Continuous and orientation-preserving correspondences via functional maps

Abstract

Fig. 1. Given a pair of shapes, our method produces a point-wise map that is orientation-preserving as well as approximately continuous and bijective. Here we show the maps produced by different methods via texture transfer: BIM [Kim et al. 2011] has a large distortion on the face and the left hand; functional maps with ICP [Ovsjanikov et al. 2012] and PMF with the Gauss kernel [Vestner et al. 2017b] give a map that is flipped left to right; for PMF with the heat kernel [Vestner et al. 2017a], the orientation in the torso region is reversed; The map produced by our method preserves the orientation consistently and has lower overall error when compared to the ground-truth.We propose a method for efficiently computing orientation-preserving and approximately continuous correspondences between non-rigid shapes, using the functional maps framework. We first show how orientation preservation can be formulated directly in the functional (spectral) domain without using landmark or region correspondences and without relying on external symmetry information. This allows us to obtain functional maps that promote orientation preservation, even when using descriptors, that are invariant to orientation changes. We then show how higher quality, approximately continuous and bijective pointwise correspondences can be obtained from initial functional maps by introducing a novel refinement technique that aims to simultaneously improve the maps both in the spectral and spatial domains. This leads to a general pipeline for computing correspondences between shapes that results in high-quality maps, while admitting an efficient optimization scheme. We show through extensive evaluation that our approach improves upon state-of-the-art results on challenging isometric and non-isometric correspondence benchmarks according to both measures of continuity and coverage as well as producing semantically meaningful correspondences as measured by the distance to ground truth maps.

Structured Regularization of Functional Map Computations

“…After initialization, we use ICP [OBCS*12] to refine the computed functional maps. BCICP refinement. After initialization, we use the recently proposed BCICP algorithm [RPWO18] to refine the computed functional and pointwise maps, using the open‐source implementation and default parameters provided by the authors.…”

“…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. More recently constraints on functional maps have been introduced to promote continuity of the recovered pointwise correspondence [PSO18], maps between curves defined on shapes [GBKS18], kernel‐based techniques aimed at extracting more information from given descriptor constraints [WGBS18], and an approach for incorporating orientation information into the functional map infererence pipeline [RPWO18] among others.…”

“…where E desc is the descriptor‐preserving term defined in Sec. 3, E mult is the descriptor‐commutativity term defined as introduced in [NO17], E orient is the orientation‐preserving term introduced in [RPWO18]. E resolvent is the our new term introduced in Eq.…”

“…We also compare to the results obtained directly using the code and the dataset provided by the authors of [RPWO18] (see Table ). The reproduced results that are reported in Table are consistent with the values reported in the Table – (“WKS + directOp + BCICP”) in the paper, and Table –3 (“WKS + directOp”) in the supplementary materials of [RPWO18]. Note that [RPWO18] split the dataset into isometric and non‐isometric categories, and here we report the results altogether.…”

“…Recall that the total energy to optimize is E ( C 12 ) = α 1 E desc + α 2 E mult + α 3 E orient + α 4 E mask . We would like to highlight the fact that the parameters in [RPWO18] are not set in the same way as in our work. In particular, in [RPWO18] the standard Laplacian mask is used, and the weight α i are set to fixed values α i *.…”