Traditionally, shape transformation using implicit functions is performed in two distinct steps: 1) creating two implicit functions, and 2) interpolating between these two functions. We present a new shape transformation method that combines these two tasks into a single step. We create a transformation between two N-dimensional objects by casting this as a scattered data interpolation problem in N + 1 dimensions. For the case of 2D shapes, we place all of our data constraints within two planes, one for each shape. These planes are placed parallel to one another in 3D. Zero-valued constraints specify the locations of shape boundaries and positive-valued constraints are placed along the normal direction in towards the center of the shape. We then invoke a variational interpolation technique (the 3D generalization of thin-plate interpolation), and this yields a single implicit function in 3D. Intermediate shapes are simply the zero-valued contours of 2D slices through this 3D function. Shape transformation between 3D shapes can be performed similarly by solving a 4D interpolation problem. To our knowledge, ours is the first shape transformation method to unify the tasks of implicit function creation and interpolation. The transformations produced by this method appear smooth and natural, even between objects of differing topologies. If desired, one or more additional shapes may be introduced that influence the intermediate shapes in a sequence. Our method can also reconstruct surfaces from multiple slices that are not restricted to being parallel to one another.

翻译：传统上，使用隐式函数进行形状变换需分两步完成：1）构建两个隐式函数，2）对这两个函数进行插值。我们提出一种新的形状变换方法，将这两个任务合并为单一步骤。通过将变换问题转化为N+1维空间中的散乱数据插值问题，我们实现了两个N维对象之间的变换。对于二维形状，我们将所有数据约束置于两个平面内（每个形状对应一个平面），这些平面在三维空间中相互平行放置。零值约束指定形状边界的位置，正数值约束则沿法线方向朝向形状中心放置。随后我们采用变分插值技术（薄板插值的三维泛化），由此生成三维空间中的单一隐式函数。中间形状即为该三维函数二维切片的零值轮廓。三维形状间的变换可通过求解四维插值问题类似实现。据我们所知，这是首个将隐式函数创建与插值任务统一起来的形状变换方法。该方法生成的变换过程平滑自然，即使对于拓扑结构不同的对象也是如此。如有需要，可引入一个或多个额外形状来影响序列中的中间形状。我们的方法还能从多个不限于相互平行的切片中重建表面。