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Graph Rigidity, Cyclic Belief Propagation and Point Pattern Matching

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Graph Rigidity, Cyclic Belief Propagation and Point Pattern Matching Graph rigidity, Cyclic Belief Propagation and Point Pattern Matching Julian J. McAuley,∗ Tib´erioS. Caetano and Marconi S. Barbosa September 26, 2018 Abstract for matching in R2), but that the graph itself is optimal. There it is shown that the maximum a posteriori (MAP) A recent paper [1] proposed a provably optimal, polyno- solution in the sparse and tractable graphical model where mial time method for performing near-isometric point pat- inference is performed is actually the same MAP solution tern matching by means of exact probabilistic inference that would be obtained if a fully connected model (which in a chordal graphical model. Their fundamental result is intractable) could be used. This is due to the so-called is that the chordal graph in question is shown to be glob- global rigidity of the chordal graph in question: when the ally rigid, implying that exact inference provides the same graph is embedded in the plane, the lengths of its edges matching solution as exact inference in a complete graph- uniquely determine the lengths of the absent edges (i.e. the ical model. This implies that the algorithm is optimal edges of the graph complement) [4]. The computational when there is no noise in the point patterns. In this pa- complexity of the optimal point pattern matching algo- per, we present a new graph which is also globally rigid rithm is then shown to be O(nm4) (both in terms of pro- but has an advantage over the graph proposed in [1]: its cessing time and memory requirements), where n is the maximal clique size is smaller, rendering inference signif- number of points in the template point pattern and m is icantly more efficient. However, our graph is not chordal the number of points in the scene point pattern (usually and thus standard Junction Tree algorithms cannot be di- with m > n in applications). This reflects precisely the rectly applied. Nevertheless, we show that loopy belief computational complexity of the Junction Tree algorithm propagation in such a graph converges to the optimal so- in a chordal graph with O(n) nodes, O(m) states per node lution. This allows us to retain the optimality guarantee in and maximal cliques of size 4. The authors present exper- the noiseless case, while substantially reducing both mem- iments which give evidence that the method substantially ory requirements and processing time. Our experimental improves on well-known matching techniques, including results show that the accuracy of the proposed solution is Graduated Assignment [5]. indistinguishable from that of [1] when there is noise in In this paper, we show how the same optimality proof the point patterns. 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 1 Introduction achieve this by identifying a new graph which is globally rigid but has a smaller maximal clique size: 3. The main Point pattern matching is a fundamental problem in pat- problem we face is that our graph is not chordal, so in tern recognition, and has been modeled in several different order to enforce the running intersection property for ap- forms, depending on the demands of the application do- plying the Junction Tree algorithm the graph should first arXiv:0710.0043v2 [cs.CV] 3 Oct 2007 main in which it is required [2, 3]. A classic formulation be triangulated; this would not be interesting in our case, which is realistic in many practical scenarios is that of since the resulting triangulated graph would have larger near-isometric point pattern matching, in which we are maximal clique size. Instead, we show that belief prop- given both a \template" (T ) and a \scene" (S) point pat- agation in this graph converges to the optimal solution, terns, and it is assumed that S contains an instance of although not necessarily in a single iteration. In practice, 0 T (say T ), apart from an isometric transformation and we find that convergence occurs after a small number of possibly some small jitter in the point coordinates. The iterations, thus improving the running-time by an order 0 goal is to identify T in S and find which points in T of magnitude. We compare the performance of our model 0 correspond to which points in T . to that of [1] with synthetic and real point sets derived Recently, a method was introduced which solves this from images, and show that in fact comparable accuracy problem efficiently by means of exact belief propagation in is obtained while substantial speed-ups are observed. 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 2 Background ∗ The authors are with the Statistical Machine Learning Program, 2 NICTA, and the Research School of Information Sciences and Engi- We consider point matching problems in R . The problem neering, Australian National University we study is that of near-isometric point pattern matching 1 (as defined above), i.e. one assumes that a near-isometric instance (T 0) of the template (T ) is somewhen \hidden" in the scene (S). By \near-isometric" it is meant that the relative distances of points in T are approximately preserved in T 0. For simplicity of exposition we assume that T , T 0, and S are ordered sets (their elements are indexed). Our aim is to find a map x : T 7! S with image T 0 that best preserves the relative distances of the points in T and T 0, i.e. ∗ 2 x = argmin kD(T ) − D(x(T ))k2 ; (1) x where D(T ) is the matrix whose (i; j)th entry is the Eu- clidean distance between points indexed by i and j in T . Note that finding x∗ is inherently a combinatorial opti- mization problem, since T 0 is itself a subset of S, the scene point pattern. In [1], a generic point in T is mod- Figure 1: An example of a 3-tree eled as a random variable (Xi), and a generic point in S is modeled as a possible realization of the random variable (xi). As a result, a joint realization of all the random vari- scene points). For the case of exact matching, i.e. when ables corresponds to a match between the template and there exists an x∗ such that the minimal value in (1) is the scene point patterns. A graphical model (see [6, 7]) is zero, then f(·) = δ(·) (where δ(·) is just the indicator then defined on this set of random variables, whose edges function 1f0g(·)). The potential function of a maximal are set according to the topology of a so-called 3-tree graph clique (Ψ) is then simply defined as the product of the 4 (any 3-tree that spans T ). A 3-tree is a graph obtained potential functions over its 6 (= C2 ) edges (which will by starting with the complete graph on 3 vertices, K3, be maximal when every factor is maximal). It should be and then adding new vertices which are connected only to noted that the potential function of each edge is included those same 3 vertices.1 Figure 1 shows an example of a 3- in no more than one of the cliques containing that edge. tree. The reasons claimed in [1] for introducing 3-trees as For the case of exact matching (i.e. no jitter), it is shown a graph topology for the probabilistic graphical model are in [1] that running the Junction Tree algorithm on the 3- that (i) 3-trees are globally rigid in the plane and (ii) 3- tree graphical model with f(·) = δ(·) will actually find a trees are chordal2 graphs. This implies (i) that the 3-tree MAP assignment which coincides with x∗, i.e. such that ∗ 2 model is a type of graph which is in some sense \optimal" kD(T ) − D(x (T ))k2 = 0. This is due to the \graph rigid- (in a way that will be made clear in the next section in ity" result, which tells us that equality of the lengths of the the context of the new graph we propose) and (ii) that 3- edges in the 3-tree and the edges induced by the matching trees have a Junction Tree with fixed maximal clique size in T 0 is sufficient to ensure the equality of the lengths of all (= 4); as a result it is possible to perform exact inference pairs of points in T and T 0. This will be made technically in polynomial time [1]. precise in the next section, when we prove an analogous Potential functions are defined on pairs of neighboring result for another graph. nodes and are large if the difference between the distance of neighboring nodes in the template and the distance be- tween the nodes they map to in the scene is small (and 3 An Improved Graph small if this difference is large). This favors isometric matchings. More precisely, Here we introduce another globally rigid graph which has the advantage of having a smaller maximal clique size. ij(Xi = xi;Xj = xj) = f(d(Xi;Xj) − d(xi; xj)) (2) Although the graph is not chordal, we will show that exact inference is tractable and that we will indeed benefit from where f(·) is typically some unimodal function peaked at the decrease in the maximal clique size. As a result we will zero (e.g. a zero-mean Gaussian function) and d(·; ·) is the be able to obtain optimality guarantees like those from [1].
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