1 Minimum Ratio Contours 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(n 2 ) 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 Initial Contour Ambient Graph Input Mesh Minimum Ratio Contour Ambient Graph Construction MRC Algorithm

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

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(n 2 )

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