1 / 18

Green Coordinates

Green Coordinates. Y. Lipman D. Levin D. Cohen-Or. What for ?. Deforming 2D/3D objects. boundary + inside . Deform. 2D image. 1. How to Deform?. Cage 2D : polygon 3D : polyhedron. Coordinates. Cage based space warping

saburo
Download Presentation

Green Coordinates

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Green Coordinates Y. Lipman D. Levin D. Cohen-Or What for ? • Deforming 2D/3D objects. • boundary + inside Deform 2D image 1

  2. How to Deform? Cage2D : polygon 3D : polyhedron Coordinates • Cage based space warping • A deformed point is represented as a relative position with the cage. 3D mesh 2

  3. Conventional Coordinates Coordinates : Linear combination of cage vertices 3

  4. An Example(Barycentric coordinates on simplex) Triangulated cage 4

  5. Problems of Conventional Methods Shape-preservation is difficult Each of (x,y,z) is deformed individually. Proposed Conventional 5

  6. Previous Work for Deformation Surface based. + Shape-preservation - Must solve (non-)linear equations. Space warping based. + Computationally efficient - Difficult to preserve shape Contribution of this paper:Shape-preserving deformation using Space warping. 6

  7. Proposed Method Coordinates : Coordinates using normals 7

  8. Interpolation vs. Approximation It is not possible to achieve to conformal mapping using interpolation methods. Interpolation(not conformal) Approximation(conformal) All coordinates are possibly non-zero. Coordinates : Sometimes negative. 8

  9. Harmonic Function As flat as possible If the boundary condition is linear, then the linearity is reproduced. Necessary condition of a conformal mapping Solve PDE : Boundary conditions 1D case 9

  10. Solution using Green’s Identity (In textbook part) (Impulse function) Green’s function of Laplacian : Next page 10

  11. Solution using Green’s Identity Polygonize the boundary Barycentric coordinates 11

  12. Calculation of Green Coordinates cage

  13. Properties of Green Coordinates Linear reproduction→ Translation invariance→ Rotation and scale invariance → We need an appropriate scaling of normals. (Next page) Shape preservation → 2D : Conformal (Proved in the other paper) → 3D : Quasi-conformal (Seems to be checked experimentally. Max. distortion = 6) Smoothness → {φ} and {ψ} are smooth and harmonic 13

  14. Scaling of Normals Conformal 14

  15. Extension to Cage’s Exterior Interior: Exterior: → Try to extend the coordinates through a specified face (exit face). 15

  16. How to Extend ? cage Should keep also in exterior : Two equations in 2D case Make unknown, then decide them by solving a linear system 16

  17. Multiple Exit Faces Separate the space. Separation curve Conformingin the cage 17

  18. Discussions Only for deformation ? FEM basis ? Interpolating a target shape ?(non-linear system must be solved) Conforming two adjacent cages ? 　　→ Maybe NO. 18

More Related