QUATERNION COLOR CURVATURE

Lilong Shi, Brian Funt, and Ghassan HamarnehSchool of Computing Science, Simon Fraser University

Overview

Motivation

1/14

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

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

Curvature Imaging

4/14

Imag

e So

urce

s

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

local shape descriptor

Principle Curvatures

e1

e2

1

λ2

Hessian-based Operator

2

22

2

2

2

),(

yI

xyI

yxI

xI

yxH2nd order structure

e1

e2 λ2

1

(eigen analysis of H)

eigenvectors: (e1, e2 )

eigenvalues: |1|<|2|

5/14

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

6/14

|||| 21 BR22

21|||| HS

Quaternion Representation of ColorQuaternions

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

Quaternion Hessian

8/14

2

22

2

2

2

yQ

xyQ

yxQ

xQ

HQ

quaternion number

real numbers

k

yB

xyB

yxB

xB

j

yG

xyG

yxG

xG

i

yR

xyR

yxR

xR

2

22

2

2

2

2

22

2

2

2

2

22

2

2

2

kBjGiRQ

Quaternion Hessian

Quaternion-valued Hessian matrix HQ

Apply QSVD to HQ

Þ non-negative singular values 1 and 2 Þ UQ contains quaternion basis vectors

9/14

k

yB

xyB

yxB

xB

j

yG

xyG

yxG

xG

i

yR

xyR

yxR

xR

HQ

2

22

2

2

2

2

22

2

2

2

2

22

2

2

2

QT

Q UVHQ

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

10/14

Experimental Results Test on photomicrographs, nature photos, and satellite

images

Input Image Frangi’s grayscale Quaternion Hessian

11/14

Experimental Results Test on photomicrographs, nature photos, and satellite

images

Input Image Frangi’s grayscale Quaternion Hessian

12/14

Experimental Results Test on photomicrographs, nature photos, and satellite

images

Input Image Frangi’s grayscale Quaternion Hessian

13/14

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

14/14

Questions

Thank you!

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