Independent Component Analysis on Images

1
Independent Component Analysis on Images

• Instructor Dr. Longin Jan Latecki
• Presented by Bo Han

2
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

3
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) Ssi(k)ai
• A mixing matrix
• q source signals s1(k), s2(k), , sq(k)

4
Assumptions

• Easy from A,S to compute XAS
• Difficult to compute A, S from X
• Assumptions
• 1. Statistical independence for source
  signals
• ps1(k), s2(k), , sq(k) ? psi(k)
• 2. Each source signal has nongauss distribution

5
Important Properties of Independent Variables

• Eh1(y1) h2(y2) Eh1(y1)Eh2(y2)
• h1, h2 are two functions
• Prove

6
Uncorrelated Partly Independent

• Uncorrelated
• E y1y2 Ey1Ey2
• Let h(y)y, Independent ? Uncorrelated

4 points (0, 1) (0, -1) (-1, 0) (1, 0) with equal
possibility ¼ E y1y2 Ey1Ey2 But E
y12y220 Ey12Ey221/4
y2
y1
7
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 make
  variance equal unity
• Advantage reduce the dimensionality

8
How ICA Compute (cont)

• 3. The appropriate rotation is sought by
  maximizing the nongaussianity
• How to measure nongaussianity
• Kurtosis Kurt(y)Ey4-3(Ey2)2
  (approach 0 for a Gaussian random var)
• Negentropy Neg(y)H(ygauss)-H(y)
• (H is entropy)
• Approximations of negentropy J(y)Ey32/12
  Kurt(y)2/48

9
Different ICA Algorithms

• With different measures on nongaussianity
• FAST ICA
• based on some nonquadratic functions
• g(u)tanh(a1u)
• g(u)uexp(-u2/2)

10
Fast ICA Steps

• Iteration procedure for maximizing nongaussianity
• Step1 choose an initial weight vector w
• Step2 Let wExg(wTx)-Eg(wTx)w (g a
  non-quadratic function)
• Step3 Let ww/w
• Step4 if not converged, go back to
• Step2

11
How ICA compute (example)
Running an example in matlab
12
Compare ICA and PCA
PCA Finds directions of maximal variance in
gaussian data ICA Finds directions of maximal
independence in nongaussian data
13
Ambiguities with ICA

• The ICA expansion
• X(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.

14
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

15
Apply ICA On Images

• Running MATLAB CODE

16
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

17
Apply ICA on Video
Source images
18
Apply ICA on Video
ICs
19
Apply ICA on Video
Source images
20
Apply ICA on Video
ICs
21
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

22
References

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