Abstract:
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.
