Fmapopt

Fmapopt, short for functional map optimization, is a computational framework used to compute and refine correspondences between two geometric domains, typically two triangle meshes or manifolds. It sits within the functional maps approach to shape analysis, where a correspondence is represented not as a direct point-to-point map but as a linear operator acting on function spaces defined on each domain. The central object is a functional map matrix C that encodes how basis functions on one shape push forward to functions on the other.

Formulation and objective liquids: The usual setup involves spectral bases on the shapes, such as Laplace-Beltrami eigenfunctions, denoted Φ_A and Φ_B, and possibly descriptor fields D_A and D_B. Fmapopt seeks C that minimizes a data fidelity term, for example ensuring that C maps basis functions on shape A to their counterparts on shape B, and that descriptors are preserved under the map. Regularizers are commonly added to promote low distortion, sparsity, and stability, along with geometric constraints such as approximate orthogonality or commutativity with the Laplacian operators. A typical objective takes the form of a regularized least-squares problem over C, balancing data fidelity with smoothness and geometric consistency.

Algorithmic approach: The method proceeds by solving for C using convex or nonconvex optimization, often with constraints or penalties that encourage a compact, well-conditioned map. Once C is estimated, a discrete point-to-point correspondence can be recovered by transferring points via the basis coefficients or by nearest-neighbor projection in a descriptor space. Variants may incorporate partial or semi-supervised information to improve accuracy.

Applications and scope: Fmapopt is used in 3D shape matching, texture transfer, morphing, and animation retargeting, as well as deformation analysis in computer graphics and geometric processing. It supports unsupervised and semi-supervised workflows and is adaptable to varying mesh resolutions and partial data.