Lilong Shi, Brian Funt, and Ghassan Hamarneh
School of Computing Science,
Simon Fraser University

Motivation
1/14

Motivation
 Existing detectors are grayscale-based
 Color increases discrimination

Goals:
 Hessian-based color curvature
 Extend Frangi’s vesselness to color

Problem
 Cancellation while converting color to gray
▪ e.g. Isoluminant images
2/14
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

1st, 2nd or higher orders derivatives
Mostly grayscale based
For color:
 process summed channels
▪ eg. isoluminance situation
 sum each individually processed channel
▪ derivatives in opposite directions cancel one other
3/14

Vessel-map as constraints for segmentation, edges, etc.

Our interest is to investigate color curvature
based on the Hessian operator
Vessel Map
Image Sources
Vessel map
4/14

local shape descriptor
2nd order
structure

 2I

2
x
H ( x, y )   2
  I
 yx

2
 I 

xy

2
 I 
2
 y 
eigenvectors: (e1, e2 )
eigenvalues: |1|<|2|
(eigen analysis of H)
Principle Curvatures
1
1
λ2
λ2
e2
e2
e1
e1
5/14
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Tubular, vessel-like structures [Frangi98]
Curvature measured by eigenvalue of Hessian
 blobness:
R B  | 1 | |  2 |
 backgroundness:
S  || H  || 
1   2
2
2

vesselness <= blobness & backgroundness

For 3-channel image, 6 λ’s/e’s, in 6 directions
 No simple way to combine them for curvature
6/14

Quaternions
 extension of real and complex numbers
 1 real and 3 imaginary components
q  a  b i  c  j  d k
 <R,G,B> color is represented as
▪ simple + effective

Q  R i  G  j  B  k
Operations:
 arithmetic, fourier transform, eigenvalue
decomposition, etc.
7/14
Q  R i  G  j  B  k
HQ
  2Q

2
x
  2
 Q
 yx

2
 Q 

xy

2
 Q 
2
 y 
 2R

2
x
  2
 R
 yx

2
 R 

xy
 i 
2
 R 
2
 y 
quaternion number
real numbers
  2G

2
x
 2
 G
 yx

2
 G 

xy
 j 
2
 G 
2
 y 
 2B

2
x
 2
 B
 yx

2
 B 

xy
k
2
 B 
2
 y 
8/14

Quaternion-valued Hessian matrix HQ
HQ

 2R

2
x
  2
 R
 yx

2
 R 

xy
 i 
2
 R 
2
 y 
  2G

2
x
 2
 G
 yx

2
 G 

xy
 j 
2
 G 
2
 y 
 2B

2
x
 2
 B
 yx

2
 B 

xy
k
2
 B 
2
 y 
Apply QSVD to HQ
H Q  V Q   U Q
T
 non-negative singular values 1 and 2
 UQ contains quaternion basis vectors
9/14
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1 and 2: 2 eigen-values instead of 6 for
principle curvatures of color tubular structure
Can therefore be used the same way for
blobness and backgroundness measure
Vessel map for color image
 separability of vessel structures from background
 vessel segmentation and enhancement
 detection of tubular structures
10/14

Test on photomicrographs, nature photos, and satellite images
Input Image
Frangi’s grayscale
Quaternion Hessian
11/14

Test on photomicrographs, nature photos, and satellite images
Input Image
Frangi’s grayscale
Quaternion Hessian
12/14

Test on photomicrographs, nature photos, and satellite images
Input Image
Frangi’s grayscale
Quaternion Hessian
13/14

Summary
 Extended Frangi’s method from scalar to color
▪ Overcomes
▪ Cancellation problem,
▪ *Isoluminance
 Used Quaternions for color representation
 Prevented info loss. Increased discrimination

Future work
 3D/4D vector-valued image/volumetric data
 Feature points/blob detector in color
14/14
?