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Bounded Biharmonic Weights For Real-Time Deformation

bounded biharmonic weights, illustration

Our deformation method supports arbitrary combinations of control handles, such as points, bones, or cages. Throughout the paper, colored frames illustrate linear transformations specified at point handles.


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.

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1. Introduction

Interactive deformation is the task of assisting the user to alter an object's shape. In the case of 2D cartoon deformation, we could ask the user to manually reposition each pixel of the image, but this is unnecessarily tedious. The space of coherent configurations of the 2D shape is much smaller than the space of all possible positions for every pixel of the image. Hence, we would rather the user provide only a few, high-level constraints like "open the mouth," "enlarge the belly," or "bend the tail" (Figure 1). The rest of the shape should immediately deform in an intuitive manner. We may interface such high-level constraints to the user with handle structures, like skeletons composed of rigid bones, enclosing cages, and selected regions or points.


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