Bi-directional Shape Correspondences (BSC): A Novel Technique for 2-d Shape Warping in Quadratic Time?

TL;DR

This paper introduces Bidirectional Shape Correspondence (BSC), a new method for 2D shape warping that allows many-to-many point matching in quadratic time, improving over existing one-to-one and cubic-time algorithms.

Contribution

The paper presents a simple, efficient approach for many-to-many shape correspondence in quadratic time, addressing limitations of previous methods like SC and dynamic programming.

Findings

Enables many-to-many shape matching in quadratic time.

Uses data clustering to handle outliers effectively.

Provides a more flexible shape warping technique.

Abstract

We propose Bidirectional Shape Correspondence (BSC) as a possible improvement on the famous shape contexts (SC) framework. Our proposals derive from the observation that the SC framework enforces a one-to-one correspondence between sample points, and that this leads to two possible drawbacks. First, this denies the framework the opportunity to effect advantageous many-to-many matching between points on the two shapes being compared. Second, this calls for the Hungarian algorithm which unfortunately usurps cubic time. While the dynamic-space-warping dynamic programming algorithm has provided a standard solution to the first problem above, it demands quintic time for general multi-contour shapes, and w times quadratic time for the special case of single-contour shapes, even after an heuristic search window of width w has been chosen. Therefore, in this work, we propose a simple method for computing "many-to-many" correspondences for the class of all 2-d shapes in quadratic time. Our approach is to explicitly let each point on the first shape choose a best match on the second shape, and vice versa. Along the way, we also propose the use of data-clustering techniques for dealing with the outliers problem, and, from another viewpoint, it turns out that this clustering can be seen as an autonomous, rather than pre-computed, sampling of shape boundary.