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

2.  Motivation 1/14

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.  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 3/14

5. 4/14 Image Sources Vessel map Vessel Map  Vessel-map as constraints for segmentation, edges, etc.  Our interest is to investigate color curvature based on the Hessian operator

6.  local shape descriptor  Principle Curvatures e1e1 e2e2 1 λ2λ2 Hessian-based Operator 2nd order structure e1e1 e2e2 λ2λ2 1 (eigen analysis of H) eigenvectors: (e 1, e 2 ) eigenvalues: | 1 |<| 2 | 5/14

7.  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

8.  Quaternions  extension of real and complex numbers  1 real and 3 imaginary components  color is represented as ▪ simple + effective  Operations:  arithmetic, fourier transform, eigenvalue decomposition, etc. 7/14

9. 8/14 quaternion number real numbers

10. Quaternion Hessian  Quaternion-valued Hessian matrix H Q  Apply QSVD to H Q  non-negative singular values  1 and  2  U Q contains quaternion basis vectors 9/14

11.   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

12.  Test on photomicrographs, nature photos, and satellite images Input Image Frangi’s grayscale Quaternion Hessian 11/14

13.  Test on photomicrographs, nature photos, and satellite images Input Image Frangi’s grayscale Quaternion Hessian 12/14

14.  Test on photomicrographs, nature photos, and satellite images Input Image Frangi’s grayscale Quaternion Hessian 13/14

15.  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

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