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190
Discrete Geometric Shapes: Matching, Interpolation, and Approximation: A Survey
- Handbook of Computational Geometry
, 1996
"... In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biolog ..."
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Cited by 138 (9 self)
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In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing. 1 Introduction The matching and analysis of geometric patterns and shapes is of importance in various application areas, in particular in computer vision and pattern recognition, but also in other disciplines concerned with the form of objects such as cartography, molecular biology, and computer animation. The general situation is that we are given two objects A, B and want to know how much they resemble each other. Usually one of the objects may undergo certain transformations like translations, rotations or scalings in order to be matched with th...
Shape Matching: Similarity Measures and Algorithms
, 2001
"... Shape matching is an important ingredient in shape retrieval, recognition and classification, alignment and registration, and approximation and simplification. This paper treats various aspects that are needed to solve shape matching problems: choosing the precise problem, selecting the properties o ..."
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Cited by 117 (1 self)
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Shape matching is an important ingredient in shape retrieval, recognition and classification, alignment and registration, and approximation and simplification. This paper treats various aspects that are needed to solve shape matching problems: choosing the precise problem, selecting the properties of the similarity measure that are needed for the problem, choosing the specific similarity measure, and constructing the algorithm to compute the similarity. The focus is on methods that lie close to the field of computational geometry.
On map-matching vehicle tracking data
- In Proc. 31st VLDB Conference
, 2005
"... Vehicle tracking data is an essential “raw ” material for a broad range of applications such as traffic management and control, routing, and navigation. An important issue with this data is its accuracy. The method of sampling vehicular movement using GPS is affected by two error sources and consequ ..."
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Cited by 109 (14 self)
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Vehicle tracking data is an essential “raw ” material for a broad range of applications such as traffic management and control, routing, and navigation. An important issue with this data is its accuracy. The method of sampling vehicular movement using GPS is affected by two error sources and consequently produces inaccurate trajectory data. To become useful, the data has to be related to the underlying road network by means of map matching algorithms. We present three such algorithms that consider especially the trajectory nature of the data rather than simply the current position as in the typical map-matching case. An incremental algorithm is proposed that matches consecutive portions of the trajectory to the road network, effectively trading accuracy for speed of computation. In contrast, the two global algorithms compare the entire trajectory to candidate paths in the road network. The algorithms are evaluated in terms of (i) their running time and (ii) the quality of their matching result. Two novel quality measures utilizing the Fréchet distance are introduced and subsequently used in an experimental evaluation to assess the quality of matching real tracking data to a road network. 1
Near-linear time approximation algorithms for curve simplification
- Proc. of the 10th European Symposium on Algorithms, 2002
, 2002
"... Abstract We consider the problem of approximating a polygonal curve P under a given error criterionby another polygonal curve P 0 whose vertices are a subset of the vertices of P. The goal is tominimize the number of vertices of P 0 while ensuring that the error between P 0 and P is belowa certain t ..."
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Cited by 64 (8 self)
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Abstract We consider the problem of approximating a polygonal curve P under a given error criterionby another polygonal curve P 0 whose vertices are a subset of the vertices of P. The goal is tominimize the number of vertices of P 0 while ensuring that the error between P 0 and P is belowa certain threshold. We consider two different error measures: Hausdorff and Fr'echet. For both error criteria, we present near-linear time approximation algorithms that, given a parameter " ? 0, compute a simplified polygonal curve P 0 whose error is less than " and size at most the sizeof an optimal simplified polygonal curve with error "=2. We consider monotone curves in R2in the case of Hausdorff error measure under the uniform distance metric and arbitrary curves
Matching Planar Maps
"... The subject of this paper are algorithms for measuring the similarity of patterns of line segments in the plane, a standard problem in, e.g. computer vision, geographic information systems, etc. More precisely, we will define feasible distance measures that reflect how close a given pattern H is to ..."
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Cited by 54 (14 self)
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The subject of this paper are algorithms for measuring the similarity of patterns of line segments in the plane, a standard problem in, e.g. computer vision, geographic information systems, etc. More precisely, we will define feasible distance measures that reflect how close a given pattern H is to some part of a larger pattern G. These distance measures are generalizations of the well known Frechet distance for curves. We will first give an efficient algorithm for the case that H is a polygonal curve and G is a geometric graph. Then, slightly relaxing the definition of distance measure we will give an algorithm for the general case where both, H and G, are geometric graphs.
Reliable and Efficient Pattern Matching Using an Affine Invariant Metric
- International Journal of Computer Vision
, 1997
"... In the field of pattern matching, there is a clear trade-off between effectiveness, accuracy and robustness on one hand and efficiency and simplicity on the other hand. For example, matching patterns more effectively by using a more general class of transformations usually results in a considera ..."
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Cited by 37 (1 self)
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In the field of pattern matching, there is a clear trade-off between effectiveness, accuracy and robustness on one hand and efficiency and simplicity on the other hand. For example, matching patterns more effectively by using a more general class of transformations usually results in a considerable increase of computational complexity. In this paper, we introduce a general pattern matching approach which will be applied to a new measure called the absolute difference. This patternsimilarity measure is affine invariant, which stands out favourably in practical use. The problem of finding a transformation mapping to the minimal absolute difference, like many pattern matching problems, has a high computational complexity. Therefore, we base our algorithm on a hierarchical subdivision of transformation space. The method applies to any affine group of transformations, allowing optimisations for rigid motion. Our implementation of the method performs well in terms of reliabilit...
Comparison of distance measures for planar curves
- Algorithmica
"... Abstract The Hausdorff distance is a very natural and straightforward distance measure for comparing geometric shapes like curves or other compact sets. Unfortunately, it is not an appropriate distance measure in some cases. For this reason, the Fr'echet distance has been investigated for measu ..."
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Cited by 34 (8 self)
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Abstract The Hausdorff distance is a very natural and straightforward distance measure for comparing geometric shapes like curves or other compact sets. Unfortunately, it is not an appropriate distance measure in some cases. For this reason, the Fr'echet distance has been investigated for measuring the resemblance of geometric shapes which avoids the drawbacks of the Hausdorff distance. Unfortunately, it is much harder to compute. Here we investigate under which conditions the two distance measures approximately coincide, i.e. the pathological cases for the Hausdorff distance cannot occur. We show that for closed convex curves both distance measures are the same. Furthermore, they are within a constant factor of each other for so-called ^-straight curves, i.e., curves where the arc length between any two points on the curve is at most a constant ^ times their Euclidean distance. Therefore, algorithms for computing the Hausdorff distance can be used in these cases to get exact or approximate computations of the Fr'echet distance, as well.
Addressing the need for map-matching speed: Localizing global curve-matching algorithms
- In SSDBM
, 2006
"... With vehicle tracking data becoming an important sensor data resource for a range of applications related to traffic assessment and prediction, fast and accurate mapmatching algorithms become a necessary means to ultimately utilize this data. This work proposes a fast mapmatching algorithm which exp ..."
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Cited by 33 (8 self)
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With vehicle tracking data becoming an important sensor data resource for a range of applications related to traffic assessment and prediction, fast and accurate mapmatching algorithms become a necessary means to ultimately utilize this data. This work proposes a fast mapmatching algorithm which exploits tracking data error estimates in a provably correct way and offers a quality guarantee for the computed result trajectory. A new model for the map-matching task is introduced which takes tracking error estimates into account. The proposed Adaptive Clipping algorithm (i) provably solves this map-matching task and (ii) utilizes the weak Fréchet distance to measure similarity between curves. The algorithm uses the error estimates in the trajectory data to reduce the search space (error-aware pruning), while offering the quality guarantee of finding a curve which minimizes the weak Fréchet distance to the vehicle trajectory among all possible curves in the road network. Moreover, this work introduces an outputsensitive variant of an existing weak Fréchet map-matching algorithm, which is also employed in the Adaptive Clipping algorithm. Output-sensitiveness paired with error-aware pruning makes Adaptive Clipping the first map-matching algorithm that provably solves a well-defined map-matching task. An experimental evaluation establishes further that Adaptive Clipping is also in a practical setting a fast algorithm that at the same time produces high-quality matching results. 1
Detecting Commuting Patterns by Clustering Subtrajectories
, 2008
"... In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Fréchet distance and the discrete Fréchet distance between subtrajectories, which are invariant under differences in spee ..."
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Cited by 30 (14 self)
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In this paper we consider the problem of detecting commuting patterns in a trajectory. For this we search for similar subtrajectories. To measure spatial similarity we choose the Fréchet distance and the discrete Fréchet distance between subtrajectories, which are invariant under differences in speed. We give several approximation algorithms, and also show that the problem of finding the ‘longest’ subtrajectory cluster is as hard as MaxClique to compute and approximate.
