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Surface Reconstruction

Surface Reconstruction. Approach. Overview of important methods Properties needed Possible solution Method Current work Next Steps Conclusions and Discussion. Voronoi cells creation. Links creation. Triangles removal. p. Fig b. Fig c. Fig d. Overview of different methods.

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Surface Reconstruction

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  1. Surface Reconstruction

  2. Approach • Overview of important methods • Properties needed • Possible solution • Method • Current work • Next Steps • Conclusions and Discussion

  3. Voronoi cells creation Links creation Triangles removal p Fig b Fig c Fig d Overview of different methods Voronoi diagrams and Delaunay triangulation / tetrahedrization Point set Fig a

  4. Delaunay Triangulation algorithms Complexity = O(n2) in 3D Can be decreased to O(n) with uniform sampling and other assumptions • Different algorithms : • Divide and conquer • Convex hull sculpting • Alpha shapes • Incremental construction

  5. Surface Deformation • Energy minimization process • Start from an initial rough shape • Deform it until a local minimum is reached

  6. Example • The liver template (Johan Montagnat, Inria) • Evolution of a model (on one slice of the whole 3D image). • Rough initialization • Successive deformations • Energy deformation is functions of liver's contour points

  7. Properties + Fast methods + initial guess easy to find • Closed surface • Local minimum • Importance of initial guess

  8. F(X) = 0 F(X) < 0 F(X)>0 Implicit functions • Concept : surface = zeroes of a function • Iso-surface • Space partition (inside/outside/on the object) • Lots of possible (and famous) solutions • Hoppe, Amenta … Ex : equation of sphere x2 + y2 + z2 – 1 = 0

  9. Example of skeleton extraction `frame" over which the ``meat" of the shape hangs locus of the centers of all tangent discs contained in the shape.

  10. Polygonization • Marching cubes (or triangles) Images from Lakshman Prasad newsletter

  11. Properties + Important Data size reduction + Primitive + Adaptable polygonization + Objects can have holes + n-dimension function - Approximating - Important computation time : Recent improvement (RBF) decreased complexity to O(n2) -- Closed surface (c’est pas tout a fait vrai mais)

  12. Patches • Numerous approaches : • Idea : locally find the “best fitting” surface patch to reconstruct the surface • Least square data-fitting • Spatial Recursive subdivision and curve fitting • Subdivide the space to reduce the number of points. • Surface fitting faster • Idea : subdivide the space to fit lower degree splines.

  13. Recursive spatial subdivision Images from Benjamin Gregorski, Bernd Hamann, and Kenneth I. Joy

  14. Patch joining Images from Benjamin Gregorski, Bernd Hamann, and Kenneth I. Joy

  15. Scheme type Topological restrictions Robustness to noise Primitive extraction Main drawback delaunay interpolating None Weak No No primitive extraction deformation approximating Simple objects (no holes) Strong Yes Very simple objects implicit Approximating (can be both) Closed objects, no boundaries Strong Yes No boundary patches both None Adjustable Yes Continuity between patches Properties?

  16. Patch Approach Goals • Speed and important data size • Primitive extraction (higher abstraction) • General approach • Manage boundaries Hierarchical

  17. Local energy extrema Possible Solution Principal direction for each point “Iso-value” lines Minimizing curvature variation Image from Meyer and Desbrun 2002

  18. Curvature energy Local curvature energy maxima t Local curvature energy minima 1 0 t Initial function F0(t) 1 0 t many parametric functions + control meshes I1 I2 Hierarchical Patch extraction

  19. Isolated point 0A Isolated point 1A Patch 1-A Isolated point 1B Patch 0-0 Patch 1-B Patch 2-A Patch 1-C Patch 3-A Isolated point 3A Result? Profile example Profile reconstructed

  20. What Now? • Segmentation of the point cloud based on curvature • “Good” Uniform sampling • Pseudo-Uniform sampling with holes or cracks • New model with non uniform sampling

  21. Next steps • Work on surface theory • What kind of surface? • Uniform or non-uniform • Quadratic, cubic, else • How to combine them? • How to merge them? • Subdivision surfaces • R-functions

  22. Conclusions and discussion • Hierarchical patch reconstruction • No topological restriction (boundaries) • Parametric approach adaptable meshing • Different levels of details • Size reduction

  23. questions?

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