Vector Field Map Representation for Near Conformal Surface Correspondence

Abstract

Based on a new spectral vector field analysis on triangle meshes, we construct a compact representation for near conformal mesh surface correspondences. Generalizing the functional map representation, our representation uses the map between the low‐frequency tangent vector fields induced by the correspondence. While our representation is as efficient, it is also capable of handling a more generic class of correspondence inference. We also formulate the vector field preservation constraints and regularization terms for correspondence inference, with function preservation treated as a special case. A number of important vector field–related constraints can be implicitly enforced in our representation, including the commutativity of the mapping with the usual gradient, curl, divergence operators or angle preservation under near conformal correspondence. For function transfer between shapes, the preservation of function values on landmarks can be strictly enforced through our gradient domain representation, enabling transfer across different topologies. With the vector field map representation, a novel class of constraints can be specified for the alignment of designed or computed vector field pairs. We demonstrate the advantages of the vector field map representation in tests on conformal datasets and near‐isometric datasets.

Unsupervised Deep Learning for Structured Shape Matching

“…Moreover, as observed by several works in this domain [30,40,21,36,9], many natural properties on the underlying pointwise correspondences can be expressed as objectives on functional maps. This includes orthonormality of functional maps, which corresponds to the local area-preservation nature of pointwise correspondences [30,21,40]; commutativity with the Laplacian operators, which corresponds to intrinsic isometries [30], preservation of inner products of gradients of functions, which corresponds to conformal maps [40,9,50]; preservation of point-wise products of functions, which corresponds to functional maps arising from point-to-point correspondences [29,28]; and slanted diagonal structure of functional map in the context of partial shapes [36,24] among others.…”

Discrete Optimization for Shape Matching

“…While computing a functional map reduces to solving a least‐squares system, the conversion from a functional map to a point‐wise map is not trivial and can lead to inaccuracy and noise [RMC15, EBC17]. To improve accuracy, several desirable map attributes have been promoted via regularizers for the functional map estimation first using geometric insights [ERGB16,RCB∗17,NO17,LRBB17,BDK17,WLZT18,RPWO18, WGBS18,GBKS18,NMR∗18,SVBC19], and more recently using learning‐based techniques [LRR∗17, HLR∗19, RSO19]. Nevertheless, despite significant progress, the reliance on descriptors and decoupling of continuous optimization and pointwise map conversion remains common to all existing methods.…”

Fast Sinkhorn Filters: Using Matrix Scaling for Non-Rigid Shape Correspondence with Functional Maps

“…In addition to the convenience of the representation itself, it has been observed by several works in this domain that many natural properties on the underlying pointwise correspondences can be expressed as objectives on functional maps [16,36,32,6]. For example, orthonormal functional map matrices correspond to locally volume preserving maps [28,16,36], near isometries must result in functional maps that commute with the Laplacian [28,46,32,20,19], while conformal maps must preserve certain functional inner products [36,6,47].…”