MapTree

Abstract

In this paper we propose an approach for computing multiple high-quality near-isometric dense correspondences between a pair of 3D shapes. Our method is fully automatic and does not rely on user-provided landmarks or descriptors. This allows us to analyze the full space of maps and extract multiple diverse and accurate solutions, rather than optimizing for a single optimal correspondence as done in most previous approaches. To achieve this, we propose a compact tree structure based on the spectral map representation for encoding and enumerating possible rough initializations, and a novel efficient approach for refining them to dense pointwise maps. This leads to a new method capable of both producing multiple high-quality correspondences across shapes and revealing the symmetry structure of a shape without a priori information. In addition, we demonstrate through extensive experiments that our method is robust and results in more accurate correspondences than state-of-the-art for shape matching and symmetry detection.

Scalable and Efficient Functional Map Computations on Dense Meshes

“…We base our approach on the ZoomOut variant of the functional map famework [MRR * 19]. However, our constructions can be easily extended to other recent functional maps methods, e.g., [RMOW20, RMWO21], which share the same general algorithmic structure. Specifically, ZoomOut and related methods are based on two main building blocks: computing the eigenfunctions of the Laplace‐Beltrami operator first, and then iterating between updating the point‐to‐point and functional maps.…”

“…Despite the simplicity of the original approach, its performance is strongly dependent on accurate descriptors and hyper‐parameter tuning. As a result, this basic strategy has been extended significantly in many follow‐up works, based both on geometric insights [KBB * 12, AK13, OMPG13, BDK17, ERGB16], improved optimization strategies [KGB16, NO17, RMOW20, RMWO21], and richer correspondence models going beyond isometries across complete shapes, [RCB * 17, ROA * 13, LRBB17], among others.…”

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.…”

Complex Functional Maps: A Conformal Link Between Tangent Bundles

“…In Figure 7 we provide qualitative results with two shape pairs with the same source, and report five maps per pair: the ground-truth, bijective ZoomOut [RMOW20], and bijective isometric ZoomOut [RMWO21], as well as their version with our modification. For both shape pairs and both methods, the original algorithms are affected by the left-right symmetry and converge to discontinuous maps.…”

“…The idea behind bijective ZoomOut is to optimize for maps in both directions (from M to N and from N to M). The energy to minimize is the so-called bijective energy E bi j = C MN C NM − I 2 F + C NM C MN − I 2 F , and [RMOW20] proposes to optimize it with the following steps : Remarking that C NM =…”