INFORMATIK Laplacian Surface Editing Olga Sorkine Daniel Cohen-Or

INFORMATIK Laplacian Surface Editing Olga Sorkine Daniel Cohen-Or Yaron Lipman Tel Aviv University Marc Alexa TU Darmstadt Christian Rössl Hans-Peter Seidel Max-Planck Institut für Informatik

Differential coordinates n n Intrinsic surface representation Allows various surface editing operations: – Detail-preserving mesh editing INFORMATIK

Differential coordinates n n Intrinsic surface representation Allows various surface editing operations: – Detail-preserving mesh editing – Coating transfer INFORMATIK

Differential coordinates n n Intrinsic surface representation Allows various surface editing operations: – Detail-preserving mesh editing – Coating transfer – Mesh transplanting INFORMATIK

What is it? INFORMATIK n Differential coordinates are defined by the discrete Laplacian operator: n For highly irregular meshes: cotangent weights [Desbrun et al. 99] average of the neighbors

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

Why differential coordinates? n n n INFORMATIK Local detail representation – enables detail preservation through various modeling tasks Representation with sparse matrices Efficient linear surface reconstruction

Overall framework INFORMATIK n Compute differential representation n Pose modeling constraints n Reconstruct the surface – in least-squares sense

Overall framework n n ROI is bounded by a belt (static anchors) Manipulation through handle(s) INFORMATIK

Related work n n n INFORMATIK Multi-resolution: [Zorin el al. 97], [Kobbelt et al. 98], [Guskov et al. 99], [Boier-Martin et al. 04], [Botsch and Kobbelt 04] 2 Laplacian smoothing: Taubin [SIGGRAPH 95] Laplacian Morphing: Alexa [TVC 03] Image editing: Perez et al. [SIGGRAPH 03] Mesh Editing: Yu et al. [SIGGRAPH 04]

Problem: invariance to transformations n n INFORMATIK The basic Laplacian operator is translation-invariant, but not rotation- and scale-invariant Reconstruction attempts to preserve the original global orientation of the details

Invariance – solutions n INFORMATIK Explicit transformation of the differential coordinates prior to surface reconstruction – Lipman, Sorkine, Cohen-Or, Levin, Rössl and Seidel, “Differential Coordinates for Interactive Mesh Editing“, SMI 2004 • Estimation of rotations from naive reconstruction – Yu, Zhou, Xu, Shi, Bao, Guo and Shum, “Mesh Editing With Poisson-Based Gradient Field Manipulation“, SIGGRAPH 2004 • Propagation of handle transformation to the rest of the ROI

Estimation of rotations n INFORMATIK [Lipman et al. 2004] estimate rotation of local frames – Reconstruct the surface with the original Laplacians – Estimate the normals of underlying smooth surface – Rotate the Laplacians and reconstruct again

Explicit assignment of rotations n INFORMATIK Disadvantages: – Heuristic estimation of the rotations – Speed depends on the support of the smooth normal estimation operator; for highly detailed surfaces it must be large almost a height field not a height field

Implicit definition of transformations n INFORMATIK The idea: solve for local transformations AND the edited surface simultaneously! Transformation of the local frame

Defining the transformations Ti n How to formulate Ti ? – Based on the local (1 -ring) neighborhood – Linear dependence on the unknown vi’ Members of the 1 -ring of i-th vertex INFORMATIK

Defining the transformations Ti n First attempt: define Ti simply by solving INFORMATIK

Defining the transformations Ti n INFORMATIK Plug the expressions for Ti into the least-squares reconstruction formula: Linear combination of the unknown vi’

Constraining Ti n n INFORMATIK Trivial solution for Ti will result in membrane surface reconstruction To preserve the shape of the details we constrain Ti to rotations, uniform scales and translations Linear constraints on tlm so that Ti is rotation+scale+translation ? ?

Constraining Ti – 2 D case n Easy in 2 D: INFORMATIK

Constraining Ti – 3 D case n Not linear in 3 D: n Linearize by dropping the quadratic term INFORMATIK

Adjusting Ti INFORMATIK n Due to linearization, Ti scale the space along the h axis by cos When is large, this causes anisotropy n Possible correction: n – Compute Ti , remove the scaling component and reconstruct the surface again from the corrected i – Apply our technique from [Lipman et al. 04] first, and then the current technique – with small .

Some results INFORMATIK

Some results INFORMATIK

Some results INFORMATIK

Some results INFORMATIK

Some results n Video. . . INFORMATIK

Detail transfer and mixing n INFORMATIK “Peel“ the coating of one surface and transfer to another

Detail transfer and mixing n INFORMATIK Correspondence: – Parameterization onto a common domain and elastic warp to align the features, if needed

Detail transfer and mixing n INFORMATIK Detail peeling: Smoothing by [Desbrun et al. 99]

Detail transfer and mixing n Changing local frames: INFORMATIK

Detail transfer and mixing n Reconstruction of target surface from INFORMATIK :

Examples INFORMATIK

Examples INFORMATIK

Mixing Laplacians n Taking weighted average of i and ‘i INFORMATIK

Mesh transplanting n The user defines – Part to transplant – Where to transplant – Spatial orientation and scale n n Topological stitching Geometrical stitching via Laplacian mixing INFORMATIK

Mesh transplanting n Details gradually change in the transition area INFORMATIK

Mesh transplanting n Details gradually change in the transition area INFORMATIK

Conclusions n n n INFORMATIK Differential coordinates are useful for applications that need to preserve local details Reconstruction by linear least-squares – smoothly distributes the error across the domain Linearization of 3 D rotations was needed in order to solve for optimal local transformations – can we do better?

Acknowledgments n n n INFORMATIK German Israel Foundation (GIF) Israel Science Foundation (founded by the Israel Academy of Sciences and Humanities) Israeli Ministry of Science Bunny, Dragon, Feline courtesy of Stanford University Octopus courtesy of Mark Pauly

INFORMATIK Thank you!

Gradual transition INFORMATIK