The use of wavelets in the image processing domain is still in its infancy, and largely associated with image compression. With the advent of the dual-tree hypercomplex wavelet transform (DHWT) and its improved shift invariance and directional selectivity, applications in other areas of image processing are more conceivable. This paper discusses the problems and solutions in developing the DHWT and its inverse. It also offers a practical implementation of the algorithms involved. The aim of this work is to apply the DHWT in machine vision. Tentative work on a possible new way of feature extraction is presented. The paper shows that 2-D hypercomplex basis wavelets can be used to generate steerable filters which allow rotation as well as translation.

1. 2007 ICSPC International Conference on Signal
Processing and Communications
TA-P1: Image and Video Processing 6
STEERABLE FILTERS GENERATED WITH THE HYPERCOMPLEX DUAL-TREE
WAVELET TRANSFORM
J. Wedekind, B. Amavasai, K. Duttona
Tuesday, November 27th 2007
Microsystems and Machine Vision Laboratory
Materials Engineering Research Institute
Sheffield Hallam University
Sheffield
United Kingdom
a
This project was supported by the Nanorobotics EPSRC Basic Technology grant GR/S85696/01

2. motivation
feature extraction
Micron Nanorobotics
1. keypoint selection
• rotation
• translation
• blur/scale
2. feature descriptor
3. RANSAC,BHT

3. state of the art
keypoint selection
Roberts cross Kanade-Lucas-Tomasi
Harris-Stephens SIFT (Lowe), MOPS (Brown et al.)
next generation feature extractor?

4. ¨
Thomas Bulow
Hypercomplex Spectral Signal Representations for Image
Processing and Analysis

5. Ivan Selesnick
The design of approximate Hilbert transform pairs of
wavelet bases

6. Nick Kingsbury
Image processing with complex wavelets

7. Ivan Selesnick, Richard G. Baraniuk, Nick Kingsbury
The dual-tree complex wavelet transform

8. 1-D basis wavelets (Selesnick)
wavelets for K = K = 5 and L = 4
wavelet wavelet
0.6 1.5
0.5 1
0.4 0.5
0.3
0
0.2
0.1 -0.5
0 -1
-0.1 -1.5
recursion 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 16
H0 (z) 2 2 H0 (z)
H1 (z) 2 2 H1 (z) H0 , G0 H1 , G 1
recursion
G0 (z) 2 2 G0 (z)
wavelet wavelet
1.4 0.6
G1 (z) 2 2 G1 (z) 1.2
0.4
1
0.8 0.2
0.6
0
0.4
0.2 -0.2
0
-0.4
-0.2
-0.4 -0.6
0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14
H0 , G0 H1 , G 1

9. dual-tree complex wavelet transform (Kingsbury)
2-d
1-d
H0 (z2 ) 2
z2
H0 (z) 2 H0 (z2 ) 2
z2
G0 (z) 2 G0 (z2 ) 2
z2
G0 (z2 ) 2
z2
H1 (z) 2
H0 (z1 ) 2 H0 (z2 ) 2
z1 z2
G1 (z) 2 G0 (z1 ) 2 H0 (z2 ) 2
z1 z2
H0 (z1 ) 2 G0 (z2 ) 2
z1 z2
G0 (z1 ) 2 G0 (z2 ) 2
z1 z2
(hyper)complex representation H1 (z1 ) 2
z1
H1 (z2 ) 2
z2
G1 (z1 ) 2 H1 (z2 ) 2
z1 z2
1 H1 (z1 ) 2 G1 (z2 ) 2
z1 z2
i G1 (z1 ) 2 G1 (z2 ) 2
z1 z2
+ H1 (z2 ) 2
z2
1 j H1 (z2 ) 2
+ z2
i k G1 (z2 ) 2
z2
G1 (z2 ) 2
z2

10. commutative hypercomplex algebra
multiplication table of H
1 i j k
1 1 i j k
i i −1 k −j
j j k −1 −i
k k −j −i 1

11. test image
require ’selesnick’
require ’kingsbury’
img=MultiArray.load_grey8("circle.png")
hwt=HWT.new(MultiArray::SFLOAT)
himg=hwt.decompose(hwt.prepare(img),3)
img2=hwt.finalise(hwt.compose(himg,3))
diff=img2-img
diff.range
# -0.00522037548944354..0.0319054499268532
diff.normalise.display
Math::sqrt((diff**2).sum/diff.size)
# 0.00274400669319458

12. hypercomplex components
1 i j k
A = H, B = H A = G, B = H A = H, B = G A = G, B = G
A0 (z1 ) B0 (z2 )
A1 (z1 ) B0 (z2 )
A0 (z1 ) B1 (z2 )
A1 (z1 ) B1 (z2 )

13. wavelet tree

14. linear combinations of 1-D basis wavelets
animating ∆x
   
1
  i
   
2 π ∆x i/2
   
0 
 
 
   
 
  
 

V (z) =
     
Ha (z) R(va )+
0 0 

 


  
 
  
v  2 π ∆x i v0 
 
0

 
v = =e ·   , v ∈ C2
     0    
0 0 a∈{0,1}  
   
 
 
   
   
 
v   
 
v 
 
 
 
 
1
 
 
 
 
i Ga (z) I(va ) 1 1
animations require Javascript

15. 2-D separable basis wavelets
       
1

 0

 i

 0

 j

 0

 k

 0




 

 

 

 

 

 

 



0       
0 0 0 0 0 0 0
       
0

 1

 0

 i

 0

 j

 0

 k




 

 

 

 

 

 

 



0       
0 0 0 0 0 0 0
       
0

 0

 0

 0

 0

 0

 0

 0




 

 

 

 

 

 

 



1       
0 i 0 j 0 k 0
       
0

 0

 0

 0

 0

 0

 0

 0




 

 

 

 

 

 

 



0       
1 0 i 0 j 0 k

16. linear combinations of 2-D basis wavelets
animation
 
2 π ∆x j/2 0
V (z1 , z2 ) =
 

Ha (z1 ) Hb (z2 ) R(va,b )+




 2 





 

2 π ∆x2
 
a,b∈{0,1} 

0 j

V =e
 
· V·
Ga (z1 ) Hb (z2 ) I(va,b )+
 
2 π ∆x i/2 0
Ha (z1 ) Gb (z2 ) J(va,b )+
 





 1 





 

2 π ∆x1
 
0 i
 
Ga (z1 ) Gb (z2 ) K(va,b )

, V ∈ C2×2
 
e
animations require Javascript

17. coping with rotations (Kingsbury) (Bharath,Ng)
Rotation-invariant local feature matching A steerable complex wavelet construction
with complex wavelets and its application to image denoising

18. wavelet editor

19. blob-like pattern
   
1 0
 + (i − j) sin(α) 1 0

 
 
 

(1 − k) cos(α) 


 


0 0


 

0 0
 

animations require Javascript

20. edge pattern
v1,0 v1,1
v0,1 ρ
   
0 1
 + (i + j) cos(α − ρ) 0 1

(1 + k) cos(α + ρ)  +


 
 
 


 
 
 

0 0  
0 
0
   
0 0
 + (i + j) sin(α − ρ) 0 0

(1 + k) sin(α + ρ)  +


 
 
 


 

 

 


1 0 1 0
   
0 0 1
1  + (1 − j) cos(α) 0 0

 
 
 

(1 − i) sin(α) 

 
 

0 1
 

2 
0 1 2  

animations require Javascript

21. cross pattern
 
0 0
(1 + i + j + k) cos(α)  +


 

 
0 1
 

 
0 0
(1 + i − j − k) sin(α)  +


 

 
1 0
 

 
0 1
(−1 + i − j + k) sin(α) 


 


 
0 0
 

cos(α) H1 +sin(α) H2 , where H1 , H2 ∈ HCA2×2
animations require Javascript

22. conclusion
• improved understanding of high frequency pattern
• enable use of hypercomplex wavelets beyond edge detection
• several fully steerable patterns (translation, rotation)
• hypercomplex matrices for representing local structure
animations require Javascript

23. future work
• complete basis of steerable patterns? rotation around any point?
• understand interaction between neighbouring coefficients
• understand interaction of different layers of wavelet pyramid (Kingsbury)
• choose keypoints, descriptors
animations require Javascript

24. Hornetseye
GPL (free and open source software)
http://rubyforge.org/projects/hornetseye/
http://sourceforge.net/projects/hornetseye/
http://vision.eng.shu.ac.uk/mediawiki/index.php/Hornetseye
http://raa.ruby-lang.org/project/hornetseye/
http://www.wedesoft.demon.co.uk/hornetseye-api/

25. filter design (Selesnick)
perfect reconstruction
recursion
H0 (z) 2 2 H0 (z) 1
2 H0 (z)X(z) + H0 (−z)X(−z) H0 (z)+
H1 (z) 2 2 H1 (z)
1
2 H1 (z)X(z) + H1 (−z)X(−z) H1 (z) =(∗) X(z)
recursion ⇔
G0 (z) 2 2 G0 (z)
H0 (z) H0 (−z) + H1 (z) H1 (−z) = 0 (1)
G1 (z) 2 2 G1 (z)
H0 (z) H0 (z) + H1 (z) H1 (z) = 2 (2)
Thiran filter (τ = 0.5)
n−1 τ−L+1+k
d(n) = L−1
n (−1)n k=0 τ+1+k ⇐
vanishing moments H1 (z) = H0 (−z)
(1)
K=1 1 −1 H1 (z) = −H0 (−z)
K=2 1 −2 1 H0 (z) = Q(z) (1 + z−1 )K D(z)
(2)· · ·
K=3 1 −3 3 −1 H0 (z) = Q(z) (1 + z−1 )K D(z−1 ) z1−L
K =4 1 −4 6 −4 1 c .... s 1 X(z) + X(−z)
(∗) [↑ 2] [↓ 2] x(n) 2
... ...

26. filter design (Selesnick)
perfect reconstruction
recursion
H0 (z) 2 2 H0 (z) 1
2 G0 (z)X(z) + G0 (−z)X(−z) G0 (z)+
H1 (z) 2 2 H1 (z)
1
2 G1 (z)X(z) + G1 (−z)X(−z) G1 (z) =(∗) X(z)
recursion ⇔
G0 (z) 2 2 G0 (z)
G0 (z) G0 (−z) + G1 (z) G1 (−z) = 0 (3)
G1 (z) 2 2 G1 (z)
G0 (z) G0 (z) + G1 (z) G1 (z) = 2 (4)
Thiran filter (τ = 0.5)
n−1 τ−L+1+k
d(n) = L−1
n (−1)n k=0 τ+1+k ⇐
vanishing moments G1 (z) = G0 (−z)
(3)
K=1 1 −1 G1 (z) = −G0 (−z)
K=2 1 −2 1 G0 (z) = Q(z) (1 + z−1 )K D(z−1 ) z1−L
(4)· · ·
K=3 1 −3 3 −1 G0 (z) = Q(z) (1 + z−1 )K D(z)
K =4 1 −4 6 −4 1 c .... s 1 X(z) + X(−z)
(∗) [↑ 2] [↓ 2] x(n) 2
... ...

27. filter design (Selesnick)
(2)· · ·

H0 (z) H0 (z) + H1 (z) H1 (z) = 2 





H1 (z) = H0 (−z)







 1 = H0 (z) H0 (z) = (1 + z )
−1 K+K
D(z) D(z−1 ) z1−L Q(z) Q(z)

H1 (z) = −H0 (−z)




H0 (z) = Q(z) (1 + z−1 )K D(z)


 S (z) R(z)





H0 (z) = Q(z) (1 + z−1 )K D(z−1 ) z1−L




28. filter design (Selesnick)
(4)· · ·

G0 (z) G0 (z) + G1 (z) G1 (z) = 2 





G1 (z) = G0 (−z)







 1 = G0 (z) G0 (z) = (1 + z )
−1 K+K
D(z) D(z−1 ) z1−L Q(z) Q(z)

G1 (z) = −G0 (−z)




G0 (z) = Q(z) (1 + z−1 )K D(z−1 ) z1−L


 S (z) R(z)





G0 (z) = Q(z) (1 + z−1 )K D(z)




29. filter design (Selesnick)
(2,4)· · ·
 .
 r  .
  
 s
 N

 0 ··· 


  1  .

   
  


 

 

   
  
 sN−2


 sN−1 sN 0 · · ·


  r2  0



  
  
  
  
 .


. . . . 

  .   

 .   

S (z) Q(z) Q(z) = 1 ⇔  . . . . .   .  = 1
  
 .


 . . . . 






  
  
  
   R(z)


 

 

   
  
  
R(z) 
 ···  rN−1  0

0 s1 s2 s3 

 
 
   
  
  

 
 
   
 .
  
 r  .

 
 

.
     
 ··· 0 s1  N
  

30. spectral factorisation (Laguerre)
spectral factorisation
Laguerre
R(z) symmetric, real-valued
Input: Polynomial p(x) Im Im
Output: zero crossing o ∈ C of p
x → 0.0 + 0.0 i; c → 100;
while c ≥ 0 do Re Re
if |p(x)| sufficiently small then
return o = x; tuples quadruples
end
g = p (x)/p(x);
h = g2 − p (x)/p(x); Im Im Im Im
if |g + d| > |g − d| then Re Re Re Re
a = n/(g + d); tuples quadruples tuples quadruples
else
Q(z) Q(z)
a = n/(g − d);
H0 (z) = Q(z) (1 + z−1 )K D(z)
end
x → x − a; c → c − 1; H0 (z) = Q(z) (1 + z−1 )K D(z−1 ) z1−L
end H1 (z) = H0 (−z), H1 (z) = −H0 (−z)
(2,4)