequence of points in S according to the

discrete connectivity defined on S. Let ),(* yxL denote the shortest path between x and y on which µc

is reached; this path is not necessarily unique and can be interpreted as a geodesic path descending as

little as possible in terms of membership degrees. Let l( ),(* yxL ) denote its length (the number of

points along the path). Then the geodesic distance in µ between x and y is defined as

( )),(

),(),(

*

yxc

yxLlyxd

µµ =

If ),( yxcµ = 0, then ∞=),( yxdµ , which corresponds to the result obtained for the classical geodesic

distance in the case where x and y belong to different connected components. The definition

corresponds to the classical geodesic distance computed at the −α cut of µ at level α = ),( yxcµ . In

this −α cut, x and y belong to the same connected component. The definition satisfies the following

set of properties:

• the distance between any two points is non-negative;

• the distance between x and y is the same as the distance between y and x;

• the distance equals zero only between two spatially identical points;

• the distance is defined by the shortest path between x and y that “goes out” of µ “as little

as possible”, and tends to infinity if it is not possible to find a path between x and y

without going through a point t such that µ (t) = 0;

• the distance decreases when µ (x) and µ (y) increase;

• the distance decreases when ),( yxcµ increases;

• the distance is equal to the classical geodesic distance if µ is crisp.

The triangular inequality is not satisfied, but from the given definition it is possible to derive a true

distance, satisfying triangular inequality, while keeping all other properties:

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( ) ( )

+=′

∈ ),(

),(

),(

),(min),(

**

ytc

ytLl

txc

txLlyxd

Stµµ

µ

where S is the whole image space.

Obviously, the geodesic distance between the centroid and fuzzy boundaries of a fuzzy shape

is a good candidate for generalizing the shape signature of discrete fuzzy shapes.

Chanussot et al. (2005) proposed two alternative generalizations of shaoe signature fo a discrete fuzzy

shape:

(1) by using the distance between the boundary points and the centroid, which consists of

the following steps:

• Compute the centroid coordinates.

• Detect the inner boundary of the lowest a-cut using 8-connectivity.

• Compute the signature using length estimation based on local steps for the corresponding

discrete straight line.

The signature of a discrete fuzzy shape, calculated using the (pseudo-)distance between the boundary

points and the centroid, is given by:

( )

( )∑=

−

⋅+

+=

kN

j

kkAk

ccAfuzzydiscrete

jyjxj

yxkCD

1

1_

)(),()(

,)(

µδ

µ

(2) as an average signature obtained from the −α cuts, which consists of the following

steps:

• Compute the centroid coordinates.

• For each α (if the data are quantized using 8 bits per pixel, the total number of α –cuts

is α total = 255).

– Compute the corresponding α –cut.

– Detect the inner boundary of the α –cut.

– Compute the shape signature of the α –cut.

• Resample all the signatures to the number of pixels of the longest obtained signature;

this does not necessary correspond to the signature of the lowest a-cut.

• Average the resampled signatures.

∑=

−− =total

kCDkCD resampledtotal

fuzzydiscrete

α

α

αα

1

2_ )(1

)(

where )(kCD resampled−α is the k th sample of the resampled signature obtained for one −α cut.

3.3.2. Distance transform

For a binary object, distance transform (DT) is a process that assigns a value at each location within

the object that is simply the shortest distance between that location and the complement of the object.

Most DT methods approximate the global Euclidean distance by propagating local distances between

neighboring pixels.

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Let S be a crisp subset of nℜ .We shall use S to denote its complement and Interior (S) to denote its

interior, which is the largest open set contained in S. The distance transform (DT) of S may be

represented as an image (x, DS(x)) | nx ℜ∈ on nℜ where DS is the DT value at x that is defined as

follows.

SyyxxDS ∈−= inf)( ,

where, inf gives the infimum of a set of positive numbers and . is the Euclidean norm. In digital

images, we always deal with bounded objects so that S is always nonempty.

3.3.3. Fuzzy distance transform in continuous spaces

However, this notion of hard DT cannot be applied on fuzzy objects in a meaningful way. The notion

of DT for fuzzy objects, called fuzzy distance transform (FDT), was introduced by Saha et al. (2002).

It is very important in many imaging applications because we often deal with situations with data

inaccuracies, graded object compositions, or limited image resolution..

Similar to ordinary DT, FDT of a fuzzy subset S in nℜ is an image on nℜ , which is denoted by a set

of pairs (x, ΩS(x)) | nx ℜ∈ ; ΩS(x) is the FDT value at x and is defined in the following way.

A path π in nℜ from a point nx ℜ∈ to another point (not necessarily distinct) ny ℜ∈ is a continuous

function π : [0, 1]→ nℜ such that π(0)=x and π(1)= y. The length of a path π in S, denoted by ΠS(π), is

the value of the following integration

( )∫=Π

1

0

)()()( dt

dt

tdtSS

ππµπ

i.e., )(πSΠ is the integral of membership values (in S) along π.

When a path passes through a low density (low membership) region, its length increases slowly and

the portion of the path in the complement of the support of S contributes no length. This approach is

useful to measure regional object depth, object thickness distribution, etc.

Let ζS(x, y) denote a subset of positive real numbers defined as

),( yxSζ = ),(|)( yxPS ∈Π ππ ,

i.e., ),( yxSζ is the set of all possible path lengths in S between x and y. The fuzzy distance from

nx ℜ∈ to ny ℜ∈ in S, denoted as ),( yxSω , is the infimum of ),( yxSζ ; i.e.,

),(inf),( yxyx SS ζω =

For any fuzzy subset S of nℜ and for any nyx ℜ∈, , the fuzzy distance Sω satisfies the metric

properties:

1. ,,0),( yxifyxS ==ω

2. ),,(),( xyyx SS ωω =

3. nSSS zanyforyzzyyx ℜ∈+≤ ),,(),(),( ωωω

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Furthermore, for any nonzero positive number θ ≤1 and for any nyx ℜ∈, , such that either x or y is in

Interior( ))(SθΘ ,

4. ,,0),( yxifyxS ≠>ω

The FDT value )(xSΩ at a point nx ℜ∈ is equal to the fuzzy distance between x and the closest (with

respect to ωS) point in )(SΘ . In other words, the value of )(xSΩ is defined as follows

)(|),(inf)( Syyxx SS Θ∈=Ω ω

Actually, the fuzzy distance Sω is a geodesic distance, which means that the shortest paths (when they

exist) in a fuzzy subset S between two points nyx ℜ∈, are not necessarily a straight line segment even

when S is convex. For example, consider a fuzzy disk where there are two possible membership values

within its support )(SΘ : points with high membership value are shown as dark gray and those with

low membership value (within the outer annular region) are shown as light gray. We also consider two

points x and y as illustrated in the figure below. Assuming the high membership value sufficiently

close to one and the low one close to zero, the shortest path between x, y should be contained within

the light gray region and therefore is not a straight line segment.

(a) Continuous space (b) Digital space

Figure 9. The shortest paths (when they exist) between two points in a convex fuzzy subset are not

necessarily a straight line segment.

3.3.4. Fuzzy distance transform in digital spaces

A fuzzy digital object O is a fuzzy subset (p, µO(p)) | p ∈ Zn, where µO :

nZ →[0, 1] specifies the

membership value at each point in the object.

The support )(OΘ of a fuzzy digital object O is the set of all points in nZ each having a nonzero

object membership value, i.e., )(OΘ = p | p ∈ nZ and µO(p) ≠ 0. A path π in a set S of points from a

point p∈S to another (not necessarily distinct) point q∈S is a sequence of points

q pp pp m == ,,, 21 … such that Spi ∈ for all mi ≤≤1 and jp is adjacent to 1+jp for all 1 ≤ j <

m. The length of the path is m.

A set of points S will be called path connected if and only if, for every two points p, q ∈ S, there is a

path in S from p to q. P(p, q) will denote the set of all paths in nZ from a point p ∈ nZ to another

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point q ∈ nZ . For the purpose of defining the length of a path, we use the notion of a link and its

length. A link is a path consisting of two points. The length of a link qp, in O may be defined in

different ways, e.g.,

(1) qpqp OO −×)(),(max µµ ,

(2) ( ) qpqp OO −×+ )()(2

1µµ , etc.

It may be noted that in both examples, the length of a link has two components—one coming from the

membership values at p and at q and the other from the distance between the two points.

The length )(πOΠ of a path q pp pp m === ,,, 21 …π is the sum of the lengths of all links on the

path, i.e.,

( )∑−

=

++ −×+=Π1

1

11)()(2

1)(

m

i

iiiOiOO pppp µµπ

πp,q ∈P(p, q) is one shortest path from p ∈ nZ to q ∈ nZ in O if )()( , ππ OqpO Π≤Π , ∀ ∈P(p, q).

Unlike the case of the continuous case, one shortest path always exists between two points in a

bounded digital fuzzy object.

Although, the existence of one shortest path is guaranteed, there may be multiple shortest paths

between two points in a fuzzy digital object between two points. We consider fuzzy digital objects

with bounded supports. The fuzzy distance, from p ∈ nZ to q ∈ nZ in O, denoted as ),( qpOω , is the

length of any shortest path in O from p to q. Therefore,

),( qpOω = )(min),(

ππ

OqpP

Π∈

.

For any digital cubic space ( )α,nZD = , for any digital object O , the fuzzy distance Oω satisfies the

metric properties for any nZqp ∈, :

1. ,,0),( qpifqpO ==ω

2. ),,(),( pqqp OO ωω =

3. nOOO Zranyforqrrpqp ∈+≤ ),,(),(),( ωωω

Furthermore, for any nZqp ∈, , such that either p or q is in )(OΘ ,

4. ,,0),( qpifqpO ≠≠ω

Consequently, the fuzzy distance Oω is a metric for )(OΘ .

In conclusion, the FDT value at a point p ∈ nZ in a fuzzy object O over a digital space, denoted by

)(pOΩ , is equal to the fuzzy distance between p and the nearest point in )(OΘ . In other words, the

value of )(pOΩ is defined as follows

),(min)()(

qpp OOq

O ωΘ∈

=Ω

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4. PROCRUSTEAN SHAPE ANALYSIS

The coordinates of selected landmarks for a crisp shape can be arranged in a pn× configuration

matrix A , or equivalently on a np×1 configuration vector )(Aveca = .

A configuration matrix for a discrete p-dimensional fuzzy shape A can be represented by vertical

concatenation of its −α level contours into a pnk × block matrix. Each one of the k sub-matrices

defined at each level α collects pn× landmarks: ( ))()()( 21 ααααAp

AA xxxA …= , where

10| 1 =<<<<=∈ kii ααααα ⋯⋯ .

A similar pnk × -dimensional configuration matrix can be defined for a crisp shape with multiple (say

k ) contours.

A configuration matrix A is not a proper shape descriptor, because it is not pose invariant. For any

similarity transformation, i.e. +ℜ∈s , )(pSOR∈ (the special orthogonal group, i.e. R is ( pp× )

matrix, s.t. IRR =′ ) and pt ℜ∈ , the configuration given by tRAs p′+1 describes the same shape as

A, where p1 is the 1×p vector )111( ′… . To obtain a true shape representation, location, scale and

rotational effects need to be filtered out. This is carried out by shape alignment, i.e. by establishing a

coordinate reference, commonly known as pose. A very popular alignment procedure is Procrustes

shape analysis, which provides a measure, Procrustes distance, that quantifies the dissimilarity of two

configurations, and which is invariant with respect to translation, scaling, and rotation. Procrustes

shape analysis also provides a way to define the average shape, the Procrustes mean shape, which can

be viewed as a representative class template.

The Extended Orthogonal Procrustes (EOP) problem is a least squares method for fitting a given

configuration matrix A to another given matrix B . It is based on the functional model

BtsARE p −′+= 1 and consists of minimizing the Procrustes distance between A and B (i.e. 2

FE ),

under choice of unknown similarity transformation parameters R , t and s . This leads to solving the

problem EEstR

′,,

min , subject to the orthogonality restriction IRR =′ .

Generalized Orthogonal Procrustes (GOP) analysis is a technique that provides least-squares

correspondence of more than two model points. The solution of the problem can be thought as the

search of the unknown optimal matrix W (also named consensus matrix), defined as follows:

mitRAsAEM ipiiiii ,,1;1ˆ…=′+==+

( )( )pni QQNEvec ⊗=Σ 2,0~)( σ

where iE is the random error matrix in normal distribution, Σ is the covariance matrix, nQ is the

cofactor matrix of the n points, pQ is the cofactor matrix of the p coordinates of each point, ⊗

stands for the Kronecker product, and 2σ is the variance factor.

Let mACm

i i∑ ==

1

ˆ be the geometrical centroid of the transformed matrices. Therefore, Generalized

Orthogonal Procrustes problem can be solved minimizing

14

( ) ( )∑∑ ==

−′

−=−m

i ii

m

i i CACAtrmCAm1

2

1

ˆˆˆ

Crosilla and Beinat (2002) proved that the shape mean (centroid) C corresponds to the least squares

estimation M of the true value M : ∑ ===

m

i iAMC1

ˆˆ .

In the case of multiple-contour crisp shapes we can benefit from the Extended Orthogonal Procrustes

method in order to find mutual distances between shape pairs and from the Generalized Orthogonal

Procrustes technique in order to estimate the Procrustes mean shape of a collection of shapes. This is

illustrated in figures 10 and 11.

On the other hand, dealing with the case of fuzzy shapes needs more advanced Procrustean techniques,

which allow us to consider weighted distances between points placed on −α level contours with

different membership degrees. This leads to solve a Weighted Orthogonal Procrustes (WOP) problem.

0 100 200 300 400 500 6000

100

200

300

400

500

1

23

4

5

6

7

8

9

10

Figure 10: Ten double-contour star-shaped 2D objects with 3, 4 and 5 “lobes”

-0.1 -0.05 0 0.05 0.1-0.1

-0.05

0

0.05

0.1

Figure 11: Procrustes mean shape (shape centroid)

A weighting matrix W of the residual E (defined above) is now introduced and the minimization

problem becomes:

2min

FEW

15

subject to orthogonality restriction IRR =′ , 1)det( =R .

Typically, an iterative method is needed to derive a solution to WOP.

5. A GENERALIZATION OF K-MEANS ALGORITHM FOR CLUSTERING FUZZY

SHAPES

K-means is a commonly used data clustering for partitioning data points into disjoint groups such that

data points belonging to same cluster are similar, while data points belonging to different clusters are

dissimilar. The main idea is to define k centroids, one for each cluster, and to take each point

belonging to a given data set and associate it to the nearest centroid. When no point is pending, an

early groupage is done. Next, we need to re-calculate k new centroids of the clusters resulting from the

previous step. After we have these k new centroids, a new binding has to be done between the same

data set points and the nearest new centroid. We continue this loop until no more changes are done.

Clustering of objects or images of objects, according to the shapes of their boundaries is of a key

importance in computer vision and pattern recognition. This paper was intended to pay attention to this

reason by proposing a generalization of K-means algorithm in order to integrate Procrustean metrics

and full mean shape estimation, in a way making it able of clustering objects with either multiple or

fuzzy contours.

We present the algorithm in pseudo-code, as follows:

• Make initial guesses for the mean shapes 1v , 2v , …, kv , by choosing the first k shapes

from a random permutation.

• While any change still exists in any mean shape

o Calculate all pair-wise Procrustes distances between shapes using the Extended

Orthogonal Procrustes algorithm

o Use the estimated mean shapes to assign the shape samples into clusters

o For i from 1 to k

Replace iv with the mean shape of all of the samples for cluster i , using the

Generalized Orthogonal Procrustes algorithm

o end_for

• end_while

The resulting mean shapes for each one of the 3 clusters are shown below.

-0.1 -0.05 0 0.05 0.1-0.1

-0.05

0

0.05

0.1

Figure 12: First cluster: 2, 3, 8. Mean shape and cluster members

16

-0.1 -0.05 0 0.05 0.1-0.1

-0.05

0

0.05

0.1

Figure 13: Second cluster: 1, 6, 7, 9. Mean shape and cluster members

-0.1 -0.05 0 0.05 0.1-0.1

-0.05

0

0.05

0.1

Figure 14: Third cluster: 4, 5, 10. Mean shape and cluster members

Our method is thus graphically validated.

Similar results can be obtained by transforming each shape contour into a time series and applying a

time series clustering techniques (see section 6.3).

6. TIME SERIES KNOWLEDGE MINING

In a recent paper we have published in Fuzzy Economic Review, we proposed a time series knowledge

mining framework, designed to favor the synergy between subsequence time series clustering and

predictive tools such as Hidden Markov Models. Many tasks for temporal data mining rely heavily on

the choice of the representation scheme and the dissimilarity measure. We first presented a detailed

representation taxonomy for numeric and symbolic time series and comprehensive categorization of

distance measures. Subsequence time series clustering methods with a sliding window were addressed

next and a generalization of Fuzzy C-Means algorithm based on the dynamic time warping distance

was proposed as a very effective solution. This involves a shape-based distance tolerant to phase shifts

in time or accelerations/decelerations along the time axis. It also allows to determine the degree to

which set-defined objects, such as subsequence time series and their cluster centroids (similar in

nature) differ from each other. Finally we discussed the integration of clustering algorithms with

probabilistic predictive tools, such as discrete Markov chains or hidden Markov models. We applied

these techniques to clustering of non-overlapping sequences extracted from Standard and Poor’s 500

stock index historical data and we suggested different integrations with markovian models to improve

the predictive power.

17

6.1. TIME SERIES REPRESENTATIONS

Time series data mining involves techniques and methods adjusted in a way that they take into

consideration the temporal nature of data. Actually, it is concerned with data mining using numeric

and symbolic time series and sequences. Preprocessing, noise filtering, outlier detection, scaling,

representation methods, (dis)similarity measures, subsequence matching, indexing, clustering,

classification, segmentation, anomaly detection, motif discovery, rule discovery and prediction are

typical tasks for time series data mining.

The selection of a suitable time series representation is interconnected with the selection of a distance

measure. On the other hand, many tasks, such as clustering and classification of time series, rely

heavily on the similarity measure and the representation scheme selected.

Most of the time series are presented using numeric representations. However, transforming the

original time series in some form other than using the actual values is mainly a way of reducing their

intrinsic high dimensionality. Therefore, many methods have been proposed in order to convert

numeric time series to symbolic representations.

6.2. TIME SERIES DISTANCES

For short time series, shape-based distances are commonly used to compare their overall appearance.

The Euclidean distance is the most widely used shape-based distance for the comparison of numeric

time series. More generally, other pL norms, i.e., Manhattan for 1=p , Euclidean for 2=p ,

Maximum for ∞=p , can be used as well, putting different emphasis on large deviations.

The usage of the Euclidean distance is subject to the constraint that both time subsequences are of the

same length w. Let 11 11,, −+= wmm yyS … and 12 22

,, −+= wmm yyS … be two subsequences with length w

of time series nyyY ,,1 …= , where 2,1,11 =+−≤≤ jwnm j . Each subsequence will be represented

as a vector in a w-dimensional vector space. Thus we can define the dissimilarity between

subsequences 1S and 2S as the Euclidean distance between the two −w dimensional vectors measured

by the 2L norm:

( )∑=

−+−+ −=w

i

imim yySSL1

2

11212 21),(

Figure 15. The Euclidean distance

There are several pitfalls when using pL distances on time series: it does not allow for different

baselines in the time sequences; it is very sensitive to phase shifts in time; it does not allow for

acceleration and deceleration along the time axis (time warping). Another problem with pL distances

of time series is when scaling and translation of the amplitudes or the time axis are considered, or

S1

S2

18

when outliers and noisy regions are present. Care has to be taken in choosing the transformations to

obtain a time series distance measure that is meaningful to the application. A common example where

a translation or scaling is not desired is the comparison of stock prices. The absolute value of a stock is

usually not as interesting as the shape of up and down movements. A linear transformation or

normalization to a fixed or zero mean and unit variance can be a satisfactory solution for this case.

However, a number of non-metric distance measures have been defined to overcome some of these

problems. Distance measures that are robust to extremely noisy data will typically violate the

triangular inequality, i.e., are non-metric. This happens because such measures do not consider equally

all parts of the time series. Examples of such non-metric distance measures are presented in what

follows.

Small distortions of the time axis are commonly addressed with non-uniform time warping, more

precisely with Dynamic Time Warping (DTW). The DTW distance is an extensively used technique in

speech recognition and allows warping of the time axes (acceleration–deceleration of signals along the

time dimension) in order to align the shapes of the two times series better. The two series can also be

of different lengths. The optimal alignment is found by calculating the shortest warping path in the

matrix of distances between all pairs of time points under several constraints. The point-wise distance

is usually the Euclidean or Manhattan distance.

Let us consider two sequences (of possibly different lengths) ,, 1 nqqQ …= and ,,, 1 mccC …= .

To align two sequences using DTW, we construct an n-by-m matrix where the (ith, j

th) element of the

matrix contains the distance d(qi , cj) between the two points qi and cj (i.e. 2)(),( jiji cqcqd −= ). Each

matrix element ),( ji corresponds to the alignment between the points qi and cj . A warping path W is

a contiguous set of matrix elements that defines a mapping between Q and C. The kth element of W is

defined as kk jiw ),(= , so we have:

,,,,,, 21 Kk wwwwW ……= 1),max( −+<≤ nmKnm

The warping path is typically subject to several constraints.

• Boundary conditions: w1 = (1, 1) and wK = (m, n).

• Continuity: Given wk = (a, b), then ),(1 bawk′′=− , where 1≤′− aa and 1≤′− bb . This restricts the

allowable steps in the warping path to adjacent cells.

• Monotonicity: Given wk = (a, b), then ),(1 bawk′′=− , where 0≥′− aa and 0≥′− bb . This forces

the points in W to be monotonically spaced in time.

There are exponentially many warping paths that satisfy the above conditions, however we are

interested only in the path which minimizes the warping cost:

KwCQDTWK

i

k

= ∑

=1

min),(

The K in the denominator is used to compensate for the fact that warping paths may have different

lengths.

19

This path can be found efficiently using dynamic programming to evaluate the following recurrence,

which defines the cumulative distance ),( jiγ as the distance d(i, j) found in the current cell and the

minimum of the cumulative distances of the adjacent elements:

),( jiγ = d(qi, cj) + min γ (i −1, j −1), γ (i −1, j), γ (i, j −1)

The Euclidean distance between two sequences can be seen as a special case of DTW where the kth

element of W is constrained such that kjijiw kk === ,),( .

The warping path is also constrained in a global sense by limiting how far it may stray from the

diagonal. The subset of the matrix that the warping path is allowed to visit is called the warping

window. The two most common constraints in the literature are the Sakoe-Chiba band and the Itakura

parallelogram. We can view a global or local constraint as constraining the indices of the warping path

kk jiw ),(= , such that j − r ≤ i ≤ j + r, where r is a term defining the allowed range of warping, for a

given point in a sequence. In the case of the Sakoe-Chiba band, r is independent of i; for the Itakura

parallelogram, r is a function of i.

Figure 16. Optimal warping path with the Sakoe-Chiba band as global constraints

Figure 17. Aligning two time sequences using DTW

6.3. TIME SERIES CLUSTERING

The idea in subsequence time series clustering is as follows. Just a single long time series is given at

the start of the clustering process, from which we extract short series with a sliding window. The

resulting set of subsequences are then clustered, such that each time series is allowed to belong to each

cluster to a certain degree, because of the fuzzy nature of the fuzzy c-means algorithm we use. The

window width and the time delay between consecutive windows are two key choices. The window

width depends on the application; it could be some larger time unit (e.g., 24 days for time series

sampled as daily S&P 500 stock index, in our application). Overlapping or non-overlapping windows

can be used. If the delay is equal to the window width, the problem is essentially converted to non-

overlapping subsequence time series clustering. We will follow this approach, being motivated by the

Keogh’s criticism, where using overlapping windows has been shown to produce meaningless results.

When using a time delay of one, almost all time points contribute equally to each position within the

sliding window. Experiments with k-Means showed that clusters of sine waves are produced, the

prototypes of which add up to the constant function. Using larger time delays for placing the windows

20

does not really solve the problem as long as there is some overlap. Also, the less overlap, the more

problematic the choice of the offsets becomes.

In a recent paper we have generalized the fuzzy c-means algorithm to subsequence time series

clustering. Since clustering relies strongly on a good choice of the dissimilarity measure, this leads to

adopting an appropriate distance, depending on the very nature of the subsequence time series: type of

representation (numeric/symbolic), presence of time axis shifts (uniform/non-uniform time warping),

etc.

Three shape-based distances between numeric time series have been used as alternative choices in our

implementation of fuzzy c-means algorithm: the 2L (Euclidian) distance, the LB-Keogh’s distance

and the DTW distance.

DTW is a much more robust distance measure for time series than 2L , allowing similar shapes to

match even if they are out of phase in the time axis.

Figure 18. Fuzzy c-means clustering of time sequences using the DTW distance

7. FUZZY TIME SERIES

Typically, in conventional time series analysis, we assume that the generating mechanism is

probabilistic and that the observed values …… ,,,, 21 txxx are realizations of stochastic processes

…… ,,,, 21 tXXX . In contrast to the conventional time series, the observations of fuzzy time series

are fuzzy sets (the observations of conventional time series are real numbers).

Fuzzy randomness arises when random variables – e.g. as a result of changing boundary conditions –

cannot be observed with exactness. Fuzzy random variables may also be interpreted as fuzzified

21

random variables, as the random event can only be observed in an uncertain manner. If the fuzzy

random function is solely dependent on time, a fuzzy random process is obtained.

Fuzzy random processes can be used for modeling time series with fuzzy data. In other words,

imprecise data at equally spaced discrete time points are modeled as fuzzy variables. The set of this

discrete fuzzy data forms a fuzzy time series.

Fuzzy time series are regarded as realizations of fuzzy random processes. Fig. 20 gives illustration of

such a realization. For example, it can be obtained by a fuzzification of numerical data based on the

histograms of each weekday. The time series thus obtained is assumed to be stationary.

Song and Chissom (1993) give a thorough treatment of time series models. They define a fuzzy time

series as follows.

Let Y(t) (t=…, 0, 1, 2,…), a subset of ℜ , be the universe of discourse on which fuzzy sets if (t) (i =

1,2,…) are defined and let F(t) be a collection of if (t). Then, F(t) is called a fuzzy time series on Y(t)

(t = …, 0, 1, 2,…).

Figure 19. Fuzzy functions as realizations of a fuzzy random function

Figure 20. Time series with fuzzy data

22

In the case of fuzzy time series, the fuzzy relational equations can be employed as models.

Analogously to the conventional time series models, it is assumed that the observation at the time t

accumulates the information of the observation at the previous times.

Let F(t) and F(t−1) be fuzzy time series on Y(t) and Y(t−1) (t = …, 0, 1, 2,…), for any jf (t)∈F(t),

where j∈J, there exists an if (t−1)∈F(t−1) where i∈I , I and J are indices sets for F(t−1) and F(t),

respectively, such that there exists a first-order fuzzy relation )1,( −ttR and

)1,()1()( −−= ttRtftf ijij , where “ ° ” is the composition, then F(t) is said to be caused by F(t-1)

only. Denote this as if (t−1) → jf (t) or equivalently F(t−1) → F(t).

8. CONCLUSIONS

The main area of interest for this paper was to extend the methods of transforming a shape closed

contour to a time series in a fuzzy context, based on recent advances in fuzzy digital geometry. For

developing this unitary methodological framework, allowing to deal with both fuzzy shapes and fuzzy

time series as equivalent objects, the concept of fuzzy geodesic distance is foundational.

However, apart from this opportunity, robust advances in fuzzy time series area of research provide us

with powerful mathematical machinery for modeling and predicting fuzzy time series generated in

terms of imprecise data as realizations of fuzzy random processes.

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