continuous-valued quaternionic hopfield neural …valle/pdffiles/talk17bracisa.pdf · fidelis...

Continuous-Valued Quaternionic Hopfield NeuralNetwork for Image Retrieval: A Color Space Study

Fidelis Zanetti de Castro andMarcos Eduardo Valle*

Department of Applied MathematicsInstitute of Mathematics, Statistics, and

Scientific ComputingUniversity of Campinas - Brazil

4 October 2017

Marcos Eduardo Valle (Unicamp) Quaternionic Hopfield Network 4 October 2017 1 / 25

Hopfield Model and Quaternionic Networks

The Hopfield neural network is a famous model that implementsassociative memory and solves optimization problems.

In the past years, the Hopfield network have been extended toquaternions.

Quaternionic neural networks treat four-dimensional data as a singleentity. Applications include control, signal and image processing,classification, and prediction.

We know only one paper on quaternionic Hopfield neural network(QHNN) for color images retrieval. Here, a color is converted into aquaternion using the RGB space.

There exists, however, other color spaces besides the RGB!


Quaternions

A quaternion is an hypercomplex number

q = q0 + q1i + q2j + q3k,

where i2 = j2 = k2 = ijk = −1 are the hyper-imaginary units.

The conjugate and the norm of a quaternion q are

q̄ = q0 − q1i− q2j− q3k,

and|q| =

√q2

0 + q21 + q2

2 + q23 .

We say that q is a unit quaternion if |q| = 1. We denote by S the setof all unit quaternions, that is,

S = {q : |q| = 1}.


Phase-Angle Representation

Alternatively, a quaternion can be written using the phase-anglerepresentation:

q = |q|eiφekψejθ,

where φ ∈ [−π, π), θ ∈[−π

2 ,π2

), and ψ ∈

[−π

4 ,π4

].

The phase-angle representation is derived from the relationshipbetween quaternions and rotations in R3.

Note thatq = eiφekψejθ,

is a unit quaternion!


Discrete-Time Bipolar Hopfield Neural Network

The discrete-time bipolar Hopfield network is defined by

xi(t + ∆t) =

{sgn

(vi(t)

), vi(t) 6= 0,

xi(t), vi(t) = 0,

where

vi(t) =n∑

j=1

wijxj(t),

is the activation potential of the i-th neuron, sgn is the signumfunction, and

xi(t) ∈ {−1,+1},

is the state of the i-th neuron.


Quaternionic Hopfield Neural Network

The discrete-time continuous-valued quaternionic Hopfield network(QHNN) is given by

xi(t + ∆t) =

{vi(t)/|vi(t)|, vi(t) 6= 0,xi(t), vi(t) = 0,

where

vi(t) =n∑

j=1

wijxj(t),

is the quaternionic activation potential of the i-th neuron and its statexi(t) is an unit quaternion.


Like traditional Hopfield network, a QHNN always settles at astationary state if the synaptic weights satisfy

wij = w̄ji and wii ≥ 0, ∀i , j .

A QHNN can implement an associative memory as follows:Given quaternionic vectors u1, . . . ,up whose components are unitquaternions, define the synaptic weight matrix by

W = UU†,

where U = [u1, . . . ,up] is a n × p matrix and U† denotes itsMoore-Penrose generalized inverse.

Given an initial state vector x(0), the network evolves to anequilibrium y = limt→∞ x(t) which corresponds to the vector recalledby the associative memory.


Suppose we want to store the following color images of size 64× 64:

We need a correspondence between colors and unit quaternions!


The RGB Color Space

In the RGB space CRGB, a color c = (r ,g,b) is characterized by theamount of red (r), green (g), and blue (b) intensities.


From CRGB to S and vice-versa

In the paper, we provide the mapping FRGB for the conversion fromCRGB to S and vice-versa. Basically, c = (r ,g,b)→ eiφekψejθ bysetting

r ∈ [0,1]→ φ ∈ [−π + τ, π − τ ] ≈ [−π, π)

g ∈ [0,1]→ ψ ∈ [−π4

+ τ,π

4− τ ] ≈ (−π

4,π

4)

b ∈ [0,1]→ θ ∈ [−π2

+ τ,π

2− τ ] ≈ [−π

2,π

2)

The red channel is mapped into a circle! Different colors aremapped to very close quaternions!

The distortion caused by the transformation of CRGB into S is

δRGB = supc1 6=c2

∆Eab(c1,c2)

|FRGB(c1)− FRGB(c2)|≈ 393,240.


Illustrative Example

We synthesized a QHNN/RGB designed for the storage of{uξRGB : ξ = 1, . . . ,12}, where uξRGB ∈ S4096

3 is obtained applyingFRGB in an entry-wise manner.

We presented images corrupted by• Gaussian noise with variance σ2 = 0.01,• Impulsive noise with probability p = 0.05.

The following shows the corrupted images and retrieved images.


Gaussian noise Impulsive noise

Inpu

tIm

age

∆Eab = 29.89 ∆Eab = 1.98

QH

NN

/RG

B

∆Eab = 19.52 ∆Eab = 6.21


The HSV Color Space

In the HSV space CHSV , a color c = (h, s, v) is characterized bythree subjective attributes: hue (H), saturation (S), and value (V).

Hue represents the dominant color as perceived by an observer. It isan angle from 0 to 360 degrees.

Saturation s ∈ [0,1], is the relative purity of white light mixed with ahue.

The value v ∈ [0,1], also called lightness, describes how dark thecolor is.


The HSV Color Space


From CHSV to S and vice-versa

In the paper, we provide the mapping FHSV for the conversion fromCHSV to S and vice-versa. Basically,

h ∈ [0,360)→ φ ∈ [−π, π)

s ∈ [0,1]→ ψ ∈ [−π4

+ τ,π

4− τ ] ≈ (−π

4,π

4)

v ∈ [0,1]→ θ ∈ [−π2

+ τ,π

2− τ ] ≈ [−π

2,π

2)

Since hue is an angle, there is a one-to-one correspondencebetween h and the phase-angle φ.

The distortion δHSV caused by the mapping FHSV is

δHSV = supc1 6=c2

∆Eab(c1,c2)

|FHSV (c1)− FHSV (c2)|≈ 6,368.

Recall that δRGB ≈ 393,240.Marcos Eduardo Valle (Unicamp) Quaternionic Hopfield Network 4 October 2017 15 / 25

Illustrative Example

For example, we synthesized a QHNN/HSV designed for the storageof 12 color images.

Upon presentation of the images corrupted by• Gaussian noise with variance σ2 = 0.01,• Impulsive noise with probability p = 0.05,we obtained the following:



Inpu

tIm

age

∆Eab = 29.89 ∆Eab = 1.98

QH

NN

/HS

V

∆Eab = 9.74 ∆Eab = 11.29


The CIE-LAB and CIE-HCL Color Space

In the CIE-LAB space, the Euclidean distance between two pointsresembles the color difference perceived by the human brain.

A color c = (L,a,b) ∈ CLAB has the following interpretation:• The coordinate L ∈ [0,100] represents the lightness,• The coordinates a and b correspond to the green-red positions,• The coordinates a and b correspond to the blue-yellow positions.

The CIE-HCL CHCL space is obtained by writing the CIE-LAB usingcylindrical coordinates.

A color is represented by c = (}, c,L), where L is the lightness, c isthe chroma, and } is the hue.


From CRGB to S and vice-versa

In the paper, we provide the mapping FHCL for the conversion fromCHCL to S and vice-versa. Basically,

} ∈ [−π, π)→ φ ∈ [−π, π)

c ∈ [−300,300]→ ψ ∈ [−π4

+ τ,π

4− τ ] ≈ (−π

4,π

4)

L ∈ [0,100]→ θ ∈ [−π2

+ τ,π

2− τ ] ≈ [−π

2,π

2)

The distortion caused by the mapping FHCL is

δHCL = supc1 6=c2

∆Eab(c1,c2)

|FHCL(c1)− FHCL(c2)|≈ 4,103.

Recall that

δRGB ≈ 393,240 and δHSV ≈ 6,368.



Inpu

tIm

age

∆Eab = 29.89 ∆Eab = 1.98

QH

NN

/HC

L

∆Eab = 27.65 ∆Eab = 6.60


Computational Experiments

We synthesized QHNN/RGB, QHNN/HSV, andQHNN/HCL,designed for the storage of the 12 color images.

We probed the memories with images corrupted by either• Gaussian noise with variance σ2 varying from 10−3 to 0.25.• Impulsive noise with probability r ∈ [10−3,1].

The following images show the average ∆Eab error by the noiseintensity.

The average ∆Eab was obtained by corrupting each original colorimage 10 times.


Gaussian Noise

10

15

20

25

30

35

40

45

10-3

10-2

10-1

Average error

Gaussian noise

InputCV-QHNN/RGBCV-QHNN/HSVCV-QHNN/HCL


Impulsive Noise

0

5

10

15

20

25

10-3

10-2

10-1

100

Average error

Impulsive noise

InputCV-QHNN/RGBCV-QHNN/HSVCV-QHNN/HCL


Concluding Remarks

In this paper, we investigated the performance of the quaternionicHopfield neural network (CV-QHNN) for the storage and retrieval ofcolor images using the color spaces: RGB, HSV, and CIE-HCL.

The color conversion plays an important role in the noise toleranceof a quaternionic network and requires further investigation.

Preliminary computational experiments showed that theCV-QHNN/HSV can be used for the removal of Gaussian noise.

In the future, we plan to investigate applications of the quaternionicHopfield neural networks for color imagery processing and analysis.

Thank you!Marcos Eduardo Valle (Unicamp) Quaternionic Hopfield Network 4 October 2017 24 / 25

Have an idea?

Submit a paper to the special session

Complex-Valued andQuaternionic Neural Networks!

organized by Marcos Eduardo Valle, Igor Aizenberg,Akira Hirose, and Danilo Mandic.


continuous-valued quaternionic hopfield neural …valle/pdffiles/talk17bracisa.pdf · fidelis...

Documents