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Shape Representation Using R-Functions Applied To Supershapes

Explore the use of R-functions to represent complex objects with a single equation using supershapes, covering the forward and inverse problems, algorithm details, results, and future work possibilities. This study delves into the combination of R-functions and supershapes, mesh refinement, CSG trees, and potential applications in shape modeling, computer vision, compression, and more.

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Shape Representation Using R-Functions Applied To Supershapes

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  1. Shape Representation Using R-Functions Applied To Supershapes Yohan FOUGEROLLE August, the 24th

  2. Outline • Introduction • R-functions and supershapes • Algorithm • Results • Conclusions and future work

  3. ? Forward problem Increasing complexity Inverse problem Is it possible to represent complex objects with one equation? Single object or multiple primitives and their combinations? If possible, how can we recover the parameters? Forward/Inverse problem of analytical geometry

  4. Outline • Introduction • R-functions and supershapes • Algorithm • Results • Conclusions and future work

  5. y x Union Equation? R-functions : Example • Theory to combine implicit functions y x

  6. Different R-functions

  7. Example using Rp functions Scalar field generated

  8. Conclusion on R-functions • Sign is treated as a Boolean variable • Decomposition in simple primitives  equation • Guaranteed differential properties • Question: what kind of primitive do we need?

  9. Supershapes • What • Recent extension of superquadrics • Why • Superquadrics widely used in computer graphics and computer vision • Compactness • Both parametric and implicit representations

  10. Inside / outside function Implicit and parametric representations

  11. Outline • Introduction • R-functions and supershapes • Algorithm • Results • Conclusions and future work

  12. Algorithm • Implicit function  characteristic function • Parametric representation • Vertices generation • Mesh refinement around the intersection curve

  13. Flag vertices using implicit function Splitting intersecting faces Transferred faces Results at different samplings Algorithm

  14. Outline • Introduction • R-functions and supershapes • Algorithm • Results • Conclusions and future work

  15. Previous Results • First results • Hierarchical global deformations • Simple CSG Trees (20 nodes at most)

  16. III I I II IV II I I Results “Axle Mesh” Clemson University, ~40k faces Cleaned/segmented CAD Model

  17. 7 3 18 2 1 14 4 6 10 9 5 17 11 8 13 12 15 16 Part I : stabilizers CSG tree structure:18 Supershapes17 Boolean operations

  18. Modeled stabilizers Supershape representation Implicit function intensity CAD model

  19. Part II : extremities CSG tree composed by ~120 nodes ~60 Supershapes ~60 Boolean operations Original CAD file

  20. Union Union Union Union Part II : Detailed CSG Tree Bolts and details Supershape representation Detailed CSG Tree

  21. Part II : extremities Original CAD model Supershape representation Implicit function intensity

  22. Part III: “Head” Original CAD Model Supershape representation (CSG tree~30 nodes) Implicit function intensity

  23. Part IV: “Body” Original CAD model Supershape representation (CSG Tree ~100 nodes) Most complicated part??

  24. Union Finally… Supershape representation (~500 nodes) Original CAD model (~40k faces)

  25. Outline • Introduction • R-functions and supershapes • Algorithm • Results • Conclusions and future work

  26. Conclusions • “Can we represent arbitrary complex shapes” • What do we want to reconstruct? • What are the properties we need? • Easy to represent? • Supershape and R-functions • Compact primitives • Guaranteed differential properties • Arbitrary topology handled (holes) • Applied to simple objects to very complex objects  next step: real data

  27. Conclusions • Avoid many computational errors and most of degenerate cases • No intersection computed • Avoid additional operations and most degenerated cases • End up with one implicit equation • n-differentiable if primitives are n-differentiable (except at intersection (only Co continuity) ) • Easy draw due to parametric representation • ε-approximation only near the intersection, other points are lying exactly on the surface

  28. Potential Applications • Shape modeling • Computer vision • Object and shape recognition • Compression • Application to solve boundary problems of mathematical physics which requires inclusion of geometric information • Sections with Free Form Surface • Heat transmission • Torsion

  29. Future work • Now • Working on 2 papers • IEEE Transactions on Visualization and Computer Graphics (2nd review) • Model real data CVPR • Automation of the process • Primitive recovery • CSG Tree recovery (?)

  30. Potential Ideas • Brute force approaches (digging/filling) using topological tool and R-functions • Segmentation + part reconstruction (Yan’s approach) • Seed growing + region merging • Binary partition + part reconstruction • Iterative insertion • Supershape packing • 2 levels of recovery needed • Primitive • CSG Tree / Boolean decomposition • R-functions used to merge everything into one implicit equation

  31. Questions?

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