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Laplacian Surface Editing
Olga Sorkine, Daniel Cohen-Or, Yaron Lipman, Marc Alexa, Christian Rössl, Hans-Peter Seidel
Encodes each vertex by the mesh Laplacian relative to its neighbourhood, an intrinsic differential representation that preserves surface detail under deformation and lets detail be transferred or transplanted between meshes.
Abstract
Surface editing operations commonly require geometric details of the surface to be preserved as much as possible. We argue that geometric detail is an intrinsic property of a surface and that, consequently, surface editing is best performed by operating over an intrinsic surface representation. We provide such a representation of a surface, based on the Laplacian of the mesh, by encoding each vertex relative to its neighborhood. The Laplacian of the mesh is enhanced to be invariant to locally linearized rigid transformations and scaling. Based on this Laplacian representation, we develop useful editing operations: interactive free-form deformation in a region of interest based on the transformation of a handle, transfer and mixing of geometric details between two surfaces, and transplanting of a partial surface mesh onto another surface. The main computation involved in all operations is the solution of a sparse linear system, which can be done at interactive rates. We demonstrate the effectiveness of our approach in several examples, showing that the editing operations change the shape while respecting the structural geometric detail.
How to read this
- Category
- Method: differential (Laplacian) coordinates for detail-preserving mesh editing and transfer
- Contributions
- Represents a surface intrinsically by the mesh Laplacian, encoding each vertex relative to its one-ring neighbourhood so geometric detail is stored explicitly rather than as absolute positions
- Makes that Laplacian representation invariant to locally linearized rigid transformation and scaling, so handle-driven edits rotate and stretch detail correctly
- Builds three operations on it: interactive handle-based free-form deformation, transfer and mixing of geometric detail between two surfaces, and transplanting a partial mesh onto another, each reduced to one sparse linear solve at interactive rates
- Context
- A root of the differential-coordinate line of mesh deformation: it formalized Laplacian coordinates that later detail-transfer, as-rigid-as-possible and biharmonic-weight methods lean on, and it is a sibling of the contemporaneous Deformation Transfer for Triangle Meshes. Mesh-Based Inverse Kinematics descends from this representation.
- Correctness
- The rigid-invariance is a local linearization, so very large rotations can distort detail and need the implicit per-vertex transform fit; results depend on clean one-ring neighbourhoods and a well-conditioned Laplacian, and the cited examples are interactive single-mesh edits rather than full articulated rigs.
- Clarity
- Readable and example-driven; a first pass conveys the encode-relative-to-neighbours idea, a second pass earns the rotation-invariance derivation and the least-squares system.
- How to read it
- First pass: the intrinsic Laplacian encoding and why detail is preserved. Second pass: the rigid/scale invariance trick and the sparse least-squares formulation, especially the detail-transfer-between-meshes section if rigging detail transfer is your interest.
Builds on
Nothing in the archive, this is a starting point.
Built upon by
Related work
- Direct Manipulation of Free-Form Deformations 1992 / SIGGRAPH
- Surface Based Motion Retargeting by Preserving Spatial Relationship 2018 / MIG
- Deformation Transfer for Triangle Meshes 2004 / SIGGRAPH
- Mesh-Based Inverse Kinematics 2005 / SIGGRAPH
Keywords
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