Quaternion Hessian - Computing Science
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Transcript Quaternion Hessian - Computing Science
Lilong Shi, Brian Funt, and Ghassan Hamarneh
School of Computing Science,
Simon Fraser University
Motivation
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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
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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
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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
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local shape descriptor
2nd order
structure
2I
2
x
H ( x, y ) 2
I
yx
2
I
xy
2
I
2
y
eigenvectors: (e1, e2 )
eigenvalues: |1|<|2|
(eigen analysis of H)
Principle Curvatures
1
1
λ2
λ2
e2
e2
e1
e1
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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
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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.
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Q R i G j B k
HQ
2Q
2
x
2
Q
yx
2
Q
xy
2
Q
2
y
2R
2
x
2
R
yx
2
R
xy
i
2
R
2
y
quaternion number
real numbers
2G
2
x
2
G
yx
2
G
xy
j
2
G
2
y
2B
2
x
2
B
yx
2
B
xy
k
2
B
2
y
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Quaternion-valued Hessian matrix HQ
HQ
2R
2
x
2
R
yx
2
R
xy
i
2
R
2
y
2G
2
x
2
G
yx
2
G
xy
j
2
G
2
y
2B
2
x
2
B
yx
2
B
xy
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
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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
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Test on photomicrographs, nature photos, and satellite images
Input Image
Frangi’s grayscale
Quaternion Hessian
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Test on photomicrographs, nature photos, and satellite images
Input Image
Frangi’s grayscale
Quaternion Hessian
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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
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?