INFORMATIK Differential Coordinates for Interactive Mesh Editing Yaron

INFORMATIK 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

Our goal: INFORMATIK Edit a surface while retaining its visual appearance

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

Our goal n INFORMATIK 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

Differential coordinates n INFORMATIK Differential coordinates are defined for triangular mesh vertices the relative coordinate vector average of the neighbors

Differential coordinates n INFORMATIK Differential coordinates are defined for triangular mesh vertices

Why differential coordinates? n INFORMATIK They represent the local detail / local shape description – The direction approximates the normal – The size approximates the mean curvature

Related work n n n INFORMATIK Multi-resolution: Zorin el al. [97], Kobbelt et al. [98], Guskov et al. [99] Laplacians smoothing Taubin [SIGGRAPH 95], Laplacians Morphing Alexa [TVC 03] Image editing: Perez et al. [SIGGRAPH 03] Mesh Editing: Zhou et al. [SIGGRAPH 04]

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

Laplacian reconstruction n INFORMATIK The operator is linear and thus can be represented by the following matrix:

Laplacian reconstruction INFORMATIK n Transforming the mesh to the differential representation: n Note that where

Laplacian reconstruction n INFORMATIK 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 , weights and are the spatial constraints. are

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

Laplacian reconstruction n n INFORMATIK Laplacian reconstruction gives smooth transformation, interactive time and ease of user interface -using few spatial constraints but doesn’t preserve details orientation and shape

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

Rotated Laplacian reconstruction n For each 1 -ring we look for rigid affine : INFORMATIK transformations

Rotated Laplacian reconstruction n The Laplacians are translation invariant: INFORMATIK

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

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

Rotated Laplacian reconstruction INFORMATIK n From our assumption that detail remain with same orientation to the underlying smooth surface: The rotations are defined by the smoothed surface. n We use the Laplacian reconstruction to evaluate the smoothed underlying surface normals.

Rotated Laplacian reconstruction n INFORMATIK In summary we have the following steps: 1. Reconstruct the surface with original Laplacians: 2. Approximate local rotations 3. Rotate each Laplacian coordinate 4. Reconstruct the edited surface: by

Some results INFORMATIK

Some results INFORMATIK

Some results INFORMATIK

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

Future work (October 2003) n n n INFORMATIK 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…

INFORMATIK Differential Coordinates for Interactive Mesh Editing