## Combinatorial and experimental methods for approximate point pattern matching (2003)

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Venue: | Algorithmica |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Gavrilov03combinatorialand,

author = {Martin Gavrilov and Piotr Indyk and Rajeev Motwani and Suresh Venkatasubramanian},

title = {Combinatorial and experimental methods for approximate point pattern matching },

journal = {Algorithmica},

year = {2003},

volume = {38},

pages = {38--2}

}

### OpenURL

### Abstract

Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high degree polynomials, and require robust calculations of intersection points of high degree surfaces. We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate. We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average case analysis and a detailed empirical study of our methods.

### Citations

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Citation Context ...algorithm of Section 9. The main subroutine is the subset matching algorithm, that involves computing convolutions of binary vectors over (·, +) [30]. We use the NTL package developed by Victor Shoup =-=[36]-=- that implements various operations on finite fields, including fast polynomial multiplication (which we use to implement convolution). 12 Running Time Analysis We present two analytical estimations o... |

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Citation Context ...ea was done by Huttenlocher, Kedem and Sharir in 1993 [28], where they presented an algorithm running in time O(kn(k + n)α(kn) log(k + n)) for PM(d�, T � , ɛ) This was generalized to rigid motions in =-=[9]-=-, yielding an algorithm running in time Õ(k � n � ). Briefly, the algorithms proceed by computing a partition of transformation space such that each cell of the partition corresponds to a fixed cost a... |

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Citation Context ...iven a set of n points in the plane, how many pairs of points can be a unit distance apart?”. Erdős showed an upper bound of O(n ��� ) and a lower bound of Ω(n ���� log log � ). Szemeredi and Trotter =-=[38]-=- reduced the upper bound to O(n ��� ). Later results [10, 37] reduced the constants and simplified the proof considerably. The lower bound remains unimproved and is conjectured to be tight. Akutsu, Ta... |

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Citation Context ...oint sets P and Q, space of transformations T , and parameter ɛ, determine A ⊆ P, B ⊆ Q such that and |A| is maximized. min �∈T d�(T(A), B) ≤ ɛ Definition 2.5 (β-approximate pattern matching PM(ɛ, β) =-=[26]-=-). Given ɛ, β > 0 and point sets P and Q, • If ɛ ∗ ≤ ɛ, then return a transformation T ∈ T such that d�(T(P), Q) ≤ (1 + β)ɛ; • If ɛ ∗ > (1 + β)ɛ, then return NONE. • Otherwise, when ɛ ∗ ∈ (ɛ, (1 + β)ɛ... |

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Citation Context ...d in combinatorial pattern matching to obtain near-linear time algorithms for ∗ This paper expands and combines the contents of papers presented at the 10th ACM-SIAM Conference on Discrete Algorithms =-=[31]-=- and the 15th ACM Symposium on Computational Geometry [23]svarious pattern matching problems. We show that geometric matching problems can be reduced to combinatorial pattern matching problems, thus o... |

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Citation Context ...blems. We show that geometric matching problems can be reduced to combinatorial pattern matching problems, thus obtaining fast approximations. This idea was independently used by Cardoze and Schulman =-=[8]-=- to provide a near-linear scheme for pattern matching; we discuss their work in relation to ours later. Subsequent to the first presentation of this work, other problems in geometric matching have bee... |

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Citation Context ... t q Figure 2: An instance of subset matching Recent work of Cole and Hariharan [11] showed that this problem can be solved in (randomized) O(n log � m) time. Subsequently, Cole, Hariharan, and Indyk =-=[12]-=- have shown that the above algorithm can be derandomized to run in O(n log � m) time. We will solve PM(d�, M � , ɛ, β) by reducing it to multiple instances of the subset matching problem. The basic id... |

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Citation Context ...o be the ratio of the largest distance between pairs of points to the smallest distance between pairs of points. This quantity, also called the spread of a point set, has been exploited in other work =-=[22]-=- to obtain fast algorithms for various problems via a careful analysis of the dependency on the diameter. Experimental Evaluation We conduct a detailed experimental evaluation of the schemes that we p... |

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Citation Context ...) is often not the most useful parameter to determine the running time of geometric algorithms. Examples of properties of a point set that influence running time include fatness [16] and aspect ratio =-=[42]-=-. The running times of our algorithms contain terms involving ∆, the diameter of a point set. More formally, we denote ∆ to be the ratio of the largest distance between pairs of points to the smallest... |

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Citation Context ...pendence on the tolerance parameters, and thus an empirical study provides deeper insight into the efficacy of our methods in practice. Much of the recent empirical work in geometric pattern matching =-=[29, 25, 33]-=- has examined algorithms based on an exhaustive search of the transformation space (with clever pruning techniques). Our evaluation naturally complements this work. However, an exact comparison of our... |

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Citation Context ...4 The Combinatorial Pattern Matching Scheme We use the algorithm of Section 9. The main subroutine is the subset matching algorithm, that involves computing convolutions of binary vectors over (·, +) =-=[30]-=-. We use the NTL package developed by Victor Shoup [36] that implements various operations on finite fields, including fast polynomial multiplication (which we use to implement convolution). 12 Runnin... |

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Citation Context ...aluating the cost of each cell. For a more detailed overview of the techniques, the reader is referred to the survey by Alt and Guibas [3]. In the approximate setting, Goodrich, Mitchell and Orletsky =-=[24]-=- developed a very simple Õ(kn � ) algorithm for PM(d�, M � , ɛ, 2 + γ) (for any γ ≥ 0). Their scheme is based on the alignment method, a popular heuristic for point set matching that we outline in Alg... |

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Citation Context ...s of the pattern match the corresponding positions in the text. Text Pattern x p q r a 1 f c d f d g 3 t p 4 y q x r 3 4 t q Figure 2: An instance of subset matching Recent work of Cole and Hariharan =-=[11]-=- showed that this problem can be solved in (randomized) O(n log � m) time. Subsequently, Cole, Hariharan, and Indyk [12] have shown that the above algorithm can be derandomized to run in O(n log � m) ... |

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Citation Context ...p�, p�), we need only consider pairs (q�, q�) such that |p� − p�| = |q� − q�|. By the combinatorial distance bounds, we know that there are O(n ��� ) such pairs, and Agarwal, Aronov, Sharir, and Suri =-=[1]-=- have given an effective way of extracting these distances in time O(n ��� log n). This immediately improves the running time of the naive algorithm to O(kn ��� log n). Using more sophisticated ideas ... |

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Citation Context ... dependence of the bound on the diameter of the point set. The connection between the diameter of a point set and such approximate combinatorial quantities has been exploited before. Valtr and others =-=[14, 39, 40, 41]-=- investigated bounds on the number of approximate incidences between families of well-separated lines and points; assuming that the point sets are dense, i.e., their diameter is O( √ n), he showed tha... |

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Image analysis and computer vision
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Citation Context ...before the distance between them is computed. This general formulation captures a wide variety of problems, arising in image registration [33], model-based object recognition [4, 21], computer vision =-=[35]-=-, molecular modelling [27], and many other areas. In the domain of computational geometry, these problems were mostly studied in an exact formulation: point coordinates are given with infinite precisi... |

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Citation Context ...time algorithms for ∗ This paper expands and combines the contents of papers presented at the 10th ACM-SIAM Conference on Discrete Algorithms [31] and the 15th ACM Symposium on Computational Geometry =-=[23]-=-svarious pattern matching problems. We show that geometric matching problems can be reduced to combinatorial pattern matching problems, thus obtaining fast approximations. This idea was independently ... |

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Citation Context ...ergo transformations like rotations, translations and scaling before the distance between them is computed. This general formulation captures a wide variety of problems, arising in image registration =-=[33]-=-, model-based object recognition [4, 21], computer vision [35], molecular modelling [27], and many other areas. In the domain of computational geometry, these problems were mostly studied in an exact ... |

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Citation Context ...iscuss their work in relation to ours later. Subsequent to the first presentation of this work, other problems in geometric matching have been solved via a reduction to combinatorial pattern matching =-=[17, 32]-=-. Diameter Dependence It has been observed in computational geometry that the parameter n (defining the number of geometric objects in the input) is often not the most useful parameter to determine th... |

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Citation Context ...higher dimensions using such bounds. More recently, Braβ [7] has used similar methods for exact point set matching in three dimensions. Interestingly, a simple construction presented by Erdős in 1967 =-=[19]-=- shows that for any d ≥ 4, there exists a set of n points in R � having Θ(n � ) pairs of points a unit distance apart. Approximate Distance Bounds A natural question would be whether we could extend t... |

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Citation Context ...es like d�. For this purpose, we need a noisy version of the above combinatorial problem. Such a 4squestion was posed and solved (in the plane) by Erdős, Makai, Pach and Spencer in 1991: Theorem 3.1 (=-=[20]-=-). Given t > 0, and a set of n points in the plane with minimum distance at least 1, the number of pairs (i, j) such that d(p�, p�) lies in the range [t, t + 1] is at most � n � /4 � , provided n is s... |

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Citation Context ...translations and scaling before the distance between them is computed. This general formulation captures a wide variety of problems, arising in image registration [33], model-based object recognition =-=[4, 21]-=-, computer vision [35], molecular modelling [27], and many other areas. In the domain of computational geometry, these problems were mostly studied in an exact formulation: point coordinates are given... |

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Citation Context ...�� ). Later results [10, 37] reduced the constants and simplified the proof considerably. The lower bound remains unimproved and is conjectured to be tight. Akutsu, Tamaki, and Tokuyama [2] and Boxer =-=[6]-=-, made the key observation that these bounds could be used to improve the analysis of simple point set matching algorithms. Consider the following naive algorithm to solve PM(d�, M � ) i.e to determin... |

3 |
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Citation Context ... dependence of the bound on the diameter of the point set. The connection between the diameter of a point set and such approximate combinatorial quantities has been exploited before. Valtr and others =-=[14, 39, 40, 41]-=- investigated bounds on the number of approximate incidences between families of well-separated lines and points; assuming that the point sets are dense, i.e., their diameter is O( √ n), he showed tha... |

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Citation Context ...iscuss their work in relation to ours later. Subsequent to the first presentation of this work, other problems in geometric matching have been solved via a reduction to combinatorial pattern matching =-=[17, 32]-=-. Diameter Dependence It has been observed in computational geometry that the parameter n (defining the number of geometric objects in the input) is often not the most useful parameter to determine th... |

2 |
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Citation Context ... before the distance between them is computed. This general formulation captures a wide variety of problems, arising in image registration [33], model-based object recognition [4, 21], computervision =-=[35]-=-, molecular modelling [27], and many other areas. In the domain of computational geometry, these problems were mostly studied in an exact formulation: point co-ordinates are given with infinite precis... |

1 |
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Citation Context ...d to higher dimensions, where the combinatorial object of interest is now a simplex in R � . Akutsu et. al provide results for PM in three and higher dimensions using such bounds. More recently, Braβ =-=[7]-=- has used similar methods for exact point set matching in three dimensions. Interestingly, a simple construction presented by Erdős in 1967 [19] shows that for any d ≥ 4, there exists a set of n point... |

1 |
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Citation Context ...way of understanding the behaviour of these methods might come from making assumptions about the range of permissible transformations. In the same spirit, the following problem was suggested by Efrat =-=[15]-=-: 24sDefinition 14.1 (Dog in the Field). Given two point sets P, Q in the plane and ɛ > 0 such that there exists a unique transformation T such that d(T(P), Q) ≤ ɛ, determine T in time O(n + k � ), wh... |

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Citation Context ...c objects in the input) is often not the most useful parameter to determine the running time of geometric algorithms. Examples of properties of a point set that influence running time include fatness =-=[16]-=- and aspect ratio [42]. The running times of our algorithms contain terms involving ∆, the diameter of a point set. More formally, we denote ∆ to be the ratio of the largest distance between pairs of ... |

1 |
line-point incidences and crossing families in dense sets
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Citation Context ... dependence of the bound on the diameter of the point set. The connection between the diameter of a point set and such approximate combinatorial quantities has been exploited before. Valtr and others =-=[14, 39, 40, 41]-=- investigated bounds on the number of approximate incidences between families of well-separated lines and points; assuming that the point sets are dense, i.e., their diameter is O( √ n), he showed tha... |

1 |
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Citation Context |

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set pattern matching in 3-D. Pattern Recognition Letters 17, 12
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Citation Context ...n4=3). Later results [10, 37] reduced the constants and simplified the proofconsiderably. The lower bound remains unimproved and is conjectured to be tight. Akutsu, Tamaki, and Tokuyama [2] and Boxer =-=[6]-=-, made the key observation that these bounds could be usedto improve the analysis of simple point set matching algorithms. Consider the following naive algorithm to solve PM(dE, M2) i.e to determine w... |