Non‐Isometric Shape Matching via Functional Maps on Landmark‐Adapted Bases

Abstract

We propose a principled approach for non-isometric landmark-preserving non-rigid shape matching. Our method is based on the functional map framework, but rather than promoting isometries we focus on near-conformal maps that preserve landmarks exactly. We achieve this, first, by introducing a novel landmark-adapted basis using an intrinsic Dirichlet-Steklov eigenproblem. Second, we establish the functional decomposition of conformal maps expressed in this basis. Finally, we formulate a conformally-invariant energy that promotes high-quality landmark-preserving maps, and show how it can be optimized via a variant of the recently proposed ZoomOut method that we extend to our setting. Our method is descriptor-free, efficient and robust to significant mesh variability. We evaluate our approach on a range of benchmark datasets and demonstrate state-of-the-art performance on non-isometric benchmarks and near state-of-the-art performance on isometric ones.

Extracting a functional representation from a dictionary for non-rigid shape matching

“…ZoomOut [26] is an iterative method for extending the size of an initial functional map estimated with few eigenvectors, while improving the quality of the estimated correspondence. Many recent algorithms build upon the ZoomOut procedure [15,39,35,40,36], which alternates conversions to point-wise maps and back to functional maps of increased size.…”

Attention And Positional Encoding Are (Almost) All You Need For Shape Matching

“…Another valuable solution is the functional map framework [OBCS*12], which focuses on the functional mapping induced by the point‐to‐point correspondence. Several variants of this framework have been proposed, adding regularizers [OCB*17,NO17,NMR*18,RPWO18,DCMO22], addressing clutter and partialities [RCB*17, CRM*16], or exploiting the relation between pointwise and functional mapping [MRR*19, HRWO20, RMOW20,PRM*21,RMWO21,PKO22].…”