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Analysis of Spatio-chromatic Decorrelation for Colour Image Reconstruction

Analysis of Spatio-chromatic Decorrelation for Colour Image Reconstruction. School of Computing Science, Simon Fraser University, Canada. Mark S. Drew and Steven Bergner. {mark/sbergner}@cs.sfu.ca. I. Overview. - Use of PCA vs. ICA — what’s the difference? - How do you do ICA?

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Analysis of Spatio-chromatic Decorrelation for Colour Image Reconstruction

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  1. Analysis of Spatio-chromatic Decorrelation for Colour Image Reconstruction School of Computing Science, Simon Fraser University, Canada Mark S. Drew and Steven Bergner {mark/sbergner}@cs.sfu.ca

  2. I. Overview - 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.

  3. 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 spatialdecomposition at once, for decorrelation, compression, and reconstruction.

  4. 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: s2 s1 x2 x1 -sources -data

  5. Can we disentangle s1, s2from measured data x1, x2 ? == The “cocktail party problem”. An example:

  6. Order and sign not determined. ICA:

  7. What about PCA?  Writing the signals in terms of reduced set of sourcess1, s2, s3, . . ., for higher-dimensional data, we can do a better job in compression.

  8. mixing matrix separating matrix III. ICA  How to do it? (xwas 2xN in the audio example.) Model:

  9. Driving idea for finding sources:s1, s2are statistically independent == information about one gives no knowledge re. the other. Not just uncorrelated: covariance = 0 ==PCA

  10. joint pdf  for any functions , ! useful for solving. If independentas well, the pdf is separable: marginal pdf’s which implies

  11. 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.

  12. 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.

  13. 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.

  14. Algorithm • whiten x • -we seek a column w of orthogonal W, with , • that maximizes kurtosis: Euler eqn.: 1. Initialize w randomly, with 2. 3. 4. stop when Code

  15. Matlab

  16. 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.

  17. ICA(162x1 greyscale data) 16 x 16 greyscale tiles ICA finds “sparse” features: localization in space

  18. With colour: PCA vs. ICA (3x1 data) (no spatial information)

  19. PCA vs. DCT(4x4 x3 data) • less axis-aligned • ordering by variance-accounted-for is different: pure colour axes appear first PCA (4x4 x3) • pure colour axes appear later, after luminance frequencies • separates colour from luminance DCT (4x4 x3) • Colour: luminance, • blue-yellow, red-green

  20. PCA vs. ICA • localization in frequency PCA (4x4 x3) again • colour less separate from spatial information • combined localization in space and frequency • patterns not rectangular more like Gabor functions (Gaussian-modulated sine functions) ICA (4x4 x3)

  21. ICA (4x4) ICA (5x5) ICA (8x8) ICA (16x16)

  22. Colourvs.Greyscale: Compression performance Better quality SNR Greyscale Colour (Generic basis) - Higher reconstruction quality (SNR) for larger patches - Colour has better quality than grey, at equal compression

  23. Better quality ICA vs. PCA (Specific basis: image = ) PCA ICA - ICA much better than PCA: higher compression for same SNR - ICA increased quality with larger patches, for equal compression

  24. ICA vs. PCA • ICA does better separating axes such that they influence each other least •  better entropy coding • Colour aids in compression • Large patch sizes and low rate encoding  At equal compression, SNR (quality) better for ICA

  25. ICA PSNR= 35.55 DCT: PSNR= 31.97 ICA vs. PCA: Image reconstruction (compression ratio: 1:12)

  26. Orig ICA DCT --blocking Another image  7:1 DCT: PSNR= 31.40 ICA PSNR= 39.69

  27. ICA (6x6x6) PCA (6x6x6) The Future: Video Bases [submitted]

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