QUATERNION COLOR CURVATURE

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

Overview

Motivation

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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

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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

Curvature Imaging

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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

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

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

Quaternion Hessian

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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

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

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!

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