Minimum Ratio Contours For Meshes

1. Minimum RatioContours For Meshes Andrew Clements Hao Zhang gruvi graphics + usability + visualization

2. Introduction • Problem: Feature extraction and segmentation of 3D mesh models for the purpose of object recognition • Object parts are delimited by contours • Given an initial contour, search for a ‘better’ contour • What are features? • Explained later • Contributions • Applying minimum ratio to meshes • Energy definition • Efficiency

3. Outline • Previous Work & Motivation • Algorithm Overview • Ambient Graph Construction • Minimum Ratio Contour (MRC) Algorithm • Results

4. Snake Methods • Widely applied for the task of feature extraction and segmentation • Techniques work with images and meshes • Formulated as a minimization problem • Snake is parameterized by v(s), and energy is defined as • Internal energy controls length and smoothness • External energy controls feature adaptation • A search for the snake with the lowest energy is performed • Gradient descent, graph minimization

5. Drawbacks of Snakes • Local in nature • When using gradient descent, snake cannot jump out of local minima • Global minimum does not yield a meaningful result • Trivial solution results with classical energy definition

6. Minimum Ratio Methods • Previously applied for the task of image segmentation • Energy of a contour is defined as a ratio • F(v) controls feature adaptation and smoothness • G(v) is a general measure of length • Removes bias towards short contours • Trivial solutions are not minimizers • Due to normalization by length

7. Minimum Ratio Methods • To find a solution, problem is discretized • Goal is to find the minimum ratio cycle in a graph • A global solution can be obtained in polynomial time • Requires at least O(n2) time to find minimizing cycle in a general graph

8. Method Differences • Snaking method uses a Total Energy • MRC uses a Ratio Energy

9. Outline • Previous Work & Motivation • Algorithm Overview • Ambient Graph Construction • Minimum Ratio Contour (MRC) Algorithm • Results

10. Algorithm Overview MRC Algorithm Ambient Graph Construction Minimum Ratio Contour Ambient Graph Input Mesh Initial Contour

11. Outline • Previous Work & Motivation • Algorithm Overview • Ambient Graph Construction • Refinement • Energy definition • Minimum Ratio Contour (MRC) Algorithm • Results

12. Ambient Graph Construction • Ambient graph models the space of admissible contours • Nodes in ambient graph correspond to directed edges of mesh • Arcs in ambient graph are inserted between nodes of successive directed edges • Weights can be assigned to arcs which encode bending between nodes • Contours on mesh map to cycles in ambient graph

13. Sample Ambient Graph Mesh Ambient Graph

14. Refined Ambient Graph • Problem: irregular mesh connectivity • Contours may not be smooth • Refine mesh before constructing ambient graph • Smoother contours are possible • Refinement scheme inserts extra chords passing through faces of mesh • Subdivision is not sufficient

15. Energy Motivation • Each arc in ambient graph is assigned a numerator and denominator weight • Energy of a contour C is defined as • Denominator weight is taken to be Euclidean length • Numerator weight controls feature adaptation • serves to attract the contour to features which are perceptually salient

16. Energy Considerations • What features should be segmented? • Minima Rule • A theory which describes where the humans perceive boundaries between parts • Boundaries consist of surface points at the negative minima of principal curvatures • Contour Steering: favour contours aligned with principle curvature directions

17. Outline • Previous Work & Motivation • Algorithm Overview • Ambient Graph Construction • Minimum Ratio Contour (MRC) Algorithm • Results

18. MRC Algorithm Overview • Initial Contour • Strip Boundaries • Edge Cut • Gate Segments • Acyclic Edge Graph • Optimization

19. MRC Algorithm Overview • Initial Contour • Strip Boundaries • Edge Cut • Gate Segments • Acyclic Edge Graph • Optimization

20. MRC Algorithm Overview • Strip Boundaries • Mimic flow of initial contour • Constructed by ‘dilating’ initial contour

21. MRC Algorithm Overview • Edge Cut • Disconnects search space • Used in the acyclic graph construction

22. MRC Algorithm Overview • Gate Segments • Help orient flow • Inserted at constrictions between adjacent strip boundaries

23. MRC Algorithm Overview • Acyclic Edge Graph • select nodes from ambient graph – orient edges in search space • Edge cut nodes are duplicated • Paths from edge cut nodes in acyclic graph correspond to contours in search space

24. MRC Algorithm Overview • Optimization • A series of Minimum Ratio Path (MRP) problems are solved, one for every edge in the edge cut • The path with minimum ratio corresponds to the contour with least ratio

25. Solving The MRP Problem • Reduces to a series of decisions determining whether a negative path exists in an acyclic graph • Can be performed in linear time • Linear vs. Binary Search • Experimentally, a constant number of iterations is needed for linear search • Affirms other researchers observations

26. Outline • Previous Work & Motivation • Algorithm Overview • Ambient Graph Construction • Minimum Ratio Contour (MRC) Algorithm • Results

27. Results – Regular vs. Refined

28. Results – Escaping Local Minima

29. Results – Iterations

30. Results – Constraints

31. Questions?

32. Future Work • Numerator weights that incorporate area • Use Stoke’s Theorem • MRP + Length • Combine ratio with length • Currently have algorithm to handle minimum mean path with length • Generalize to MRP + length • Reduce running time from O(n2)

33. MRC Algorithm Overview • Initial Contour • Strip Boundaries • Edge Cut • Gate Segments • Acyclic Edge Graph • Optimization