Tetrahedral Meshes Interpolation

We present a method for volumetric shape interpolation with unique shape preserving features. 

The input to our algorithm are two or more 3-manifolds, immersed in R3 and discretized as tetrahedral meshes with shared connectivity. The output is a continuum of shapes that naturally blends the input shapes,

while striving to preserve the geometric character of the input. The basis of our approach relies on the fact that the space of metrics with bounded isometric and angular distortion is convex. The convexity of the space implies that a linear blend of the (squared) edge lengths of the input tetrahedral meshes is a simple yet powerful and natural choice. Linearly blending metric tensors result in a new metric which is, in general, not  flat, and cannot be immersed into three dimensional space. We further design an efficient optimization procedures to completely flatten the metric. The flattening procedure strives to preserve the low distortion exhibited in the blended metric phase, while guaranteeing the validity of the metric, resulting in a locally injective map with low distortion. 

Our method leads to volumetric interpolation with superb quality, demonstrating significant improvement over the state-of-the-art methods.

* M.Sc. research supervised by Prof. Weber Ofir