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Color Imaging Analysis of Spatio-chromatic Decorrelation for Colour Image Reconstruction Mark S. Drew and Steven Bergner School of Computing Science, Simon Fraser University, Canada

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Color Imaging /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

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Color Imaging /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.

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Color Imaging /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: s1s1 s2s2 -sources x1x1 x2x2 -data

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Color Imaging /27 Can we disentangle s 1, s 2 from measured data x 1, x 2 ? == The “cocktail party problem”. An example:

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Color Imaging /27 ICA: Order and sign not determined.

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Color Imaging /27 What about PCA? Writing the signals in terms of reduced set of sources s 1, s 2, s 3,..., for higher-dimensional data, we can do a better job in compression.

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Color Imaging /27 III. ICA How to do it? Model: ( x was 2xN in the audio example.) mixing matrix separating matrix

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Color Imaging /27 Driving idea for finding sources: s 1, s 2 are statistically independent == information about one gives no knowledge re. the other. Not just uncorrelated: covariance = 0 ==PCA

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Color Imaging /27 If independent as well, the pdf is separable: joint pdf marginal pdf’s which implies for any functions, ! useful for solving.

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Color Imaging /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 is as non-Gaussian as possible.

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Color Imaging /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 maximizes kurtosis.

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Color Imaging /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.

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Color Imaging /27 Algorithm -whiten x -we seek a column w of orthogonal W, with, that maximizes kurtosis: Euler eqn.: Code 1. Initialize w randomly, with stop when

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Color Imaging /27 Matlab

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Color Imaging /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.

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Color Imaging /27 16 x 16 greyscale tiles ICA finds “sparse” features: ICA (16 2 x1 greyscale data) localization in space

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Color Imaging /27 PCA vs. ICA (3x1 data) (no spatial information) With colour:

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Color Imaging /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

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Color Imaging /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

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Color Imaging /27 ICA (4x4) ICA (5x5) ICA (8x8) ICA (16x16)

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Color Imaging /27 SNRSNR Colour vs. Greyscale: Compression performance (Generic basis) Colour Greyscale - Higher reconstruction quality (SNR) for larger patches - Colour has better quality than grey, at equal compression Better quality

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Color Imaging /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 ICA PCA Better quality

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Color Imaging /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

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Color Imaging /27 ICA vs. PCA: Image reconstruction (compression ratio: 1:12) ICA PSNR= DCT: PSNR= 31.97

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Color Imaging /27 Another image ICA PSNR= DCT: PSNR= 7:1 Orig ICA DCT --blocking

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Color Imaging /27 The Future: Video Bases [submitted] ICA (6x6x6) PCA (6x6x6)

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