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Title: Mesh editing with affine-invariant laplacian coordinates
Authors: Fu, Hongbo
Tai, Chiew-Lan
Keywords: Laplacian coordinate
Affine-invariant transformation
Mesh deformation
Pose-independent mesh merging
Pose-dependent mesh merging
Issue Date: Jan-2005
Series/Report no.: Computer Science Technical Report ; HKUST-CS05-01
Abstract: Differential coordinates as an intrinsic surface representation capture geometric details of surface. However, differential coordinates alone cannot achieve desirable editing results, because they are not affine invariant. In this paper, we present a novel method that makes the Laplacian coordinates completely affine-invariant during editing. For each vertex of a surface to be edited, we compute the Laplacian coordinate and implicitly define a local affine transformation that is dependent on the unknown edited vertices. During editing, both the resulting surface and the implicit local affine transformations are solved simultaneously through a constrained optimization. The underlying mathematics of our method is a set of linear Partial Differential Equations (PDEs) with a generalized boundary condition. The main computation involved comes from factorizing the resulting sparse system of linear equations, which is performed only once. After that, back substitutions are executed to interactively respond to user manipulations. We propose a new editing technique, called pose-independent merging, to demonstrate the advantages of the affine-invariant Laplacian coordinates. In the the same framework, large-scale mesh deformation and pose-dependent mesh merging are also presented.
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