INDEPENDENT COMPONENT

ANALYSIS OF

TEXTURES

based on the articleR.Manduchi, J. Portilla, ICA of Textures, The Proc. of the 7th IEEE Int. Conf. On

Comp. Vision, 1999

Ramūnas Girdziušas, 30.11.2000

Outline

Step-1 Markov Random Fields as texture models

Step-2Combining MRF with steerable pyramids

Step-3 Optimizing representation by ICA

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FilteringIm age C lass m ap

M APY i

Z ikL i

Filtering

C lass m ap

IC A M AP

Im age Z ik Z'ik

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 tasksRestoration, Segmentation, Classification, Synthesis

ToolsRandom Fields Co-occurrence matricesReaction-diffusion equations Mosaic models Fractal parameters Subband decompositionsHigher order statistics

We focus onthe 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 classifierAccording to the Bayes’ Theorem:

MAP: Find L that maximizes .

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Y

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p

ppp

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The 1st assumption:

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

i

iii LYpLYp )|()|(

L2 L3

L9

L8

L1

L7

L4

L5

L6

Y2 Y3

Y9

Y8

Y1

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Y2 Y3

Y9

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Y6

L2 L3

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L6

Yi - im age p ixe l Li - labe l o f the p ixe l sta tis tica l dependence

L2 L3

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L8

L1

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L4

L5

L6

Y2 Y3

Y9

Y8

Y1

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Y2 Y3

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F irst o rder M arkovian ity

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The Iterated Conditional Modes (ICM) algorithm (J. Besag, 1983)

- Fast alternative to MAP. - Local deterministic relaxation.

AlgorithmInitialize 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.

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M AP ICM

Step2: Combining MRF modeling with multiresolution approachesWhy?-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 Gif any transformation of f can be written asas a linear combination of a fixed, finite set of basisfunctions :

Gg )(

)},({ yxi

y)Φ(x,aT

n

iii yxayxfg

1),()(),()(

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

H0(-w )

L0(-w ) B0(-w )

B1(-w )

H1(-w ) D2 B0(-w )

B1(-w )

H1(-w ) 2D

wy

wx

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B 00

B 01

B 00

B 01

L1B 10 B 10

B 11

B 11

256x256

64x64

128x128

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256x256

256x256

128x128

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

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i

Filtering

C lass m ap

IC A M AP

Im age Z ik Z'ik

The algorithm

-TrainingFor each texture l,Filter texture with a fixed filter bankDemix filter outputs by using ICACompute the channel histograms

-ClassificationApply the fixed filter bank to the test imageFor 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 .

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)|( lzp

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Few words about texture synthesis

Problem: generate an image that matches the appearance of a given texture sample

Histogram matching

Texture synthesis algorithm

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255

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Texture Im agecdf

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N oise Im age

Texture

N oise Im agem ore like texture

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E stim ated by using IC A

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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 readSimilar 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 anindependence is violated:Domingos P., Pazzani, M., Beyond Independence: Conditions for the Optimality of the Simple 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

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