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Dynamic Programming (DP), Shortest Paths (SP)

Dynamic Programming (DP), Shortest Paths (SP). CS664 Lecture 22 Thursday 11/11/04 Some slides care of Yuri Boykov, Dan Huttenlocher. Level sets. [Donald Tanguay]. Level sets and curve evolution. Z. A. Shortest path problem. Lecture theme. B. A.

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Dynamic Programming (DP), Shortest Paths (SP)

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  1. Dynamic Programming (DP),Shortest Paths (SP) CS664 Lecture 22 Thursday 11/11/04 Some slides care of Yuri Boykov, Dan Huttenlocher

  2. Level sets [Donald Tanguay]

  3. Level sets and curve evolution

  4. Z A Shortest path problem Lecture theme

  5. B A - processed nodes (distance to A is known) - active nodes (front) - active node with the smallest distance value Dijkstra algorithm

  6. Example: 1 4 3 2 Graph edges are “cheap” in places with high intensity gradients Shortest paths segmentation

  7. Shortest paths approach p Compute the shortest pathp ->p for a point p. Shortest paths segmentation Example: find the shortest closed contour in a given domain of a graph Repeat for all points on the black line. Then choose the optimal contour.

  8. Disparities of pixels in the scan line regularization photoconsistency DP (SP) for stereo

  9. Discrete snakes • Represent the snake as a set of points • Curve as spline, e.g. (particle method) • Local update problem can be solved exactly (compute global min) • Do this repeatedly • Problems with collisions, change of topology

  10. Discrete snake energy Best location of the last vertex vn depends only the location of vn-1

  11. First-order interactions Discrete snakes example control points Fold data term into smoothness term

  12. B A Energy minimization by SP sites states 1 2 … m

  13. Distance transform (DT) Note: can be generalized beyond 1P (DT of arbitrary f)

  14. Computing distance transforms • Depends on the distance measure (L1 or L2 distance) • Linear time algorithms based on dynamic programming • Fast in practice • Can think of this as smoothing in feature space

  15. Distance transform applications • Primarily used in recognition • Represent the model as a set of points • Edges, or maybe corners • Compare model to image • Under some transformation of the model • Chamfer matching: L1 distance on distance transform • Not robust at all

  16. Hausdorff distance • Defined between two sets of points • h(A,B)= if every point in A lies within  of the nearest point in B •  is the smallest value for which this holds

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