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Curvature Regularization for Curves and Surfaces in a Global Optimization Framework

Petter Strandmark Fredrik Kahl . Curvature Regularization for Curves and Surfaces in a Global Optimization Framework. Centre for Mathematical Sciences, Lund University. Length Regularization. Segmentation. Segmentation by minimizing an energy:. Data term. Length of boundary.

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Curvature Regularization for Curves and Surfaces in a Global Optimization Framework

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  1. Petter Strandmark Fredrik Kahl Curvature Regularization for Curves andSurfaces in a Global Optimization Framework Centre for Mathematical Sciences, Lund University

  2. Length Regularization Segmentation Segmentation by minimizing an energy: Dataterm Length of boundary

  3. Long, thin structures Example from Schoenemann et al. 2009 Dataterm Squared curvature Length of boundary

  4. Important papers Motivation from a psychological/biological standpoint Improved multi-label formulation • Schoenemann, Kahl and Cremers, ICCV 2009 • Schoenemann, Kahl, Masnou and Cremers, arXiv 2011 • Schoenemann, Masnou and Cremers, arXiv 2011 Continuous formulation Global optimization of curvature • Schoenemann, Kuang and Kahl, EMMCVPR 2011 • Goldluecke and Cremers, ICCV 2011 • Kanizsa, Italian Journal of Psychology 1971 • Dobbins, Zucker and Cynader, Nature 1987 Correct formulation, efficiency, • This paper: 3D

  5. Approximating Curves

  6. Approximating Curves • Start with a mesh of all possible line segments variable for each region variables for each pair of edges Restricted to {0,1}

  7. Linear Objective Function variable for each region; 1 meansforeground, 0 background Incorporate curvature: variables for each pair of edges

  8. Linear Constraints Boundary constraints: then Surface continuation constraints: then

  9. New Constraints • Problem with the existing formulation: Nothing prevents a ”double boundary”

  10. New Constraints Existing formulation Simple fix? Global solution! Require that Not correct! Not optimal (fractional)

  11. New Constraints • Consistency: then

  12. New Constraints New constraints Existing formulation Global solution! Global + correct! Not optimal (fractional) Not correct!

  13. Mesh Types Too coarse! 90° 60° 45° 27° 30° 32 regions, 52 lines 12 regions, 18 lines

  14. Mesh Types

  15. Adaptive Meshes Always split the most important region; use a priority queue

  16. Adaptive Meshes p. 69

  17. Adaptive Meshes

  18. Does It Matter? 16-connectivity

  19. Does It Matter? 8-connectivity

  20. Curvature of Surfaces Approximate surface with a mesh of faces Want to measure how much the surface bends: Willmore energy

  21. 3D Mesh One unit cell (5 tetrahedrons) 8 unit cells

  22. 3D Results Problem: “Wrapping a surface around a cross” Area regularization Curvature regularization

  23. Surface CompletionResults Problem: “Connecting two discs” Area regularization Curvature regularization 491,000 variables 637,000 variables 128 seconds

  24. Conclusions • Curvature regularization is now more practical • Adaptive meshes • Hexagonal meshes • New constraints give correct formulation • Surface completion Source code available online (2D and 3D)

  25. The end

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