Discrete Optimization for Shape Matching

Abstract

We propose a novel discrete solver for optimizing functional map‐based energies, including descriptor preservation and promoting structural properties such as area‐preservation, bijectivity and Laplacian commutativity among others. Unlike the commonly‐used continuous optimization methods, our approach enforces the functional map to be associated with a pointwise correspondence as a hard constraint, which provides a stronger link between optimized properties of functional and point‐to‐point maps. Under this hard constraint, our solver obtains functional maps with lower energy values compared to the standard continuous strategies. Perhaps more importantly, the recovered pointwise maps from our discrete solver preserve the optimized for functional properties and are thus of higher overall quality. We demonstrate the advantages of our discrete solver on a range of energies and shape categories, compared to existing techniques for promoting pointwise maps within the functional map framework. Finally, with this solver in hand, we introduce a novel Effective Functional Map Refinement (EFMR) method which achieves the state‐of‐the‐art accuracy on the SHREC'19 benchmark.

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

“…In [MRR*19] it is observed that a particular case of such problems can be efficiently solved by considering the two maps as being independent variables. This observation was recently extended to a wide range of energies in [RMWO21]. Following this line of work we will move all of the difficult optimization on the side of the vertex-to-vertex map and use Equation (10) to restore the relationship between the maps.…”

“…Specifically, the first term promotes $$F_{{\mathcal {MN}}}$$ being a proper functional map (i.e. the pullback of a vertex‐to‐vertex map), as recently defined in [RMWO21], and the second promotes the invertibility of $$F_{{\mathcal {MN}}}$$ [ERGB16].…”

“…Our main tool is the following result, standard in functional maps literature [EB17, RMWO21], which allows one to reduce optimization problems of a certain form to nearest neighbor searches. Lemma Let A be a symmetric positive‐definite matrix inducing the matrix norm $$\Vert M\Vert ^2_A = \text{Tr}(M^T A M)$$.…”

“…Proof See [EB17] for a proof of a special case and [RMWO21] (Lemma 4.1) for the general statement.$$\Box$$…”

“…Our main tool is the following result, standard in functional maps literature [EB17,RMWO21], which allows one to reduce optimization problems of a certain form to nearest neighbor searches.…”