On the Selection of an optimal wavelet basis for texture characterization Vision lab 구경모.

On the Selection of an optimal wavelet basis for texture characterization

Vision lab 구경모

Contents

1. Introduction2. Review of the wavelet transform3. Shift variance of the wavelet transform4. Regularity and number of vanishing

moments5. Texture classification-methodology6. Filter design7. Experiment and result

1 Introduction

the choice of filter bank in texture processing remain unresolved criteria in predicting the texture classification

performance has not been established

the scope of this paper is to investigate whether the properties of decomposition filter play an important role in texture description Properties of filter bank

shift-variance degree, regularity and number of vanishing moments, linear phase

2 Review of the wavelet transform decomposition of a signal onto the

family of functions

the mother wavelet is constructed from scaling function as follows

)2(2 2/, ntmmnm

)2()(2)(

)2()(2)(

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Cont..2 Review of the wavelet transform

In DWT, decomposition and reconstruction can be computed as:

filter highpass:

filter lowpass:

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)()2()(

)()2()(

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Cont..2 Review of the wavelet transform

linear phase (symmetry) and orthogonality are incompatible

to overcome, biorthogonal bases that use different filter for decomposition and reconstruction are introduced,

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Cont..2 Review of the wavelet transform

the simplest way to computer 2D DWT is to apply 1D DWT over rows and columns separately

3 Shift variance of the WT

shift invariance is satisfied

In DWT shift invariance not achievable, because

of the downsampling with the factor N periodically shift invariant with factor N

)()(),()( mnymnxnynx

)()(),()( mnymNnxnynx

3.1 the impact of shift variance System-identification

System withlinear operator

input

System behavior?input : unit impulse

1

output : impulse response

output

Cont..3.1 the impact of shift variance

compactly supported output

DWT system

input : unit impulse with shifts

The number of difference output means the degree of shift variance

Cont..3.1 the impact of shift variance

to examine the impact response (IR) of the wavelet filter bank for various shifts ni

at k’th decomposition, is formed as the convolution with , followed by the downsampling with factor 2 , specially

)()( innnx

)(2nf k

)(12nf k )(0 nh

)()(02nfnf

Cont..3.1 the impact of shift variance

k iterations of LP branch can be expressed as FIR filter

for various shifts , can be expressed as samples of the compactly supported piecewise constant function

)2/(**)4/(*)2/(*)()( kk nhnhnhnhnh

)(2nf k in

)(nf k

kik

i

kkkkk

nnfnnf

nx

nnhxf

k

2),(

2

1

2 ),(2)(

2

Cont..3.1 the impact of shift variance

)(2 nh

Cont..3.1 the impact of shift variance

Cont..3.1 the impact of shift variance

result at the kth decomposition level 2k difference im

pulse responses exist for symmetric filters, 2k-1 (for even length filter)

and 2k-1+1 for (odd length filter) difference impulse responses exist

this illustrates an enormous variety in the impulse response shift variance depending on the choice of decomposition filters

4 Regularity and number of Vanishing moments the regularity of a function f(t) is closely relate

d to its differentiability. more higher-order differentiability implies highe

r regularity determine smoothness of filters

4.1 Some definition of regularity regularity is defined as a maximum

value of such that

Implies that is m-times continuously differentiable, where

determines smoothness of scaling function and associated wavelet

R

,1

1)( 1rF

r

mr m

Cont.. 4.1 Some definition of regularity

regularity using Lipschitz(Holder) exponent A function is called Lipschitz of order

, if for any and some small

higher-orders of Lipschitz exponent implies higer-order regularity

ctftf )()(

)(tf

t10,

)1(

4.2 Vanishing moment

The divisibility of filter by means that the associated will have vanishing moments

If wavelet has vanishing moments, then the wavelet coefficients of a function have high compression potential

1,...,1,0 ,)( Lldttt l

)(zH Lz )1( 1L

L

5 Texture classification Methodology estimation of texture quality

4-level DWT with 13 energies

distance function used simplified Mahalanobis distance

M

x

N

yii yxI

NMe

1 1

2 ),(1

ji

j

jjij cmxixD ,

1

2, /)(),(

ijc

ijm

xjx

ji

ji

j

classin element th ' of ce varian

class ofvector -mean ofelement th '

vector feature ofelement th '

,

,

Cont.. 5 Texture classification-methodology

classification texture is assigned to class if

like nearest neighbor

jjxDixD |)},(min{),(

i

6 Filter design

some constraint for filter design perfect reconstruction finite impulse response orthogonality linear phase some regularity

orthogonality and linear phase are incompatible so biothogonal filters are selected

7. Experiments and Results

environment filter families

Haar, Daubechies

Daub1, Daub2 eight different biothogonal filter pair

spline filter(biort1 1.3, 1.5, 2.2, 2.4, 3.1, 3.3, 3.5, biort2 4.4)

feature vector k length, correspond to largest amount of signal e

nergy among 13 energy

Cont.. 7. Experiments and Results

symmetric even biorthogonal filter have less shift variance

Cont.. 7. Experiments and Results

shift variance degree of decomposition filters is much more important than the regularity

Cont.. 7. Experiments and Results

regularity of the LP filter is more important than regularity of the HP filter

Cont.. 7. Experiments and Results

in case of biorthogonal filters, a better filter should be placed in LP channel, whereas it’s biorthogonal pair should be modulated and placed in the HP channel

Cont.. 7. Experiments and Results

the number of vanishing moments of the lowpass filter is another important criteria

Cont.. 7. Experiments and Results

Effect of Linear phase linear phase filter has lower shift variance

at kth decomposition level, 2k-1 or 2k-1 +1 distinct impulse response (vs. 2k with nolinear phase filters)

Nonlinear phase can have a major effect on the shape of output signals

cause the decrease discrimination ability

Cont.. 7. Experiments and Results

Experiments with Noise Data number of vanishing moment for the

lowpass filter become more important Shift variance of the impulse response is

still important

Cont.. 7. Experiments and Results

8 CONCLUDE even length biorthogonal filters are more suitable for

texture analysis degree of the impulse response shift variance is more

important than the regularity reasonable number of vanishing moments for the

lowpass filter is desirable regularity of the LP filter is more important than

regularity of the HP filter in case of biorthogonal filters, a better filter should be

placed in LP channel, whereas it’s biorthogonal pair should be modulated and placed in the HP channel

shift variance of the impulse response is still important criterion

orthogonal filters should be used, as well as filters which ensure the aliasing cancellation

수식 모음

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