On the Selection of an optimal wavelet basis for texture characterization

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

2. Contents • Introduction • Review of the wavelet transform • Shift variance of the wavelet transform • Regularity and number of vanishing moments • Texture classification-methodology • Filter design • Experiment and result

3. 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

4. 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

5. Cont..2 Review of the wavelet transform • In DWT, decomposition and reconstruction can be computed as:

6. 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,

7. Cont..2 Review of the wavelet transform • the simplest way to computer 2D DWT is to apply 1D DWT over rows and columns separately

8. 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

9. output input System with linear operator System behavior? input : unit impulse output : impulse response 3.1 the impact of shift variance • System-identification

10. input : unit impulse with shifts compactly supported output DWT system The number of difference output means the degree of shift variance Cont..3.1 the impact of shift variance

11. 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

12. 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

13. Cont..3.1 the impact of shift variance

14. Cont..3.1 the impact of shift variance

15. Cont..3.1 the impact of shift variance • result • at the kth decomposition level 2k difference impulse 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

16. 4 Regularity and number of Vanishing moments • the regularity of a function f(t) is closely related to its differentiability. • more higher-order differentiability implies higher regularity • determine smoothness of filters

17. 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

18. 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

19. 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

20. 5 Texture classification Methodology • estimation of texture quality • 4-level DWT with 13 energies • distance function used simplified Mahalanobis distance

21. Cont.. 5 Texture classification-methodology • classification • texture is assigned to class if • like nearest neighbor

22. 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

23. 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 energy among 13 energy

24. Cont.. 7. Experiments and Results • symmetric even biorthogonal filter have less shift variance

25. Cont.. 7. Experiments and Results • shift variance degree of decomposition filters is much more important than the regularity

26. Cont.. 7. Experiments and Results • regularity of the LP filter is more important than regularity of the HP filter

27. 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

28. Cont.. 7. Experiments and Results • the number of vanishing moments of the lowpass filter is another important criteria

29. 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

30. 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

31. Cont.. 7. Experiments and Results

32. 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

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