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22
Graphical models and point pattern matching
 IEEE Trans. PAMI
, 2006
"... Abstract—This paper describes a novel solution to the rigid point pattern matching problem in Euclidean spaces of any dimension. Although we assume rigid motion, jitter is allowed. We present a noniterative, polynomial time algorithm that is guaranteed to find an optimal solution for the noiseless c ..."
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Cited by 30 (5 self)
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Abstract—This paper describes a novel solution to the rigid point pattern matching problem in Euclidean spaces of any dimension. Although we assume rigid motion, jitter is allowed. We present a noniterative, polynomial time algorithm that is guaranteed to find an optimal solution for the noiseless case. First, we model point pattern matching as a weighted graph matching problem, where weights correspond to Euclidean distances between nodes. We then formulate graph matching as a problem of finding a maximum probability configuration in a graphical model. By using graph rigidity arguments, we prove that a sparse graphical model yields equivalent results to the fully connected model in the noiseless case. This allows us to obtain an algorithm that runs in polynomial time and is provably optimal for exact matching between noiseless point sets. For inexact matching, we can still apply the same algorithm to find approximately optimal solutions. Experimental results obtained by our approach show improvements in accuracy over current methods, particularly when matching patterns of different sizes. Index Terms—Point pattern matching, graph matching, graphical models, Markov random fields, junction tree algorithm. 1
Gromovhausdorff distances in Euclidean spaces
 In Proc. Computer Vision and Pattern Recognition (CVPR
"... The purpose of this paper is to study the relationship between measures of dissimilarity between shapes in Euclidean space. We first concentrate on the pair GromovHausdorff distance (GH) versus Hausdorff distance under the action of Euclidean isometries (EH). Then, we (1) show they are comparable i ..."
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Cited by 12 (6 self)
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The purpose of this paper is to study the relationship between measures of dissimilarity between shapes in Euclidean space. We first concentrate on the pair GromovHausdorff distance (GH) versus Hausdorff distance under the action of Euclidean isometries (EH). Then, we (1) show they are comparable in a precise sense that is not the linear behaviour one would expect and (2) explain the source of this phenomenon via explicit constructions. Finally, (3) by conveniently modifying the expression for the GH distance, we recover the EH distance. This allows us to uncover a connection that links the problem of computing GH and EH and the family of Euclidean Distance Matrix completion problems. The second pair of dissimilarity notions we study is the so called LpGromovHausdorff distance versus the Earth Mover’s distance under the action of Euclidean isometries. We obtain results about comparability in this situation as well. 1.
Graph rigidity, cyclic belief propagation and point pattern matching
 IEEE Transansactions on Pattern Analysis and Machine Intelligence
"... A recent paper [1] proposed a provably optimal, polynomial time method for performing nearisometric point pattern 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 globally rigid, implying t ..."
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Cited by 12 (7 self)
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A recent paper [1] proposed a provably optimal, polynomial time method for performing nearisometric point pattern 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 globally rigid, implying that exact inference provides the same matching solution as exact inference in a complete graphical model. This implies that the algorithm is optimal when there is no noise in the point patterns. In this paper, 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 significantly more efficient. However, our graph is not chordal and thus standard Junction Tree algorithms cannot be directly applied. Nevertheless, we show that loopy belief propagation in such a graph converges to the optimal solution. This allows us to retain the optimality guarantee in the noiseless case, while substantially reducing both memory 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
DynamicsAware Similarity of Moving Objects Trajectories
, 2007
"... This work addresses the problem of obtaining the degree of similarity between trajectories of moving objects. Typically, a Moving Objects Database (MOD) contains sequences of (location,time) points describing the motion of individual objects, however, they also have an implicit information about the ..."
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Cited by 3 (1 self)
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This work addresses the problem of obtaining the degree of similarity between trajectories of moving objects. Typically, a Moving Objects Database (MOD) contains sequences of (location,time) points describing the motion of individual objects, however, they also have an implicit information about the velocity, which is an important attribute describing the dynamics of a particular object. Our main goal is to extend the MOD functionalities with the capability of reasoning about how similar are the trajectories of objects that, possibly, move along geographically different routes. Towards this, we use a distance function which balances the lack of temporalawareness of the Hausdorff distance with the generality (and complexity of calculation) of the Fréchet distance. Based on the observation that, as a firstapproximation in practice, the individual segments of trajectories are assumed to have constant speed, we provide efficient algorithms for: (1) optimal matching between trajectories; and (2) approximate matching between trajectories, both under translations and rotations, where the approximate algorithm guarantees a bounded errorquality with respect to the optimal one.
Improved Approximation Bounds for Planar Point Pattern Matching
"... We consider the well known problem of matching two planar point sets under rigid transformations so as to minimize the directed Hausdorff distance. This is a well studied problem in computational geometry. Goodrich, Mitchell, and Orletsky [GMO94] presented a very simple approximation algorithm for t ..."
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Cited by 2 (1 self)
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We consider the well known problem of matching two planar point sets under rigid transformations so as to minimize the directed Hausdorff distance. This is a well studied problem in computational geometry. Goodrich, Mitchell, and Orletsky [GMO94] presented a very simple approximation algorithm for this problem, which computes transformations based on aligning pairs of points. They showed that their algorithm achieves an approximation ratio of 4. We consider a minor modification to their algorithm, which is based on aligning midpoints rather than endpoints. We show that this algorithm achieves an approximation ratio not greater than 3.13. Our analysis is sensitive to the ratio between the diameter of the pattern set and the Hausdorff distance, and we show that the approximation ratio approaches 3 in the limit. Finally, we provide lower bounds that show that our approximation bounds are nearly tight.
Partial LeastSquares Point Matching under Translations
"... We consider the problem of translating a given pattern set B of size m, and matching every point of B to some point of a larger ground set A of size n in an injective way, minimizing the sum of the squared distances between matched points. We show that when B can only be translated along a line, the ..."
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Cited by 2 (2 self)
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We consider the problem of translating a given pattern set B of size m, and matching every point of B to some point of a larger ground set A of size n in an injective way, minimizing the sum of the squared distances between matched points. We show that when B can only be translated along a line, there can be at most m(n − m) + 1 different matchings as B moves along the line, and hence the optimal translation can be found in polynomial time. 1
A Sweep Line Algorithm for Nearest Neighbour Queries
"... We introduce a novel algorithm for solving the nearest neighbour problem when the query points are known in advance, which is based on Fortune's plane sweep algorithm. The crucial idea is to use the wavefront for solving the nearest neighbour queries as the Voronoi diagram is being computed, instead ..."
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Cited by 1 (0 self)
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We introduce a novel algorithm for solving the nearest neighbour problem when the query points are known in advance, which is based on Fortune's plane sweep algorithm. The crucial idea is to use the wavefront for solving the nearest neighbour queries as the Voronoi diagram is being computed, instead of storing it in an auxiliary data structure, as the algorithm presented by Lee and Yang [9] does, and then querying that data structure. Although
REFERENCES CS 5237 Computational Geometry (Fall 2004) 13 Registration in R 3
"... Registration means matching two simlar shapes in different orientations in either R 2 or R 3. The main goal is to find the optimum transformation (hopefully it may be optimal) that moves one of the objects to match the other. There are in general two approaches, namely discrete and numerical methods ..."
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Registration means matching two simlar shapes in different orientations in either R 2 or R 3. The main goal is to find the optimum transformation (hopefully it may be optimal) that moves one of the objects to match the other. There are in general two approaches, namely discrete and numerical methods. The discrete methods [1,3–5,7] solve the problem by trying every ‘discrete ’ possibility (transformation). However, it is not practical in real situation. The other methods are based on the iterative closest point (ICP) algorithm [2] with improvements [6,8]. In this section we will focus on the ICP algorithm. ICP algorithm. Generally, two objects KR and
REFERENCES CS 5237 Computational Geometry (Spring 2010) 13 Registration in R 3
"... Registration means matching two simlar shapes in different orientations in either R 2 or R 3. The main goal is to find the optimum transformation (hopefully it may be optimal) that moves one of the objects to match the other. There are in general two approaches, namely discrete and numerical methods ..."
Abstract
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Registration means matching two simlar shapes in different orientations in either R 2 or R 3. The main goal is to find the optimum transformation (hopefully it may be optimal) that moves one of the objects to match the other. There are in general two approaches, namely discrete and numerical methods. The discrete methods [1,3–5,7] solve the problem by trying every ‘discrete ’ possibility (transformation). However, it is not practical in real situation. The other methods are based on the iterative closest point (ICP) algorithm [2] with improvements [6,8]. In this section we will focus on the ICP algorithm. ICP algorithm. Generally, two objects KR and
New Approaches to Robust, PointBased Image Registration
"... We consider various algorithmic solutions to image registration based on the alignment of a set of feature points. We present a number of enhancements to a branchandbound algorithm introduced by Mount, Netanyahu, and Le Moigne (Pattern Recognition, Vol. 32, 1999, pp. 17–38), which presented a regi ..."
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We consider various algorithmic solutions to image registration based on the alignment of a set of feature points. We present a number of enhancements to a branchandbound algorithm introduced by Mount, Netanyahu, and Le Moigne (Pattern Recognition, Vol. 32, 1999, pp. 17–38), which presented a registration algorithm based on the partial Hausdorff distance. Our enhancements include a new distance measure, the discrete Gaussian mismatch, and a number of improvements and extensions to the above search algorithm. Both distance measures are robust to the presence of outliers, that is, data points from either set that do not match any point of the other set. We present experimental studies, which show that the new distance measure considered can provide significant improvements over the partial Hausdorff distance in instances where the number of outliers is not known in advance. These experiments also show that our other algorithmic improvements can offer tangible improvements. We demonstrate the algorithm’s efficacy by considering images involving different sensors and different spectral bands, both in a traditional framework and in a multiresolution framework.