1. Geodesic Minimal Paths Vida Movahedi Elder Lab, January 2010

2. Contents What is the goal? Minimal Path Algorithm Challenges How can Elderlab help? Results

3. Goal Finding boundary of salient objects in images of natural scenes

4. Minimal Path Inputs: –Two key points – A potential function to be minimized along the path Output: –The minimal path

5. Minimal Path- problem formulation Global minimum of the active contour energy: C(s): curve, s: arclength, L: length of curve Surface of minimal action U: minimal energy integrated along a path between p 0 and p A p0,p : set of all paths between p0 and p

6. Fast Marching Algorithm Computing U by frontpropagation: evolving a front starting from an infinitesimal circle around p0 until each point in image is reached

7. Challenges Can the minimal path algorithm solve the boundary detection problem? –Key points? –Potential Function? Idea: Use York’s multi-scale algorithm (MS)

8. MS Algorithm We have a set of contour hypotheses at each scale These contours can be used to find good candidates for key points These contours (and some other cues) can also be used to build potential functions. Multi-scale model (coarse to fine) can also help

10. Key Points Simplest approach: 3 key points, equally spaced on the MS contour (prior) Maximize product of probabilities (MS unary cue)

11. Rotating Key Points Consider multiple hypothesis for key points Obtain multiple contours Next step: Find which contour is the best –Distribution model for contour lengths –Distribution model for average Pb value –Improve method to find simple contours only

12. Rotating Key Points

13. Potential Function Ideas: –The Sobel edge map –Distance transform of MS contour (prior) –Distance transform of several overlapped MS contours –Berkeley’s Pb map –Likelihood based on Pb and distance to prior contour

14. Sobel Edge Map

15. Can use the MS prior to emphasize or de-emphasize map

16. Distance Transform

17. Distance transform Too much emphasis on MS prior

18. Distance transform of 10 overlapped MS contours

19. Challenge: If MS contours are not good

21. Berkeley’s Pb map

22. Combining Pb and Distance Next step: learning models

23. Summary The MP algorithm provides global minimal paths The MS algorithm provides contour hypothesis The MS contours can be used to obtain key points and potential functions for MP algorithm Next steps: –Learning models for better potential functions –Obtaining simple contours –Ranking contours –Evaluate multi-scale model

24. References Laurent D. Cohen (2001), “Multiple Contour Finding and Perceptual Grouping using Minimal Paths”, Journal of Mathematical Imaging and Vision, vol. 14, pp. 225-236. Estrada, F.J. and Elder, J.H. (2006) “Multi-scale contour extraction based on natural image statistics”, Proc. IEEE Workshop on Perceptual Organization in Computer Vision, pp. 134-141. J. H. Elder, A. Krupnik and L. A. Johnston (2003), "Contour grouping with prior models," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, pp. 661-674.