|Title:||Nonlinear Interpolation between Slices
|Abstract:||The topic of interpolation between slices has been an intriguing problem for many years, as it offers means to visualize and investigate a 3D object given only by its level sets. A slice consists of multiple non-intersecting simple contours, each defined by a cyclic list of vertices. An interpolation solution matches between a number of such slices (two or more at a time), providing means to create a closed surface connecting these slices, or the equivalent morph from one slice to another. Most interpolation methods either interpolate the surface between two slices based on these slices alone, which can cause abrupt changes in the surface, or solve the problem as a global optimization problem, which is computationally expensive.
We offer a method to incorporate the influence of more than two slices at each point in the reconstructed surface. We investigate the flow of the surface from one slice to the next by matching vertices and extracting differential geometric quantities from that matching. Interpolating these quantities with surface patches then allows a nonlinear reconstruction which produces a free-form, non-intersecting surface. No assumptions are made about the input, such as on the number of contours in each slice, their geometric similarity, their nesting hierarchy, etc., and the proposed algorithm handles automatically all branching and hierarchical structures. Unlike polyhedral-reconstruction methods, the resulting surface is smooth, and it does not require further subdivision measures.
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