Independent Component Analysis on Images

Instructor: Dr. Longin Jan Latecki

Presented by: Bo Han

Motivation

• Decomposing a mixed signal into independent sources Ex.

Given: Mixed Signal Our Objective is to gain: Source1 News Source2 Song

• ICA (Independent Component Analysis) is a quite powerful technique to separate independent sources

What is ICA (From Math View)

• Given h measured mixture signals x1(k), x2(k), …, xh(k)

k is the discrete time index or pixels in images

• Assume a linear combination matrix form of q source signals:

X(k) = As(k) = Σsi(k)ai

A: mixing matrix

q source signals s1(k), s2(k), …, sq(k)

Assumptions

• Easy from A,S to compute X=AS

Difficult to compute A, S from X• Assumptions 1. Statistical independence for source signals

p[s1(k), s2(k), …, sq(k)] = П p[si(k)]

2. Each source signal has nongauss distribution

Important Properties of Independent Variables

• E[h1(y1) h2(y2)] = E[h1(y1)]E[h2(y2)]

h1, h2 are two functions

Prove:

)]y(h[E)]y(h[E

dy)y(p)y(hdy)y(p)y(h

dydy)y(p)y(p)y(h)y(h

dydy)y,y(p)y(h)y(h)]y(h)y(h[E

2211

22221111

21212211

212122112211

Uncorrelated: Partly Independent

• Uncorrelated:

E[ y1y2] = E[y1]E[y2]

Let h(y)=y, Independent Uncorrelated

y1

y24 points (0, 1) (0, -1) (-1, 0) (1, 0) with equal possibility ¼

E[ y1y2] = E[y1]E[y2]

But E[ y12y2

2]=0 E[y1

2]E[y22]=1/4

How ICA Compute

• Basic idea: X(k)=AS(k) Solution S(k)=A-1X(k)=WX(k)• 1. Centering: resulting a variable with 0-

mean value

• 2. Whiten the data Remove any correlations in the data and m

ake variance equal unity Advantage: reduce the dimensionality

How ICA Compute (cont)

• 3. The appropriate rotation is sought by maximizing the nongaussianity

How to measure nongaussianity Kurtosis: Kurt(y)=E[y4]-3(E[y2])2 (approac

h 0 for a Gaussian random var)

Negentropy: Neg(y)=H(ygauss)-H(y) (H is entropy)

Approximations of negentropy: J(y)=E[y3]2/12 + Kurt(y)2/48

Different ICA Algorithms

• With different measures on nongaussianity

FAST ICA

based on some nonquadratic functions

g(u)=tanh(a1u)

g(u)=uexp(-u2/2)

Fast ICA Steps

Iteration procedure for maximizing nongaussianity

Step1: choose an initial weight vector wStep2: Let w+=E[xg(wTx)]-E[g’(wTx)]w (g:

a non-quadratic function)Step3: Let w=w+/||w+||Step4: if not converged, go back to Step2

How ICA compute (example)

Running an example in matlab

Compare ICA and PCA

PCA: Finds directions of maximal variance in gaussian dataICA: Finds directions of maximal independence in nongaussian data

Ambiguities with ICA

• The ICA expansionX(k) = AS(k)

• Amplitudes of separated signals cannot be determined.

• There is a sign ambiguity associated with separated signals.

• The order of separated signals cannot be determined.

Apply ICA On Images

• Objective: Gain independent information from images

• 1. To get X, change each image into a vector• 2. Generate a series of images which share

some common information but changing other fixed parts

• 3. Apply ICA• 4. Convert the ICs to images• 5. Sensitive to the position change

Apply ICA On Images

Running MATLAB CODE

Apply ICA on Video

• Video is a good application of ICA

1) Little information change between neighborhood frames

Easy to detect independent parts in images

2) Time series data

Apply ICA on Video

Source images

Apply ICA on Video

ICs

Apply ICA on Video

Source images

Apply ICA on Video

ICs

Conclusions

• ICA can be used to detect independent changing/moving parts in

images and videos

• But ICA is very sensitive to the position change

• ICA simplify the work of motion detection

References

• Aapo Hyvärinen and Erkki Oja, Independent Component Analysis: Algorithms and Applications. Neural Netw

orks, 13(4-5):411-430, 2000 • Alphan Altinok, Independent Component Analysis. • Aapo Hyvärinen – Survey on ICA

• D. Pokrajac and L. J. Latecki: Spatiotemporal Blocks-Based Moving Objects Identification and Tracking, IEEE Visual Surveillance and Performance Evaluation of Tracking and Surveillance (VS-PETS), October 2003.