On the Selection of an
optimal wavelet basis for
texture characterization
Vision lab 구경모
Contents
1. Introduction
2. Review of the wavelet transform
3. Shift variance of the wavelet transform
4. Regularity and number of vanishing
moments
5. Texture classification-methodology
6. Filter design
7. 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
 m,n  2m / 2 (2m t  n)
 the mother wavelet is constructed from
scaling function as follows

 (t )  2  h0 (k ) (2t  k )
k  

 (t )  2  h1 (k ) (2t  k )
k  
V2  V1 W1
L2 space
V1  V0  W0
W1
W0
V0
Cont..2 Review of the wavelet transform
 In DWT, decomposition and reconstruction
can be computed as:
f 2 j (t ) 
d 2 j (t ) 

 h ( k  2k ) f
k  
0
2 j 1
(k )

 h ( k  2k ) f
k  
f 2 j1 (k ) 
1
2 j 1
(k )

 [h (k  2k ) f
k  
0
2j
(k )  h1 (k  2k )d 2 j (t )]
 h0 : lowpass filter

 h1 : highpass filter



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,
f 2 j (t ) 
d 2 j (t ) 

 h ( k  2k ) f
k  
0
2 j 1
(k )

 g ( k  2k ) f
k  
f 2 j 1 ( k ) 
1

 [g
k  
0
2 j 1
(k )
(k  2k ) f 2 j (k )  h1 (k  2k )d 2 j (t )]
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
x(n)  y (n), x(n  m)  y (n  m)
 In DWT
 shift invariance not achievable, because of
the downsampling with the factor N
 periodically shift invariant with factor N
x(n)  y (n), x(n  mN )  y (n  m)
3.1 the impact of shift variance
 System-identification
output
input
System with
linear operator
System behavior?
input : unit impulse
1
output : impulse response
Cont..3.1 the impact of shift variance
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
 to examine the impact response (IR) of the
wavelet filter bank for various shifts ni
x(n)   (n  ni )
 at k’th decomposition, f 2 (n) is formed as
the convolution f 2 (n) with h0 (n) , followed
k
k 1
by the downsampling with factor 2 ,
specially f 2 (n)  f (n)
0
Cont..3.1 the impact of shift variance
 k iterations of LP branch can be expressed
as FIR filter
h k (n)  h(n) * h(n / 2) * h(n / 4) ** h(n / 2k )
 f 2 (n) for various shifts ni , can be
k
expressed as samples of the compactly
supported piecewise constant function f k (n)
n
n 1
f ( x)  2 h (n),
x k
k
2
2
n 

f 2k (n, ni )  f k  n  ki 
2 

k
k
k
Cont..3.1 the impact of shift variance
h 2 ( n)
Cont..3.1 the impact of shift variance
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
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
4.1 Some definition of regularity
 regularity is defined as a maximum value
of
r
such that
F ( ) 


1
1 
r 1
,
 R
Implies that m is m-times continuously
differentiable, where r  m
determines smoothness of scaling function
and associated wavelet
Cont.. 4.1 Some definition of regularity
 regularity using Lipschitz(Holder) exponent
 A function f (t ) is called Lipschitz of
order  ,0    1 , if for any t and some
small 
f (t )  f (t   )  c 


higher-orders (  1) of Lipschitz exponent
implies higer-order regularity
4.2 Vanishing moment
 The divisibility of filter H (z ) by (1  z 1 ) L
means that the associated  will have L
vanishing moments
l
t
  (t )dt ,
l  0,1,..., L  1
 If wavelet has L vanishing moments, then
the wavelet coefficients of a function have
high compression potential
5 Texture classification
Methodology
 estimation of texture quality
 4-level DWT with 13 energies
M N
1
2
ei 
I

i ( x, y )
M  N x 1 y 1
 distance function used simplified
Mahalanobis distance
j
 xj

j ' th element of feature vector x


i , j  mi , j j ' th element of mean - vector of class i 


c

varian
ce
of
j
'
th
element
in
class
i
 i, j

D( x, i )   ( x j  mi , j ) / c
2
j 1
Cont.. 5 Texture classification-methodology
 classification
 texture is assigned to class
D( x, i )  min{ D( x, j )} | j

like nearest neighbor
i if
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 energy 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
L
 1  z 1 
 P ( z )
H 0 ( z )  
 2 
i.e. H 0 (e j )  cos

 (t )  2  h0 (k ) (2t  k )

L
P ( e j )
2
2
 


H 0 (e j )   cos 2  f  sin 2 
2 
2

k  
L

 (t )  2  h1 (k ) (2t  k )
2
2
H 0 ( e j )  H 0 ( e j (    ) )  1
k  
(1  x) L f ( x)  x L f (1  x)  1
f 2 j (t ) 
d 2 j (t ) 

 h ( k  2k ) f
k  
2 j 1
0
(k )
L 1 L  1  k

 k
L
f ( x)   
x  x R (1  2 x)
k
k 0 

(k )
H 0 (e j )  e  jk ph0 (cos  )

 h ( k  2k ) f
k  
f 2 j 1 ( k ) 
1
2 j 1
G0 (e j )  e  jk p g 0 (cos  )

 [h (k  2k ) f
k  
0
2j
(k )  h1 (k  2k )d 2 j (t )]
H 0 (e j )G0 (e j )  H 0 (e j (  ) )G0 (e j (  ) )
ph0 ( x) p g 0 ( x)  ph0 ( x) p g 0 ( x)  1
 


H 0 (e j )G0 (e j )   cos   cos  q (cos  )q~ (cos  )
2 
2

LG
f 2 j (t ) 
d 2 j (t ) 

 h ( k  2k ) f
k  
0
2 j 1
(k )

 g ( k  2k ) f
k  
f 2 j 1 ( k ) 
1

 [g
k  
0
2 j 1
(k )
(k  2k ) f 2 j (k )  h1 (k  2k )d 2 j (t )]


H 0 (e j )G0 (e j )   cos 
2


2l
Lh
2l
 l 1  l  1  k     2 k

 
 cos   cos  R ( )
 
2
 2
 k 0  k


1
F (e j ( / 2 ) ) H 0 (e j ( / 2 ) )  F (e j (( / 2 )  ) ) H 0 (e j (( / 2 )  ) )
2
1
F3 (e j ) 
F (e j ( / 8) ) H 0 (e j ( / 8) ) H 0 (e j ( / 4 ) ) H 0 (e j ( / 2 ) )  7" aliasing terms"
2
F1 (e j ) 

