Faster graphical models for point-pattern matching

Abstract

It has been shown that isometric matching problems can be solved exactly in polynomial time, by means of a Junction Tree with small maximal clique size. Recently, an iterative algorithm was presented which converges to the same solution an order of magnitude faster. Here, we build on both of these ideas to produce an algorithm with the same asymptotic running time as the iterative solution, but which requires only a single iteration of belief propagation. Thus our algorithm is much faster in practice, while maintaining similar error rates.

Unified graph matching in Euclidean spaces

“…It is shown in [14,15] that in the case of exact isometric matching, one need not consider all edges in V 2 , but only a subset of edges that constitute a 'globally rigid' graph: by definition, preserving the distances of the edges in such a subgraph implies that the distances between all edges in the complete graph will be preserved. Thus, for a globally rigid subgraph R = (V, E R ) of G, we need to solve…”

“…2) can be modelled as inference in a tractable graphical model (whose nodes and assignments correspond to points in V and V respectively); [14] reports running times and memory requirements of O(|V||V | n+1 ), where n is the number of dimensions. We use a similar topology which replaces the 'ring' structure from [14] with a 'junction-tree', which has the advantage that only a single iteration is required for exact inference [15]. The topology of this graph is shown in Figure 2 (for n = 2), and it is rigid by Theorem 2.1.…”

Multi-modal optical microscopy image analysis and matching techniques for spatially encoded bead-based microarrays

“…While this graph does not have a chordal tree structure, the clique graph extracted from it is cyclic, and it was shown that loopy belief propagation in a cyclic clique graph converges to the optimal MAP assignment when iterated multiple times. Caetano and McAuley [154] then modified the graph representation into a chain graph structure, with a clique size of 2, augmented with an additional binary variable to encode reflections. This eliminated the need for multiple iterations of loopy belief propagation.…”

“…Caetano and McAuley[153,154,155,156] developed a series of globally rigid graph representations of point sets, on which probabilistic graphical model based inferencing can be performed to do matching. In their framework, template points represent graph nodes and query points represent possible assignments (class labels).…”