Bounded Biharmonic Weights For Real-Time Deformation

Changing an object's shape is a basic operation in computer graphics, necessary for transforming raster images, vector graphics, geometric models, and animated characters. The fastest approaches for such object deformation involve linearly blending a small number of given affine transformations, typically each associated with bones of an internal skeleton, vertices of an enclosing cage, or a collection of loose point handles. Unfortunately, linear blending schemes are not always easy to use because they may require manually painting influence weights or modeling closed polyhedral cages around the input object. Our goal is to make the design and control of deformations simpler by allowing the user to work freely with the most convenient combination of handle types. We develop linear blending weights that produce smooth and intuitive deformations for points, bones, and cages of arbitrary topology. Our weights, called bounded biharmonic weights, minimize the Laplacian energy subject to bound constraints. Doing so spreads the influences of the handles in a shape-aware and localized manner, even for objects with complex and concave boundaries. The variational weight optimization also makes it possible to customize the weights so that they preserve the shape of specified essential object features. We demonstrate successful use of our blending weights for real-time deformation of 2D and 3D shapes.