
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 30, NO. 11, NOVEMBER 2008 2047 Graph Rigidity, Cyclic Belief Propagation, is intractable) could be used. This is due to the so-called global and Point Pattern Matching rigidity of the chordal graph in question: When the graph is embedded in the plane, the lengths of its edges uniquely determine Julian J. McAuley, Tibe´ rioS.Caetano,and the lengths of the absent edges (i.e., the edges of the graph Marconi S. Barbosa complement) [9]. The computational complexity of the optimal point pattern matching algorithm is then shown to be Oðnm4Þ (both in terms of processing time and memory requirements), Abstract—A recent paper [1] proposed a provably optimal polynomial time where n is the number of points in the template point pattern and method for performing near-isometric point pattern matching by means of exact m is the number of points in the scene point pattern (usually with probabilistic inference in a chordal graphical model. Its fundamental result is that m>n in applications). This reflects precisely the computational the chordal graph in question is shown to be globally rigid, implying that exact inference provides the same matching solution as exact inference in a complete complexity of the Junction Tree algorithm in a chordal graph with graphical model. This implies that the algorithm is optimal when there is no noise OðnÞ nodes, OðmÞ states per node, and maximal cliques of size 4. in the point patterns. In this paper, we present a new graph that is also globally The authors present experiments which give evidence that the rigid but has an advantage over the graph proposed in [1]: Its maximal clique size method substantially improves on well-known matching techni- is smaller, rendering inference significantly more efficient. However, this graph is ques, including Graduated Assignment [10]. not chordal, and thus, standard Junction Tree algorithms cannot be directly In this paper, we show how the same optimality proof can be applied. Nevertheless, we show that loopy belief propagation in such a graph 3 converges to the optimal solution. This allows us to retain the optimality guarantee obtained with an algorithm that runs in Oðnm Þ time per iteration. in the noiseless case, while substantially reducing both memory requirements and In addition, memory requirements are precisely decreased by a processing time. Our experimental results show that the accuracy of the proposed factor of m. We are able to achieve this by identifying a new graph, solution is indistinguishable from that in [1] when there is noise in the point which is globally rigid but has a smaller maximal clique size: three. patterns. The main problem we face is that our graph is not chordal, so in order to enforce the running intersection property for applying the Index Terms—Point pattern matching, graph matching, graphical models, belief Junction Tree algorithm, the graph should first be triangulated; propagation, global rigidity, chordal graphs. this would not be interesting in our case, since the resulting Ç triangulated graph would have larger maximal clique size. Instead, we show that belief propagation in this graph converges to the 1INTRODUCTION optimal solution, although not necessarily in a single iteration. In POINT pattern matching is a fundamental problem in pattern practice, we find that convergence occurs after a small number of recognition and has been modeled in several different forms iterations, thus improving the runtime by an order of magnitude. depending on the demands of the application domain in which it is We compare the performance of our model to that in [1] with required [2], [3], [4], [5]. A classic formulation that is realistic in synthetic and real point sets derived from images and show that in many practical scenarios is that of near-isometric point pattern fact comparable accuracy is obtained while substantial speedups matching, in which we are given both a “template” ðT Þ and a are observed. “scene” ðSÞ point pattern, and it is assumed that S contains an instance of T (say, T 0) apart from an isometric transformation and possibly some small jitter in the point coordinates. The goal is to 2BACKGROUND 0 identify T in S and find which points in T correspond to which We consider point matching problems in IR 2. The problem we 0 points in T . study is that of near-isometric point pattern matching (as defined Recently, a method was introduced that solves this problem above), i.e., one assumes that a near-isometric instance ðT 0Þ of efficiently by means of exact belief propagation in a certain the template ðT Þ is somewhere “hidden” in the scene ðSÞ.By graphical model [1].1 The approach is appealing because it is “near-isometric,” it is meant that the relative distances of points 0 optimal not only in that it consists of exact inference in a graph in T are approximately preserved in T . For simplicity of 0 with small maximal clique size (¼ 4 for matching in IR 2) but also in exposition, we assume that T , T , and S are ordered sets (their that the graph itself is optimal. There it is shown that the maximum elements are indexed). Our aim is to find a map x : T 7!S with T 0 a posteriori (MAP) solution in the sparse and tractable graphical image that best preserves the relative distances of the points in T and T 0, i.e., model where inference is performed is actually the same MAP solution that would be obtained if a fully connected model (which à 2 x ¼ argminkkDðT Þ À DxðÞðT Þ 2; ð1Þ x 1. See also [6] for further treatment of shape matching in the Bayesian th setting; see [7] and [8] for further treatment using tree-structured models. where DðT Þ is the matrix whose ði; jÞ entry is the euclidean distance between points indexed by i and j in T and kÁk2 is the Frobenius norm. Note that finding xà is inherently a combinatorial . The authors are with the Statistical Machine Learning group, NICTA, optimization problem since T 0 is itself a subset of S, the scene Locked Bag 8001 Canberra ACT 2601 Australia, and the Research School of point pattern. In [1], a generic point in T is modeled as a random Information Sciences and Engineering, Australian National University, variable ðX Þ and a generic point in S is modeled as a possible ACT 0200. i E-mail: {julian.mcauley, Tiberio.Caetano}@nicta.com.au, realization of the random variable ðxiÞ. As a result, a joint [email protected]. realization of all the random variables corresponds to a match Manuscript received 29 Sept. 2007; revised 16 Jan. 2008; accepted 29 Apr. between the template and the scene point patterns. A graphical 2008; published online 14 May 2008. model (see [11] and [12]) is then defined on this set of random Recommended for acceptance by M. Pelillo. variables whose edges are set according to the topology of a so- For information on obtaining reprints of this article, please send e-mail to: called 3-tree graph (any 3-tree that spans T ). A 3-tree is a graph [email protected], and reference IEEECS Log Number TPAMI-2007-09-0655. obtained by starting with the complete graph on three vertices, Digital Object Identifier no. 10.1109/TPAMI.2008.124. K3, and then adding new vertices that are connected only to those 0162-8828/08/$25.00 ß 2008 IEEE Published by the IEEE Computer Society 2048 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 30, NO. 11, NOVEMBER 2008 Fig. 1. An example of a 3-tree. Each node corresponds to a point in the Fig. 2. The general form of the graph we consider, with n nodes. template set. precise in Section 3, when we prove an analogous result for 2 same three vertices. Fig. 1 shows an example of a 3-tree; it should another graph.4 be noted that the points in the template set correspond to the nodes in this graph, whereas the points in the scene correspond to their 3AN IMPROVED GRAPH assignments. The reasons claimed in [1] for introducing 3-trees as a graph topology for the probabilistic graphical model are that Here, we introduce another globally rigid graph that has the 1) 3-trees are globally rigid in the plane and 2) 3-trees are chordal3 advantage of having a smaller maximal clique size. Although the graphs. This implies 1) that the 3-tree model is “optimal” (in a graph is not chordal, we will show that exact inference is tractable and that we will indeed benefit from the decrease in the maximal way that will be made clear in Section 3 in the context of the new clique size. As a result, we will be able to obtain optimality graph we propose) and 2) that 3-trees have a Junction Tree with guarantees like those from [1]. fixed maximal clique size ð¼ 4Þ; as a result, it is possible to Our graph is constructed using Algorithm 1. perform exact inference in polynomial time [1]. Potential functions are defined on pairs of neighboring nodes Algorithm 1. Graph generation for G. and are large if the difference between the distance of neighboring 1 Create a cycle graph by traversing all the nodes in T (in any nodes in the template and the distance between the nodes they order) map to in the scene is small (and small if this difference is large). 2 Connect all nodes whose distance in this cycle graph is two This favors isometric matchings. More precisely, (i.e., connect each node to its neighbor’s neighbor) This algorithm will produce a graph like the one shown in ijðXi ¼ xi;Xj ¼ xjÞ¼fdðXi;XjÞÀdðxi;xjÞ ; ð2Þ Fig.
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