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Biharmonic Coordinates
Generalizes harmonic coordinates by interpolating boundary derivatives, yielding smoother 2D cage deformations with fewer global distortions.
Abstract
Barycentric coordinates are an established mathematical tool in computer graphics and geometry processing, providing a convenient way of interpolating scalar or vector data from the boundary of a planar domain to its interior. Many different recipes for barycentric coordinates exist, some offering the convenience of a closed‐form expression, some providing other desirable properties at the expense of longer computation times. For example, harmonic coordinates, which are solutions to the Laplace equation, provide a long list of desirable properties (making them suitable for a wide range of applications), but lack a closed‐form expression. We derive a new type of barycentric coordinates based on solutions to the biharmonic equation. These coordinates can be considered a natural generalization of harmonic coordinates, with the additional ability to interpolate boundary derivative data. We provide an efficient and accurate way to numerically compute the biharmonic coordinates and demonstrate their advantages over existing schemes. We show that biharmonic coordinates are especially appealing for (but not limited to) 2D shape and image deformation and have clear advantages over existing deformation methods.
How to read this
- Category
- Method: a new family of barycentric (cage) coordinates
- Contributions
- Derives biharmonic coordinates as solutions to the biharmonic equation, a natural generalization of harmonic coordinates
- Adds the ability to interpolate boundary derivative data, not just boundary values
- Provides an efficient, accurate numerical scheme to compute them and shows advantages for 2D shape and image deformation
- Context
- Generalizes harmonic coordinates and relates to the bounded-biharmonic-weights work of Jacobson et al., extending the barycentric-coordinate toolkit used for cage-based deformation.Builds on: Bounded Biharmonic Weights for Real-Time Deformation
- Correctness
- Demonstrated primarily on 2D shape and image deformation; like harmonic coordinates the scheme lacks a closed-form expression and requires numerical solution, and the abstract scopes the benefits to (though not strictly limited to) the planar case, so 3D generality should not be assumed from this read.
- Clarity
- Mathematically dense but well-motivated; a first pass conveys what the coordinates buy you, a second pass is needed to follow the biharmonic derivation.
- How to read it
- First pass to understand why interpolating boundary derivatives matters; budget a careful second/third pass on the derivation and numerical computation if you need to implement or compare against harmonic coordinates.
Builds on
Built upon by
Nothing yet.
Related work
- Somigliana Coordinates: an Elasticity-Derived Approach for Cage Deformation 2023 / SIGGRAPH
- Mean Value Coordinates for Closed Triangular Meshes 2005 / SIGGRAPH
- Green Coordinates 2008 / SIGGRAPH
- Green Coordinates for Triquad Cages in 3D 2022 / SIGGRAPH Asia
Keywords
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