Fully Spectral Partial Shape Matching

Abstract

Figure 1: Examples of dense correspondence computed with our method on real 3D scans (left pair, the areas of contact are glued together), missing parts (middle) and strong topological artifacts (right, touching parts are glued together). Corresponding points are encoded with the same color. AbstractWe propose an efficient procedure for calculating partial dense intrinsic correspondence between deformable shapes performed entirely in the spectral domain. Our technique relies on the recently introduced partial functional maps formalism and on the joint approximate diagonalization (JAD) of the Laplace-Beltrami operators previously introduced for matching non-isometric shapes. We show that a variant of the JAD problem with an appropriately modified coupling term (surprisingly) allows to construct quasi-harmonic bases localized on the latent corresponding parts. This circumvents the need to explicitly compute the unknown parts by means of the cumbersome alternating minimization used in the previous approaches, and allows performing all the calculations in the spectral domain with constant complexity independent of the number of shape vertices. We provide an extensive evaluation of the proposed technique on standard non-rigid correspondence benchmarks and show state-of-the-art performance in various settings, including partiality and the presence of topological noise.

Improved Functional Mappings via Product Preservation

“…Since the pull‐back of functions is linear and since functional spaces can be endowed with a multi‐scale basis, this leads to a representation of correspondences as moderately‐sized matrices, which can be directly manipulated and optimized for. Although initially introduced as a tool for shape matching, functional maps have been used for relating tangent vector fields [ABCCO13], extending the Generalized Multi‐Dimensional Scaling to the spectral domain [AK13], computing maps between symmetric [OMPG13] and partial [RCB*17, LRB*16, LRBB17] shapes, coupled bases [KBB*13] and even consistent quadrangulation [ACBCO17] among others. Most recently, functional maps were integrated as differentiable layers into intrinsic deep learning architectures [LRR*17].…”

Structured Regularization of Functional Map Computations

“…As observed by several works in this domain, [KBB*13, ROA*13, RCB*17, BDK17] many natural properties on the underlying pointwise correspondences can be expressed as objectives on functional maps. Most notably, this includes: orthonormality of functional maps, which corresponds to the local area‐preservation nature of pointwise correspondences [OBCS*12, KBB*13, ROA*13]; preservation of inner products of gradients of functions, which corresponds to conformal maps [ROA*13, BDK17, WLZT18]; preservation of pointwise products of functions, which corresponds to functional maps arising from point‐to‐point correspondences [NO17, NMR*18]; slanted diagonal structure of functional maps, which corresponds to correspondences between partial shapes [RCB*17, LRBB17]. Similarly, several other regularizers have been proposed, including using robust norms and matrix completion techniques [KBB*13, KBBV15], exploiting the relation between functional maps in different directions [ERGB16], the map adjoint [HO17], and powerful cycle‐consistency constraints [HWG14] in the context of shape collections, among many others.…”

“…Among all of these, perhaps the most widely‐used building block for regularizing functional maps, introduced in [OBCS*12] and extended in follow‐up works, including [WHG13, RCB*17, LRB*16, LRBB17], is based on the commutativity with the Laplacian operators, which implies a diagonal (or slanted diagonal in the case of partial correspondence) structure for functional maps. To promote this structure, the most common method (see also Chapter 2.4 in [OCB*17]) consists in adding an energy to the functional map estimation pipeline, which penalizes the failure of the unknown functional map to commute with the Laplace‐Beltrami operators on the source and target shapes.…”

Deblurring and Denoising of Maps between Shapes

“…[KBBV15, PBB*13, RRBW*14, SK14, KBB*13, ERGB16]) and new consistent descriptors have been suggested [COC14, GSTOG16], these methods did not adjust the point‐wise recovery method. Recently, this framework was extended to computing partial correspondence [RCB*16, LRB*16, LRBB17], and to computing correspondences in shape collections [SBC14, HWG14, KGB16]. In addition, functional maps have been used for analysis and visualization of maps [OBCCG13, ROA*13], and image segmentation [WHG13].…”