An optimization approach for extracting and encoding consistent maps in a shape collection

Abstract

We introduce a novel approach for computing high quality point-topoint maps among a collection of related shapes. The proposed approach takes as input a sparse set of imperfect initial maps between pairs of shapes and builds a compact data structure which implicitly encodes an improved set of maps between all pairs of shapes. These maps align well with point correspondences selected from initial maps; they map neighboring points to neighboring points; and they provide cycle-consistency, so that map compositions along cycles approximate the identity map.The proposed approach is motivated by the fact that a complete set of maps between all pairs of shapes that admits nearly perfect cycleconsistency are highly redundant and can be represented by compositions of maps through a single base shape. In general, multiple base shapes are needed to adequately cover a diverse collection. Our algorithm sequentially extracts such a small collection of base shapes and creates correspondences from each of these base shapes to all other shapes. These correspondences are found by global optimization on candidate correspondences obtained by diffusing initial maps. These are then used to create a compact graphical data structure from which globally optimal cycle-consistent maps can be extracted using simple graph algorithms.Experimental results on benchmark datasets show that the proposed approach yields significantly better results than state-of-theart data-driven shape matching methods.

Learning part-based templates from large collections of 3D shapes

“…These approaches are sensitive to multi-modal variations in collections that contain clusters that are not connected by a smooth path of rigid alignments (e.g., Figure 10: Correspondence benchmark. This figure demonstrates quality of correspondences produced by our method (red,magenta) relative to prior work: Huang et al [2012] (black,gray), Kim et al [2012a] (blue). Each point on a curve indicates fraction of correctly predicted correspondences for a given Euclidean error threshold.…”

“…We demonstrate results for our method trained on all data (red), our method only trained on models that have ground truth correspondences (magenta). We further compare the method proposed by Huang et al [2012] trained on ground truth models (black), and on the largest dataset it could handle (gray): it successfully analyzed all bikes, a subset of 1000 planes, a subset of 1000 seats, and crashed for larger subsets of helicopter dataset. We also execute the method of Kim et al [2012a] (blue), which is not able to handle collections much larger than one hundred models.…”

“…Many researchers have investigated how to compute consistent alignments and correspondences within surface pairs [van Kaick et al 2011b] or more recently on collections of 3D models [Nguyen et al 2011;Huang et al 2012;Kim et al 2012a]. For example, Kim et al [2012a] propose a diffusionbased method to compute "fuzzy correspondences" between every pair of surface points in a collection.…”

Graduated Consistency-Regularized Optimization for Multi-graph Matching

“…However, whatever path order is chosen, a single error in the corresponding sequence will typically create a large number of erroneous pairwise matches. To fully explore the information across the whole graph pool G, it is perhaps more robust to compute all or part of pairwise matching results independently, and then leave the calculation of the final solution to several post-steps [36,24]. Compared with computing a pairwise matching chain like G 1 →G 2 →.…”

“….→G N , such an exhaustive matching strategy would cause the problem of redundancy, or in another word, inconsistency, as the node mapping between two graphs cannot be uniquely determined by different pairwise matching paths. Formally, we use the term "cycle-consistency" as introduced and described in [48,26,24], i.e., that composition of correspondences between two graphs should be independent of the connecting path chosen.…”