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Abstract An implementation of a “rectilinear” geodesic lying in the Gromov–Hausdorff space is constructed in the form of the shortest geodesic with respect to the Hausdorff distance in some ambient metric space.
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Consider the set K of all nonempty compact subsets of a compact metric space (M, d), endowed with the Hausdorff metric. In this paper, we prove that K is isometric to a subset of l(infinity)(R). An approximation result is also pro...
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Consider the set K of all nonempty compact subsets of a compact metric space (M, d), endowed with the Hausdorff metric. In this paper, we prove that K is isometric to a subset of l(infinity)(R). An approximation result is also proved. (C) 2002 Elsevier Science Ltd. All rights reserved. [References: 4]
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Medial axis transform (MAT) is very sensitive to noise, in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. But it turns ...
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Medial axis transform (MAT) is very sensitive to noise, in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. But it turns out that MAT is stable, if we view this phenomenon with the one-sided Hausdorff distance, rather than with the two-sided Hausdorff distance. In this paper, we show that, if the original domain is weakly injective, which means that the MAT of the domain has no end point which is the center of an inscribed circle osculating the boundary at only one point, the one-sided Hausdorff distance of the original domain's MAT with respect to that of the perturbed one is bounded linearly with the Hausdorff distance of the perturbation. We also show by example that the linearity of this bound cannot be achieved for the domains which are not weakly injective. In particular, these results apply to the domains with sharp corners, which were excluded in the past. One consequence of these results is that we can clarify theoretically the notion of extracting "the essential part of the MAT", which is the heart of the existing pruning methods.
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If A and B are two bounded, closed, non-empty, and convex subsets of a normed space X, then the Hausdorff distance between A and B is the same as the Hausdorff distance between the boundary of A and the boundary of B.
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Engineers often use similitude analyses to design small scale models for experimental tests or to design size ranges of mechanical structures such as drive technology systems. This paper is concerned with similitude analysis metho...
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Engineers often use similitude analyses to design small scale models for experimental tests or to design size ranges of mechanical structures such as drive technology systems. This paper is concerned with similitude analysis methods for vibration analyses of rectangular plates. If their geometry is scaled by different factors (distorted similitude), the scaling laws approximate the actual vibration responses with a certain accuracy only. This paper introduces a performance measure that reliably assesses how well the scaling laws approximate the actual vibration responses of rectangular plates. This measure, the so-called Mahalanobis distance, applies in a-posteriori analyses, where the vibration responses obtained from the scaling laws are compared to the actual ones. Numerical and experimental investigations on vibrating rectangular plates validate that the Mahalanobis distance is suitable to assess the performance of similitude analyses. The Mahalanobis distance can be linked to the geometrical properties of the rectangular plates in order to define a maximum permissible distortion of the geometry. Scaling laws approximate the vibration responses of the rectangular plates sufficiently well up to this maximum permissible distortion. Furthermore, the performance of two different state-of-the-art similitude analysis methods is compared. Both similitude analysis methods are found to perform well up to the maximum permissible amount of geometrical distortion.
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We prove that for any compact set E subset of R-2, dim(H)(E) > 1, there exists x is an element of E such that the Hausdorff dimension of the pinned distance set
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Despite its usefulness in many applications, the medial axis transform (MAT) is very sensitive to the change of the boundary in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of...
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Despite its usefulness in many applications, the medial axis transform (MAT) is very sensitive to the change of the boundary in the sense that, even if a shape is perturbed only slightly, the Hausdorff distance between the MATs of the original shape and the perturbed one may be large. However, it is known that MATs of 2D domains are stable if we view this phenomenon with the one-sided Hausdorff distance. This result depends on the fact that MATs are stable if the differences between them are measured with the recently introduced hyperbolic Hausdorff distance. In this paper, we extend the result for the one-sided stability of the MAT to a class of 3D domains called weakly injective, which contains many important 3D shapes typically appearing in solid modeling. Especially, the weakly injective 3D domains can have sharp features like corners or edges. In fact, by using the stability of the MAT under the hyperbolic Hausdorff distance, we obtain an explicit bound for the one-sided Hausdorff distance of the MAT of a weakly injective 3D domain with respect to that of a perturbed domain, which is linear with respect to the domain perturbation. We discuss some consequences of this result concerning the computation and the approximation of the MAT of 3D objects.
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In this paper, we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear sys...
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In this paper, we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are rational and are at finite Hausdorff distance among them. As a consequence, we provide a projective linear subspace where all (irreducible) elements are solutions of the approximate parametrization problem for a given algebraic plane curve. Furthermore, we identify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization. Therefore, we present a generic answer to the approximate parametrization problem. In addition, we introduce the notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve.
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Image sharpening based on the partial differential equations plays an important role in the fields of image processing. It is an effective technique to clear and sharpen image features, and provides a higher resolution for the sub...
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Image sharpening based on the partial differential equations plays an important role in the fields of image processing. It is an effective technique to clear and sharpen image features, and provides a higher resolution for the subsequent processing. This paper makes the first attempt to employ the Hausdorff derivative Laplacian operator to sharpen the images. In terms of the visual quality of details, contours and edges, the original images and noisy images were sharpened by using an appropriate Hausdorff derivative order. Numerical results indicate that the Hausdorff derivative Laplacian operator outperforms the high-pass filtering, the Roberts operator and the traditional integer-order Laplacian operator. In comparison with the existing methods for the image sharpening, the proposed new methodology could be considered as a competitive alternative.
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With the emergence of digital libraries, more and more documents are stored and transmitted through the Internet in the format of compressed images. It is of significant meaning to develop a system which is capable of retrieving d...
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With the emergence of digital libraries, more and more documents are stored and transmitted through the Internet in the format of compressed images. It is of significant meaning to develop a system which is capable of retrieving documents from these compressed document images. Aiming at the popular compression standard-CCITT Group 4 which is widely used for compressing document images, we present an approach to retrieve the documents from CCITT Group 4 compressed document images in this paper. The black and white changing elements are extracted directly from the compressed document images to act as the feature pixels, and the connected components are detected simultaneously. Then the word boxes are bounded based on the merging of the connected components. Weighted Hausdorff distance is proposed to assign all of the word objects from both the query document and the document from database to corresponding classes by an unsupervised classifier, whereas the possible stop words are excluded. Document vectors are built by the occurrence frequency of the word object classes, and the pair-wise similarity of two document images is represented by the scalar product of the document vectors. Nine groups of articles pertaining to different domains are used to test the validity of the presented approach. Preliminary experimental results with the document images captured from students' theses show that the proposed approach has achieved a promising performance. (C) 2002 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved. [References: 20]
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