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Differential Coordinates for Interactive Mesh Editing

Edit a surface while preserving its visual appearance by deforming the details while preserving their shape. Use of Laplacian reconstruction and rotated Laplacian reconstruction methods.

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Differential Coordinates for Interactive Mesh Editing

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  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

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