Bijective Deformations in Rn via Integral Curve Coordinates




Huynh, Lisa

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Shape deformation is a widely studied problem in computer graphics, with applications to animation, physical simulation, parameterization, interactive modeling, and image editing. In one instance of this problem, a \cage" (polygon in 2D and polyhedra in 3D) is created around a shape or image region. As the vertices of the cage are moved, the interior deforms. The cage may be identical to the shape's boundary, which has one fewer dimension than the shape itself, and is typically more convenient, as the cage may be simpler (fewer vertices) or be free of undesirable properties (such as a non-manifold mesh or high topological genus). We introduce Integral Curve Coordinates and use them to create shape deformations that are bijective, given a bijective deformation of the shape's boundary or an enclosing cage. Our approach can be applied to shapes in any dimension, provided that the boundary of the shape (or cage) is topologically equivalent to an n-sphere. Integral Curve Coordinates identify each point in a domain with a point along an integral curve of the gradient of a function f, where f has exactly one critical point, a maximum, in the domain, and the gradient of f on the boundary points inward. By identifying every point inside a domain (shape) with a point on its boundary, Integral Curve Coordinates provide a natural mapping from one domain to another given a mapping of the boundary. We evaluate our deformation approach in 2D. Our algorithm is based on the following three steps: (i) choosing a maximum via a grass re algorithm; (ii) computing a suitable function f on a discrete grid via a construct called the cousin tree; (iii) tracing integral curves. We conclude with a discussion of limitations arising from piecewise linear interpolation and discretization to a grid.



Coordinates, Gradient, Deformation, Integral curve, Bijection