lilong shi, brian funt, and ghassan hamarneh school of computing science, simon fraser university
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![Page 1: Lilong Shi, Brian Funt, and Ghassan Hamarneh School of Computing Science, Simon Fraser University](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d795503460f94a5d52b/html5/thumbnails/1.jpg)
QUATERNION COLOR CURVATURE
Lilong Shi, Brian Funt, and Ghassan HamarnehSchool of Computing Science, Simon Fraser University
![Page 2: Lilong Shi, Brian Funt, and Ghassan Hamarneh School of Computing Science, Simon Fraser University](https://reader036.vdocuments.net/reader036/viewer/2022062407/56649d795503460f94a5d52b/html5/thumbnails/2.jpg)
Overview
Motivation
1/14
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Overview
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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Existing Detectors
1st, 2nd or higher orders derivativesMostly grayscale basedFor color:
process summed channels▪ eg. isoluminance situation
sum each individually processed channel▪ derivatives in opposite directions cancel one
other3/14
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Curvature Imaging
4/14
Image
Sourc
es
Vessel map
Vess
el M
ap
Vessel-map as constraints for segmentation, edges, etc.
Our interest is to investigate color curvature based on the Hessian operator
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local shape descriptor
Principle Curvatures
e1
e2
1
λ2
Hessian-based Operator
2
22
2
2
2
),(
y
I
xy
Iyx
I
x
I
yxH
2nd order
structure
e1
e2 λ2
1
(eigen analysis of H)
eigenvectors: (e1, e2 )
eigenvalues: |1|<|2|
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Hessian-based Approach
Tubular, vessel-like structures [Frangi98]
Curvature measured by eigenvalue of Hessian blobness: backgroundness:
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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|||| 21 BR22
21|||| HS
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Quaternion Representation of Color
Quaternions extension of real and complex numbers 1 real and 3 imaginary components
<R,G,B> color is represented as▪ simple + effective
Operations: arithmetic, fourier transform, eigenvalue
decomposition, etc.7/14
kBjGiRQ
kdjcibaq
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Quaternion Hessian
8/14
2
22
2
2
2
y
Q
xy
Qyx
Q
x
Q
HQ
quaternion number
real numbers
k
y
B
xy
Byx
B
x
B
j
y
G
xy
Gyx
G
x
G
i
y
R
xy
Ryx
R
x
R
2
22
2
2
2
2
22
2
2
2
2
22
2
2
2
kBjGiRQ
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Quaternion Hessian
Quaternion-valued Hessian matrix HQ
Apply QSVD to HQ
Þ non-negative singular values 1 and 2
Þ UQ contains quaternion basis vectors9/14
k
y
B
xy
Byx
B
x
B
j
y
G
xy
Gyx
G
x
G
i
y
R
xy
Ryx
R
x
R
HQ
2
22
2
2
2
2
22
2
2
2
2
22
2
2
2
QT
Q UVHQ
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Color Curvature Measure
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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Experimental Results
Test on photomicrographs, nature photos, and satellite images
Input Image Frangi’s grayscale Quaternion Hessian
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Experimental Results
Test on photomicrographs, nature photos, and satellite images
Input Image Frangi’s grayscale Quaternion Hessian
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Experimental Results
Test on photomicrographs, nature photos, and satellite images
Input Image Frangi’s grayscale Quaternion Hessian
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Conclusion
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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Questions
Thank you!
?