Color Imaging 2004 1 Analysis of Spatio-chromatic Decorrelation for Colour Image Reconstruction Mark S. Drew and Steven Bergner {mark/sbergner}@cs.sfu.ca.

Color Imaging 2004 1

Analysis of Spatio-chromatic Decorrelation for Colour Image Reconstruction

Mark S. Drew and Steven Bergner

{mark/sbergner}@cs.sfu.ca

School of Computing Science, Simon Fraser University, Canada

Color Imaging 2004 2 2/27

- Use of PCA vs. ICA — what’s the difference?

- How do you do ICA?

- What does this have to do with images?

- The objective: best characterize image blocks using ICA on color image block data == spatio (blocks are 16x16, say)-chromatic (x3); assign bits in bit allocation according to the importance of each ICA coefficient data compression.

I. Overview

Color Imaging 2004 3 3/27

Best characterize image colour and spatial information.

Colour: we think of using PCA (Principal Component Anaysis):discover main colour axes. Is this best, given our objective?

Spatial: use spatial Fourier filters? Gabor wavelets? Etc.

Here, we’ll use ICA (Independent Component Anaysis) to derive best colour and spatial decomposition at once, for decorrelation, compression, and reconstruction.

Color Imaging 2004 4 4/27

II. ICA What is it?ICA is a form of “Blind Source Separation” To explain, consider audio signals (in an Imaging conference!).Consider 2 speakers, and 2 microphones:

s1s2

-sourcesx1

x2

-data

Color Imaging 2004 5 5/27

Can we disentangle s1, s2 from measured data x1, x2 ?

== The “cocktail party problem”.

An example:

Color Imaging 2004 6 6/27

ICA:

Order and sign not determined.

Color Imaging 2004 7 7/27

What about PCA?

Writing the signals in terms of reduced set of

sources s1, s2, s3, . . ., for higher-dimensional data,

we can do a better job in compression.

Color Imaging 2004 8 8/27

III. ICA How to do it?

Model: sAx (x was 2xN in the audio example.)

mixing matrix

xWs separating matrix

Color Imaging 2004 9 9/27

Driving idea for finding sources: s1, s2 are

statistically independent == information about one gives no knowledge re. the other.

Not just uncorrelated: covariance = 0

==PCA

Color Imaging 2004 1010/27

If independent as well, the pdf is separable:

joint pdf marginal pdf’s

which implies

for any functions , ! useful for solving.

Color Imaging 2004 1111/27

So, to do ICA, start with uncorrelated signals (using PCA) == simplifies.

Main tool:Non-Gaussian is independent.

Central Limit Theorem: the sum of two independents is more like a Gaussian than is either one.

So we have sums .

To get s, make a linear combination of x’s that isas non-Gaussian as possible.

sAx

Color Imaging 2004 1212/27

One way: (…many others)A Gaussian has zero kurtosis.

For zero mean y,

Rescale y to variance=1:

just use

We seek a signal that maximizeskurtosis.

Color Imaging 2004 1313/27

Algorithm “whiten” the data: zero mean, + linear transform to make uncorrelated, variance=1.

First, PCA: orthogonal U with

In the new coordinate system,

Why? Now

with orthogonal simpler to search for.

Color Imaging 2004 1414/27

Algorithm

-whiten x-we seek a column w of orthogonal W, with , that maximizes kurtosis:

1|||| w

Euler eqn.:

Code 1. Initialize w randomly, with 2. 3.

4. stop when

1|||| w

Color Imaging 2004 1515/27

Matlab

Color Imaging 2004 1616/27

IV. ICA for Images

Previous work:

Greyscale and colour imagery using PCA and ICA .For colour images, x could be 3-vector pixels.But get spatial as well if use n n tiles(nice illustration in Süsstrunk et al., CGIV’04 [using PCA on raw CFA data])

We show here that compression is better using ICA+colour+spatial info.

Color Imaging 2004 1717/27

16 x 16 greyscale tiles

ICA finds “sparse” features:

ICA (162x1 greyscale data)

localization in space

Color Imaging 2004 1818/27

PCA vs. ICA (3x1 data)

(no spatial information)

With colour:

Color Imaging 2004 1919/27

PCA (4x4 x3)

DCT (4x4 x3)

-less axis-aligned-ordering by variance-accounted-for is different: pure colour axes appear first

-pure colour axes appear later, after luminance frequencies-separates colour from luminance

PCA vs. DCT (4x4 x3 data)

-Colour: luminance, blue-yellow, red-green

Color Imaging 2004 2020/27

ICA (4x4 x3)

PCA (4x4 x3) again

PCA vs. ICA

-colour less separate from spatial information-combined localization in space and frequency-patterns not rectangular more like Gabor functions (Gaussian-modulated sine functions)

-localization in frequency

Color Imaging 2004 2121/27

ICA (4x4)

ICA (5x5)

ICA (8x8)

ICA (16x16)

Color Imaging 2004 2222/27

SNR

Colour vs. Greyscale:Compression performance

(Generic basis)

ColourGreyscale

- Higher reconstruction quality (SNR) for larger patches- Colour has better quality than grey, at equal compression

Better quality

Color Imaging 2004 2323/27

ICA vs. PCA

(Specific basis: image = )

- ICA much better than PCA: higher compression for same SNR- ICA increased quality with larger patches, for equal compression

ICAPCA

Better quality

Color Imaging 2004 2424/27

ICA vs. PCA

A. ICA does better separating axes such that they influence each other least

better entropy coding

B. Colour aids in compression

C. Large patch sizes and low rate encoding At equal compression, SNR (quality) better for ICA

Color Imaging 2004 2525/27

ICA vs. PCA: Image reconstruction(compression ratio:

1:12)ICAPSNR= 35.55

DCT:PSNR= 31.97

Color Imaging 2004 2626/27

Another image

ICAPSNR= 39.69

DCT:PSNR= 31.40

7:1

Orig ICA DCT--blocking

Color Imaging 2004 2727/27

The Future: Video Bases [submitted]

ICA (6x6x6)

PCA (6x6x6)