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

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

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 FieldsCo-occurrence matrices Reaction-diffusion equationsMosaic models Fractal parametersSubband decompositions Higher order statistics We focus on the classification of image textures using MRF modeling of steerable pyramid image representations filtered by ICA.

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.

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

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, 1.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.

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 :

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.

Example: three scales and two orientations

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

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.

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

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,...

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 Representation vs. separation: T. Randen, Filter and filter bank design for image texture recognition, PhD.Thesis, Naive Bayes can be optimal even when an independence is violated: Domingos P., Pazzani, M., Beyond Independence: Conditions for the Optimality of the Simple Bayesian Classifier, Proc. ICML, Everything about the steerable pyramids: