Quaternion Color Curvature

1. Lilong Shi, Brian Funt, and Ghassan Hamarneh School of Computing Science, Simon Fraser University Quaternion Color Curvature

2. Overview • Motivation ?

3. 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

4. Existing Detectors • 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

5. Curvature Imaging • Vessel-map as constraints for segmentation, edges, etc. • Our interest is to investigate color curvature based on the Hessian operator Image Sources Vessel Map Vessel map

6. Hessian-based Operator 1 1 λ2 λ2 e2 e2 e1 e1 2nd order structure eigenvectors: (e1, e2 ) eigenvalues: |1|<|2| (eigenanalysis of H) local shape descriptor Principle Curvatures

7. 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

8. 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.

9. Quaternion Hessian quaternion number real numbers

10. Quaternion Hessian • Quaternion-valued Hessian matrix HQ • Apply QSVD to HQ • non-negative singular values 1 and 2 • UQ contains quaternion basis vectors 9/14

11. 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 backgroundnessmeasure • Vessel map for color image • separability of vessel structures from background • vessel segmentation and enhancement • detection of tubular structures

12. Experimental Results • Test on photomicrographs, nature photos, and satellite images Input Image Frangi’sgrayscale Quaternion Hessian

13. Experimental Results • Test on photomicrographs, nature photos, and satellite images Input Image Frangi’s grayscale Quaternion Hessian

14. Experimental Results • Test on photomicrographs, nature photos, and satellite images Input Image Frangi’s grayscale Quaternion Hessian

15. 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

16. Questions ? Thank you!