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W J Rucklidge - One of the best experts on this subject based on the ideXlab platform.
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efficiently locating objects using the Hausdorff Distance
International Journal of Computer Vision, 1997Co-Authors: W J RucklidgeAbstract:The Hausdorff Distance is a measure defined between two point sets, here representing a model and an image. The Hausdorff Distance is reliable even when the image contains multiple objects, noise, spurious features, and occlusions. In the past, it has been used to search images for instances of a model that has been translated, or translated and scaled, by finding transformations that bring a large number of model features close to image features, and vice versa. In this paper, we apply it to the task of locating an affine transformation of a model in an image; this corresponds to determining the pose of a planar object that has undergone weak-perspective projection. We develop a rasterised approach to the search and a number of techniques that allow us to locate quickly all transformations of the model that satisfy two quality criteria; we can also efficiently locate only the best transformation. We discuss an implementation of this approach, and present some examples of its use.
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efficient visual recognition using the Hausdorff Distance
1996Co-Authors: W J RucklidgeAbstract:The Hausdorff Distance.- Exact computation.- Rasterisation.- Efficient computation.- Implementations and examples.- Applications.- Conclusions.
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Lower bounds for the complexity of the graph of the Hausdorff Distance as a function of transformation
Discrete & Computational Geometry, 1996Co-Authors: W J RucklidgeAbstract:The Hausdorff Distance is a measure defined between two sets in some metric space. This paper investigates how the Hausdorff Distance changes as one set is transformed by some transformation group. Algorithms to find the minimum Distance as one set is transformed have been described, but few lower bounds are known. We consider the complexity of the graph of the Hausdorff Distance as a function of transformation, and exhibit some constructions that give lower bounds for this complexity. We exhibit lower-bound constructions for both sets of points in the plane, and sets of points and line segments; we consider the graph of the directed Hausdorff Distance under translation, rigid motion, translation and scaling, and affine transformation. Many of the results can also be extended to the undirected Hausdorff Distance. These lower bounds are for the complexity of the graph of the Hausdorff Distance, and thus do not necessarily bound algorithms that search this graph; however, they do give an indication of how complex the search may be.
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locating objects using the Hausdorff Distance
International Conference on Computer Vision, 1995Co-Authors: W J RucklidgeAbstract:The Hausdorff Distance is a measure defined between two point sets representing a model and an image. In the past, it has been used to search images for instances of a model that has been translated or translated and scaled by finding transformations that bring a large number of model features close to image features, and vice versa. The Hausdorff Distance is reliable even when the image contains multiple objects, noise, spurious features, and occlusions. We apply it to the task of locating an affine transformation of a model in an image; this corresponds to determining the pose of a planar object that has undergone weak perspective projection. We develop a rasterised approach to the search and a number of techniques that allow us to quickly locate all transformations of the model that satisfy two quality criteria; we can also quickly locate only the best transformation. We discuss an implementation of this approach, and present some examples of its use. >
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efficient computation of the minimum Hausdorff Distance for visual recognition
1994Co-Authors: W J RucklidgeAbstract:We have developed a method, using the minimum Hausdorff Distance, for visually locating an object in an image. This method is very reliable, and fast enough for real-world applications. A visual recognition system takes an image and a model of an object which may occur in that image; these images and models are composed of features (points, line segments, etc.). The system locates instances of the model in the image by determining transformations of the model which bring a large number of model features close to image features. One of the unique strengths of the Hausdorff Distance is the reverse Distance which reduces the frequency of erroneous matching between a model and a cluttered portion of the image. The Hausdorff Distance is a measure defined between two point sets representing a model and an image. Its properties make it attractive for model-based recognition; one of these properties is that the Hausdorff Distance is a metric. The minimum Hausdorff Distance is used to find a transformation of the model which brings it into closest correspondence with the image. This can be done by searching over a space of allowable transformations. In some cases, the minimum Hausdorff Distance is also a metric. The Hausdorff Distance can be modified so that it is reliable even when the image contains multiple objects, noise, spurious features, and occlusions. We construct lower bounds which show that finding the exact transformation that minimises the Hausdorff Distance may be quite expensive. We develop a rasterised approach to the search and a number of techniques which allow this search to be performed efficiently. The principal search technique used is transformation space subdivision. The space of transformations is searched in a tree-like fashion: a large region is examined as a unit, and if the results of this examination are good, it is subdivided and each of the subregions examined in turn; if the results are not good, then the region is discarded. We discuss some implementations of this approach, together with their applications to practical problems such as motion tracking and mobile robot navigation.
Wei Wu - One of the best experts on this subject based on the ideXlab platform.
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Quantized Gromov–Hausdorff Distance
Journal of Functional Analysis, 2006Co-Authors: Wei WuAbstract:Abstract A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov–Hausdorff Distance. We show that two quantized metric spaces are completely isometric if and only if their quantized Gromov–Hausdorff Distance is zero. We establish a completeness theorem. As applications, we show that a quantized metric space with 1-exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized Gromov–Hausdorff Distance, and that matrix algebras converge naturally to the sphere for quantized Gromov–Hausdorff Distance.
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Quantized Gromov-Hausdorff Distance
arXiv: Operator Algebras, 2005Co-Authors: Wei WuAbstract:A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov-Hausdorff Distance. We show that two quantized metric spaces are completely isometric if and only if their quantized Gromov-Hausdorff Distance is zero. We establish a completeness theorem. As applications, we show that a quantized metric space with 1-exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized Gromov-Hausdorff Distance, and that matrix algebras converge naturally to the sphere for quantized Gromov-Hausdorff Distance.
Yusu Wang - One of the best experts on this subject based on the ideXlab platform.
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computing the gromov Hausdorff Distance for metric trees
International Symposium on Algorithms and Computation, 2015Co-Authors: Pankaj K Agarwal, Abhinandan Nath, Anastasios Sidiropoulos, Yusu WangAbstract:The Gromov-Hausdorff Distance is a natural way to measure Distance between two metric spaces. We give the first proof of hardness and first non-trivial approximation algorithm for computing the Gromov-Hausdorff Distance for geodesic metrics in trees. Specifically, we prove it is \(\mathrm {NP}\)-hard to approximate the Gromov-Hausdorff Distance better than a factor of 3. We complement this result by providing a polynomial time \(O(\min \{n, \sqrt{rn}\})\)-approximation algorithm where r is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an \(O(\sqrt{n})\)-approximation algorithm.
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ISAAC - Computing the Gromov-Hausdorff Distance for Metric Trees
Algorithms and Computation, 2015Co-Authors: Pankaj K Agarwal, Abhinandan Nath, Anastasios Sidiropoulos, Kyle Fox, Yusu WangAbstract:The Gromov-Hausdorff Distance is a natural way to measure Distance between two metric spaces. We give the first proof of hardness and first non-trivial approximation algorithm for computing the Gromov-Hausdorff Distance for geodesic metrics in trees. Specifically, we prove it is \(\mathrm {NP}\)-hard to approximate the Gromov-Hausdorff Distance better than a factor of 3. We complement this result by providing a polynomial time \(O(\min \{n, \sqrt{rn}\})\)-approximation algorithm where r is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an \(O(\sqrt{n})\)-approximation algorithm.
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Hausdorff Distance under translation for points and balls
ACM Transactions on Algorithms, 2010Co-Authors: Pankaj K Agarwal, Micha Sharir, Sariel Har-peled, Yusu WangAbstract:We study the shape matching problem under the Hausdorff Distance and its variants. In the first part of the article, we consider two sets A,B of balls in Rd, de2,3, and wish to find a translation t that minimizes the Hausdorff Distance between A+t, the set of all balls in A shifted by t, and B. We consider several variants of this problem. First, we extend the notion of Hausdorff Distance from sets of points to sets of balls, so that each ball has to be matched with the nearest ball in the other set. We also consider the problem in the standard setting, by computing the Hausdorff Distance between the unions of the two sets (as point sets). Second, we consider either all possible translations t (as is the standard approach), or consider only translations that keep the balls of A+t disjoint from those of B. We propose several exact and approximation algorithms for these problems. In the second part of the article, we note that the Hausdorff Distance is sensitive to outliers, and thus consider two variants that are more robust: the root-mean-square (rms) and the summed Hausdorff Distance. We propose efficient approximation algorithms for computing the minimum rms and the minimum summed Hausdorff Distances under translation, between two point sets in Rd. In order to obtain a fast algorithm for the summed Hausdorff Distance, we propose a deterministic efficient dynamic data structure for maintaining an e-approximation of the 1-median of a set of points in Rd, under insertions and deletions.
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Hausdorff Distance under translation for points and balls
Symposium on Computational Geometry, 2003Co-Authors: Pankaj K Agarwal, Sariel Harpeled, Micha Sharir, Yusu WangAbstract:We study the shape matching problem under the Hausdorff Distance and its variants. Specifically, we consider two sets A,B of balls in Rd, d=2,3, and wish to find a translation t that minimizes the Hausdorff Distance between A+t, the set of all balls in A shifted by t, and B. We consider several variants of this problem. First, we extend the notion of Hausdorff Distance from sets of points to sets of balls, so that each ball has to be matched with the nearest ball in the other set. We also consider the problem in the standard setting, by computing the Hausdorff Distance between the unions of the two sets (as point sets). Second, we consider either all possible translates t (as is the standard approach), or consider only translations that keep the balls of A+t disjoint from those of B. We propose several exact and approximation algorithms for these problems. Since the Hausdorff Distance is sensitive to outliers, we also propose efficient approximation algorithms for computing the minimum root mean-square (rms) and the minimum summed Hausdorff Distance, under translation, between two point sets in Rd. In order to obtain a fast algorithm for the summed Hausdorff Distance, we propose a deterministic efficient dynamic data structure for maintaining an e-approximation of the 1-median of a set of points, under insertion and deletion.
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Symposium on Computational Geometry - Hausdorff Distance under translation for points and balls
Proceedings of the nineteenth conference on Computational geometry - SCG '03, 2003Co-Authors: Pankaj K Agarwal, Micha Sharir, Sariel Har-peled, Yusu WangAbstract:We study the shape matching problem under the Hausdorff Distance and its variants. Specifically, we consider two sets A,B of balls in Rd, d=2,3, and wish to find a translation t that minimizes the Hausdorff Distance between A+t, the set of all balls in A shifted by t, and B. We consider several variants of this problem. First, we extend the notion of Hausdorff Distance from sets of points to sets of balls, so that each ball has to be matched with the nearest ball in the other set. We also consider the problem in the standard setting, by computing the Hausdorff Distance between the unions of the two sets (as point sets). Second, we consider either all possible translates t (as is the standard approach), or consider only translations that keep the balls of A+t disjoint from those of B. We propose several exact and approximation algorithms for these problems. Since the Hausdorff Distance is sensitive to outliers, we also propose efficient approximation algorithms for computing the minimum root mean-square (rms) and the minimum summed Hausdorff Distance, under translation, between two point sets in Rd. In order to obtain a fast algorithm for the summed Hausdorff Distance, we propose a deterministic efficient dynamic data structure for maintaining an e-approximation of the 1-median of a set of points, under insertion and deletion.
Gang Xiao - One of the best experts on this subject based on the ideXlab platform.
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study on an improved Hausdorff Distance for multi sensor image matching
Communications in Nonlinear Science and Numerical Simulation, 2012Co-Authors: Jianming Wu, Zhongliang Jing, Zheng Wu, Yan Feng, Gang XiaoAbstract:Abstract A new modifying Hausdorff Distance image matching algorithm was proposed in this paper. After the corners of two images was extracted using Harris corner detector, a kind of Hausdorff Distance integrating points set coincidence numbers was presented to aim at the traditional Hausdorff Distance. The accuracy of matching was improved by this modifying. Hausdorff Distance coefficient matrix is calculating by corners neighborhood’s related matching. The initial matching point-pairs are obtained by the rule that the small coefficient is good matching. Finally the wrong matching point-pairs are deleted by the Distance-ration invariant, the right matching point-pairs are acquired. Experimental results show that the proposed method can be easily and quickly to process the multiple sensor images.
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study on an improved Hausdorff Distance for multi sensor image matching
Communications in Nonlinear Science and Numerical Simulation, 2012Co-Authors: Jianming Wu, Zhongliang Jing, Zheng Wu, Yan Feng, Gang XiaoAbstract:Abstract A new modifying Hausdorff Distance image matching algorithm was proposed in this paper. After the corners of two images was extracted using Harris corner detector, a kind of Hausdorff Distance integrating points set coincidence numbers was presented to aim at the traditional Hausdorff Distance. The accuracy of matching was improved by this modifying. Hausdorff Distance coefficient matrix is calculating by corners neighborhood’s related matching. The initial matching point-pairs are obtained by the rule that the small coefficient is good matching. Finally the wrong matching point-pairs are deleted by the Distance-ration invariant, the right matching point-pairs are acquired. Experimental results show that the proposed method can be easily and quickly to process the multiple sensor images.
Wan-chi Siu - One of the best experts on this subject based on the ideXlab platform.
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Human face recognition based on spatially weighted Hausdorff Distance
Pattern Recognition Letters, 2003Co-Authors: Baofeng Guo, Kwan-ho Lin, Kin-man Lam, Wan-chi SiuAbstract:The edge map of a facial image contains abundant information about its shape and structure, which is useful for face recognition. To compare edge images, Hausdorff Distance is an efficient measure that can determine the degree of their resemblance, and does not require a knowledge of correspondence among those points in the two edge maps. In this paper, a new modified Hausdorff Distance measure is proposed, which has a better discriminant power. As different facial regions have different degrees of significance for face recognition, a new modified Hausdorff Distance is proposed which is weighted according to a weighted function derived from the spatial information of the human face; hence crucial regions are emphasized for face identification. Experimental results show that the Distance measure can achieve recognition rates of 80%, 87%, and 91% for the first, the first five, and the first seven likely matched faces, respectively.Department of Electronic and Information Engineerin
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Human face recognition using a spatially weighted modified Hausdorff Distance
Proceedings of 2001 International Symposium on Intelligent Multimedia Video and Speech Processing. ISIMP 2001 (IEEE Cat. No.01EX489), 1Co-Authors: Kwan-ho Lin, Baofeng Guo, Kin-man Lam, Wan-chi SiuAbstract:Hausdorff Distance is an efficient measure of the similarity of two point sets. We propose a modified Hausdorff Distance measure for human face recognition. This modified Hausdorff Distance measure incorporates information about the location of important facial features when comparing the edge maps of two facial images. The Distance measure is weighted according to a weighted function derived from the spatial information of the human face. This Distance measure, namely spatially weighted Hausdorff Distance (SWHD), is further improved by combining it with the 'doubly' modified Hausdorff Distance (M2HD). This new Hausdorff Distance measure is called spatially weighted 'doubly' Hausdorff Distance (SW2HD), which can alleviate the effect of facial expressions in human face recognition. Experiment results show that both the SWHD and SW2HD outperform the M2HD in recognition rate. The SW2HD can achieve the best recognition rate among the different Hausdorff Distance measures. In our experiment, its recognition rates are 89%, 94%, and 98% for the first one, the first five, and the first ten likely matched faces, respectively. The average processing time for recognition of a human face is less than one second.
