1 / 27

Differential Coordinates for Interactive Mesh Editing

Differential Coordinates for Interactive Mesh Editing. Yaron Lipman Olga Sorkine Daniel Cohen-Or David Levin Tel-Aviv University Christian Rössl Hans-Peter Seidel Max-Planck Institut f ür Informatik. Our goal:.

guido
Download Presentation

Differential Coordinates for Interactive Mesh Editing

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. Differential Coordinates for Interactive Mesh Editing Yaron Lipman Olga Sorkine Daniel Cohen-Or David Levin Tel-Aviv University Christian Rössl Hans-Peter Seidel Max-Planck Institut für Informatik

  2. Our goal: Edit a surface while retaining its visual appearance

  3. Edit a surface while retaining its visual appearance The details are deformed The details shape is preserved Original surface

  4. Our goal • Editing a surface while retaining its visual appearance • Smooth deformation • Smooth transition • Preserve relative local directions of the details • Minimal user interaction • Interactive time response T

  5. Differential coordinates • Differential coordinates are defined for triangular mesh vertices average of the neighbors the relative coordinate vector

  6. Differential coordinates • Differential coordinates are defined for triangular mesh vertices

  7. Why differential coordinates? • They represent the local detail / local shape description • The direction approximates the normal • The size approximates the mean curvature

  8. Related work • Multi-resolution: Zorin el al.[97], Kobbelt et al.[98], Guskov et al.[99] • Laplacianssmoothing Taubin [SIGGRAPH95], • LaplaciansMorphing Alexa [TVC03] • Image editing: Perez et al. [SIGGRAPH03] • Mesh Editing: Zhou et al. [SIGGRAPH 04]

  9. Laplacian reconstruction • Denote by a triangular mesh with geometry , embedded in R³ . • For each vertex we define the Laplacian vector: • The Laplacians represents the details locally.

  10. Laplacian reconstruction • The operator is linear and thus can be represented by the following matrix:

  11. Laplacian reconstruction • Transforming the mesh to the differential representation: • Note that where

  12. Laplacian reconstruction • Thus for reconstructing the mesh from the Laplacian representation: add constraints to get full rank system and therefore unique solution, i.e. unique minimizer to the functional where is the index set of constrained vertices , are weights and are the spatial constraints.

  13. Laplacian reconstruction The use of Laplacian (differential) representation and least squares solution forces local detail preserving

  14. Laplacian reconstruction • Laplacian reconstruction gives smoothtransformation, interactive time and ease of user interface -using few spatial constraints • but doesn’t preserve details orientation and shape

  15. Rotated Laplacian reconstruction • We’d like to perform deformation which preserves the detail orientation and shape: • We’d like to estimate the target shape Laplacians

  16. Rotated Laplacian reconstruction • For each 1-ring we look for rigid affine transformations :

  17. Rotated Laplacian reconstruction • The Laplacians are translation invariant:

  18. Rotated Laplacian reconstruction • Laplacians are not rotational invariant (they represent detail with orientation) • Note that the Laplacian operator commute with linear rotations : R

  19. Rotated Laplacian reconstruction • Therefore we get: • So all we need is to estimate the local rotations.

  20. Rotated Laplacian reconstruction • From our assumption that detail remain with sameorientation to the underlying smooth surface: The rotations are defined by the smoothed surface. • We use the Laplacian reconstruction to evaluate the smoothed underlying surface normals.

  21. Rotated Laplacian reconstruction • In summary we have the following steps: Reconstruct the surface with original Laplacians: Approximate local rotations Rotate each Laplacian coordinate by Reconstruct the edited surface:

  22. Some results

  23. Some results

  24. Some results

  25. Implementation • We solve the normal equations via factorization. • The factorization is done once for each ROI. • And back substitution for each new handle location.

  26. Future work (October 2003) • The main problem of the Laplacian coordinates are the need to estimate the rotation explicitly (also in Zhou et al. SIGGRAPH 2004). • Instead those rotations can be computed implicitly so that the final shape is defined in one step! • To be presented in SGP 2004 in Nice next month…

  27. Differential Coordinates for Interactive Mesh Editing

More Related