Graph rigidity, Cyclic Belief Propagation and Point Pattern Matching

Graph rigidity, Cyclic Belief Propagation and Point Pattern Matching Julian J. McAuley∗, Tib´erio S. Caetano and Marconi S. Barbosa September 26, 2018 Abstract A recent paper [1] proposed a provably optimal, polyno- mial time method for performing near-isometric point pat- tern matching by means of exact probabilistic inference in a chordal graphical model. Their fundamental result is that the chordal graph in question is shown to be glob- ally rigid, implying that exact inference provides the same matching solution as exact inference in a complete graph- ical model. This implies that the algorithm is optimal when there is no noise in the point patterns. In this pa- per, we present a new graph which is also globally rigid but has an advantage over the graph proposed in [1]: its maximal clique size is smaller, rendering inference signif- icantly more efficient. However, our graph is not chordal and thus standard Junction Tree algorithms cannot be di- rectly applied. Nevertheless, we show that loopy belief propagation in such a graph converges to the optimal so- lution. This allows us to retain the optimality guarantee in the noiseless case, while substantially reducing both mem- ory requirements and processing time. Our experimental results show that the accuracy of the proposed solution is indistinguishable from that of [1] when there is noise in the point patterns. 1 Introduction Point pattern matching is a fundamental problem in pat- tern recognition, and has been modeled in several different forms, depending on the demands of the application do- main in which it is required [2, 3]. A classic formulation which is realistic in many practical scenarios is that of near-isometric point pattern matching, in which we are given both a “template” (T ) and a “scene” (S) point pat- terns, and it is assumed that S contains an instance of T (say T ′), apart from an isometric transformation and possibly some small jitter in the point coordinates. The goal is to identify T ′ in S and find which points in T correspond to which points in T ′. Recently, a method was introduced which solves this problem efficiently by means of exact belief propagation in a certain graphical model [1]. The approach is appealing because it is optimal not only in that it consists of exact inference in a graph with small maximal clique size (= 4 ∗The authors are with the Statistical Machine Learning Program, NICTA, and the Research School of Information Sciences and Engi- neering, Australian National University for matching in R2), but that the graph itself is optimal. There it is shown that the maximum a posteriori (MAP) solution in the sparse and tractable graphical model where inference is performed is actually the same MAP solution that would be obtained if a fully connected model (which is intractable) could be used. This is due to the so-called global rigidity of the chordal graph in question: when the graph is embedded in the plane, the lengths of its edges uniquely determine the lengths of the absent edges (i.e. the edges of the graph complement) [4]. The computational complexity of the optimal point pattern matching algo- rithm is then shown to be O(nm4) (both in terms of pro- cessing time and memory requirements), where n is the number of points in the template point pattern and m is the number of points in the scene point pattern (usually with m > n in applications). This reflects precisely the computational complexity of the Junction Tree algorithm in a chordal graph with O(n) nodes, O(m) states per node and maximal cliques of size 4. The authors present exper- iments which give evidence that the method substantially improves on well-known matching techniques, including Graduated Assignment [5]. In this paper, we show how the same optimality proof can be obtained with an algorithm that runs in O(nm3) time per iteration. In addition, memory requirements are precisely decreased by a factor of m. We are able to achieve this by identifying a new graph which is globally rigid but has a smaller maximal clique size: 3. The main problem we face is that our graph is not chordal, so in order to enforce the running intersection property for ap- plying the Junction Tree algorithm the graph should first be triangulated; this would not be interesting in our case, since the resulting triangulated graph would have larger maximal clique size. Instead, we show that belief prop- agation in this graph converges to the optimal solution, although not necessarily in a single iteration. In practice, we find that convergence occurs after a small number of iterations, thus improving the running-time by an order of magnitude. We compare the performance of our model to that of [1] with synthetic and real point sets derived from images, and show that in fact comparable accuracy is obtained while substantial speed-ups are observed. 2 Background We consider point matching problems in R2. The problem we study is that of near-isometric point pattern matching 1 arXiv:0710.0

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