Functional map networks for analyzing and exploring large shape collections

Abstract

Consistent basis functionsCo-segmentation Map-based exploration Figure 1: We introduce a computational framework for constructing functional map networks that can be used to capture structural similarities within heterogeneous shape collections. The shared structure emerges in the network as consistent basis functions across the shape collection. This network representation enables many joint shape analysis tasks, including co-segmentation and shape exploration. AbstractThe construction of networks of maps among shapes in a collection enables a variety of applications in data-driven geometry processing. A key task in network construction is to make the maps consistent with each other. This consistency constraint, when properly defined, leads not only to a concise representation of such networks, but more importantly, it serves as a strong regularizer for correcting and improving noisy initial maps computed between pairs of shapes in isolation. Up-to-now, however, the consistency constraint has only been fully formulated for point-based maps or for shape collections that are fully similar.In this paper, we introduce a framework for computing consistent functional maps within heterogeneous shape collections. In such collections not all shapes share the same structure -different types of shared structure may be present within different (but possibly overlapping) sub-collections. Unlike point-based maps, functional maps can encode similarities at multiple levels of detail (points or parts), and thus are particularly suitable for coping with such diversity within a shape collection. We show how to rigorously formulate the consistency constraint in the functional map setting. The formulation leads to a powerful tool for computing consistent functional maps, and also for discovering shared structures, such as meaningful shape parts. We also show how to adapt the procedure for handling very large-scale shape collections. Experimental results on benchmark datasets show that the proposed framework significantly improves upon state-of-the-art data-driven techniques. We demonstrate the usefulness of the framework in shape cosegmentation and various shape exploration tasks.

Data-driven shape analysis and processing

“…Experimentally, it generates comparable results with supervised method [KHS10] on the Princeton segmentation benchmark. Recently, Huang et al [HWG14] formulates the same idea under the framework of functional maps [OBCS * 12] and gain improved segmentation quality and computational efficiency.…”

“…This is true for maps that are represented as transformations (e.g., rotations and rigid/affine transformations), or full point-wise maps that can be described as permutation matrices). We can easily modify the constraint to handle partial maps, namely each point, when transformed along a loop, either disappears or goes back to the original point (See [HWG14] for details).…”

“…In this matrix representation, the cycle-consistency constraint can be equivalently described as simple properties of X, i.e., depending on the types of maps, X is either positive semidefinite or low-rank (c.f. [HG13,HWG14]). In addition, we may view the initial pair-wise maps as noisy measurements of the entries of X.…”

“…For example, one can embed shapes in a low-dimensional space based on shape differences, or use shape difference to interpolate variations by showing "intermediate" shapes between two re- gions of interest. To extend these technique to man-made objects, Huang et al [HWG14] construct consistent functional basis for shape collections that exhibit large geometric and topological variability. They show that resulting consistent maps can capture discrete topological variability, such as variance in number of bars in the back of a chair.…”

Supervised Descriptor Learning for Non-Rigid Shape Matching

“…This gives us a way to not only obtain better functional correspondences, but also to associate a confidence value to the different parts of the mappings. Our approach is also quite general since it can be used as a preprocessing step of other methods using functional maps [24,14,3] in order to improve the quality of the results and help to handle difficult deformation. Note that in this paper we focus on the shape matching problem which is the most developed application of the functional maps.…”