INDEPENDENT COMPONENT ANALYSIS OF TEXTURES

1. INDEPENDENT COMPONENT ANALYSIS OF TEXTURES based on the article R.Manduchi, J. Portilla, ICA of Textures, The Proc. of the 7thIEEE Int. Conf. On Comp. Vision, 1999 Ramūnas Girdziušas, 30.11.2000

2. Outline Step-1 Markov Random Fields as texture models Step-2 Combining MRF with steerable pyramids Step-3 Optimizing representation by ICA

3. Introduction What is texture? [Pickett,1970]: ”...large number of elements, each in some degree visible, and, on the whole densely and evenly (possibly randomly) arranged over the field of view such that there is a distinct spatial repetitiveness in the pattern.”[Cross and Jain,1983]: ”...stochastic, possibly periodic, two-dimensional image field.” Main tasks Restoration, Segmentation, Classification, Synthesis Tools Random Fields Co-occurrence matrices Reaction-diffusion equations Mosaic models Fractal parameters Subband decompositions Higher order statistics We focus on the classification of image textures using MRF modeling of steerable pyramid image representations filtered by ICA.

4. Step-1: Markov Random Field modeling • of texture • - Systematic approach based on sound principles. • Modeling of image through local interaction of • pixels. • Texture classification (MAP) • Problem: given an image consisting of more than • one texture, determine whether the particular pixel • comes from the l-th texture . • MAP classifier • According to the Bayes’ Theorem: • MAP: Find L that maximizes .

5. The 1st assumption: The 2nd assumption L is a locally dependent Markov Random Field (MRF) with pdf p(L):

6. The Iterated Conditional Modes (ICM) algorithm • (J. Besag, 1983) • - Fast alternative to MAP. • - Local deterministic relaxation. • Algorithm • Initialize labeling L according to ML decision. • For every epoch k, • For every image pixel i, • Choose label L(i) that maximizes : • 2. Repeat step 1 until no label changes occur. • Pros and cons of ICM • - avoids the large scale deficiencies; • - easily stucks in a local minima.

7. Step2: Combining MRF modeling with multiresolution approaches Why? -MRF is only suitable for micro-texture. -Biological relevance ? -Invariance properties? -High computational complexity. -Robustness to noise ? What kind of feature spaces to consider? -Invariance to slow-varying bias. -Energy separation while preserving locality. -Steerability (Shiftability). Steerability [Teo, 1998] A function f(x,y) is steerable under Lie group G if any transformation of f can be written as as a linear combination of a fixed, finite set of basis functions :

8. Steerable pyramids • - Introduced to remove some deficiencies of wavelets • - The code in Matlab and C is available on the web • The Steerable Pyramid is a linear, non-orthogonal, • overcomplete, self – inverting, multi-scale, • multi-orientation image decomposition. • Why is it useful? • - The power contained within a subband is invariant • under translation of the signal. • At the same scale and position the power in each • orientation subband is rotation invariant.

9. Example: three scales and two orientations

10. Step-3: Selection of the ”optimal” basis Motivation - Texture is characterized by joint feature pdf.- Typical filter based algorithms do not estimate joint description, marginal statistics are used. - Does a marginal set represent joint pdf well? Approach -Find the basis of a given filter space which generates the most informative marginals for a given texture in a sense that the product of marginal densities most closely approximates the joint pdf

11. The algorithm • -Training • For each texture l, • Filter texture with a fixed filter bank • Demix filter outputs by using ICA • Compute the channel histograms • -Classification • Apply the fixed filter bank to the test image • For texture model l, • Multiply the filter output vectors by the model • ICA matrix W and from channel histograms • obtain marginal likelihoods . • Compute the conditional likelihoods . • Use ICM to obtain pixel labels from .

12. Few words about texture synthesis Problem: generate an image that matches the appearance of a given texture sample Histogram matching Texture synthesis algorithm

13. Conclusions • Texture classification can be performed pixelwise • using MAP classifier: • - conditional independence together with Markov • property attacks MAP computational problem; • ICM is fast deterministic approximate MAP. • It is better to consider MRF under different • scales, for ex. by decomposing an image using SP. • - Classification results can be improved by making • features as independent as possible. • More textures can be synthesized using shifted • versions of filters and then performing ICA. • In general, ICA application in texture analysis • makes sense: • Textures are non-gaussian intensity processes • Wavelet representations are non-gaussian too. • In particular,...

14. Is the most informative likelihood the desired criterion of optimality? Ex. [Randen,1997]: PCA: ->0.01%. MOT: -> 67%. More to read Similar ideas without ICA: D. Heeger, J. Bergen, Pyramid based texture analysis/synthesis, Proc. SIGGRAPH, August 1995. Representation vs. separation: T. Randen, Filter and filter bank design for image texture recognition, PhD.Thesis, 1997. Naive Bayes can be optimal even when an independence is violated: Domingos P., Pazzani, M., Beyond Independence: Conditions for the Optimality of theSimple Bayesian Classifier, Proc. ICML, 1996. http://www.cs.washington.edu/homes/pedrod/ Everything about the steerable pyramids: http://www.cis.upenn.edu/~eero/steerpyr.html