The Experts below are selected from a list of 116619 Experts worldwide ranked by ideXlab platform
Rekha R Thomas - One of the best experts on this subject based on the ideXlab platform.
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the Euclidean Distance degree of orthogonally invariant matrix varieties
Israel Journal of Mathematics, 2017Co-Authors: Dmitriy Drusvyatskiy, Honleung Lee, Giorgio Ottaviani, Rekha R ThomasAbstract:The Euclidean Distance degree of a real variety is an important invariant arising in Distance minimization problems. We show that the Euclidean Distance degree of an orthogonally invariant matrix variety equals the Euclidean Distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances.
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the Euclidean Distance degree of orthogonally invariant matrix varieties
arXiv: Optimization and Control, 2016Co-Authors: Dmitriy Drusvyatskiy, Honleung Lee, Giorgio Ottaviani, Rekha R ThomasAbstract:We show that the Euclidean Distance degree of a real orthogonally invariant matrix variety equals the Euclidean Distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances.
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The Euclidean Distance Degree of an Algebraic Variety
Foundations of Computational Mathematics, 2015Co-Authors: Jan Draisma, Giorgio Ottaviani, Emil Horobeţ, Bernd Sturmfels, Rekha R ThomasAbstract:The nearest point map of a real algebraic variety with respect to Euclidean Distance is an algebraic function. For instance, for varieties of low-rank matrices, the Eckart---Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean Distance degree of a variety is the number of critical points of the squared Distance to a general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.
Owen Robert Mitchell - One of the best experts on this subject based on the ideXlab platform.
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A Euclidean Distance transform using grayscale morphology decomposition
IEEE Transactions on Pattern Analysis and Machine Intelligence, 1994Co-Authors: C.t. Huang, Owen Robert MitchellAbstract:A fast and exact Euclidean Distance transformation using decomposed grayscale morphological operators is presented. Applied on a binary image, a Distance transformation assigns each object pixel a value that corresponds to the shortest Distance between the object pixel and the background pixels. It is shown that the large structuring element required for the Euclidean Distance transformation can be easily decomposed into 3/spl times/3 windows. This is possible because the square of the Euclidean Distance matrix changes uniformly both in the vertical and horizontal directions. A simple extension for a 3D Euclidean Distance transformation is discussed. A fast Distance transform for serial computers is also presented. Acting like thinning algorithms, the version for serial computers focuses operations only on the potential changing pixels and propagates from the boundary of objects, significantly reducing execution time. Nonsquare pixels can also be used in this algorithm. An example application, shape filtering using arbitrary sized circular dilation and erosion, is discussed. Rotation-invariant basic morphological operations can be done using this example application. >
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A mathematical morphology approach to Euclidean Distance transformation
IEEE Transactions on Image Processing, 1992Co-Authors: Frank Yeong-chyang Shih, Owen Robert MitchellAbstract:A Distance transformation technique for a binary digital image using a gray-scale mathematical morphology approach is presented. Applying well-developed decomposition properties of mathematical morphology, one can significantly reduce the tremendous cost of global operations to that of small neighborhood operations suitable for parallel pipelined computers. First, the Distance transformation using mathematical morphology is developed. Then several approximations of the Euclidean Distance are discussed. The decomposition of the Euclidean Distance structuring element is presented. The decomposition technique employs a set of 3 by 3 gray scale morphological erosions with suitable weighted structuring elements and combines the outputs using the minimum operator. Real-valued Distance transformations are considered during the processes and the result is approximated to the closest integer in the final output image. >
Frank Yeong-chyang Shih - One of the best experts on this subject based on the ideXlab platform.
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The efficient algorithms for achieving Euclidean Distance transformation
IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 2004Co-Authors: Frank Yeong-chyang ShihAbstract:Euclidean Distance transformation (EDT) is used to convert a digital binary image consisting of object (foreground) and nonobject (background) pixels into another image where each pixel has a value of the minimum Euclidean Distance from nonobject pixels. In this paper, the improved iterative erosion algorithm is proposed to avoid the redundant calculations in the iterative erosion algorithm. Furthermore, to avoid the iterative operations, the two-scan-based algorithm by a deriving approach is developed for achieving EDT correctly and efficiently in a constant time. Besides, we discover when obstacles appear in the image, many algorithms cannot achieve the correct EDT except our two-scan-based algorithm. Moreover, the two-scan-based algorithm does not require the additional cost of preprocessing or relative-coordinates recording.
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Size-invariant four-scan Euclidean Distance transformation
Pattern Recognition, 1998Co-Authors: Frank Yeong-chyang Shih, Jenny J. LiuAbstract:Abstract Distance transform (DT)(1) is used to convert a binary image that consists of object (foreground) and nonobject (background) pixels into another image in which each object pixel has a value corresponding to the minimum Distance from the background by a predefined Distance function. The Euclidean Distance is more accurate than the others, such as city-block, chessboard and chamfer, but it takes more computational time due to its nonlinearity. By using the relative X and Y coordinates computed from the object pixel to the source mapping pixel of its neighbors as well as correction of particular cases, the Euclidean Distance transformation (EDT) can be correctly obtained in just four scans of an image. In other words, the new algorithm achieves the computational complexity of EDT to be linear to the size of an image.
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A mathematical morphology approach to Euclidean Distance transformation
IEEE Transactions on Image Processing, 1992Co-Authors: Frank Yeong-chyang Shih, Owen Robert MitchellAbstract:A Distance transformation technique for a binary digital image using a gray-scale mathematical morphology approach is presented. Applying well-developed decomposition properties of mathematical morphology, one can significantly reduce the tremendous cost of global operations to that of small neighborhood operations suitable for parallel pipelined computers. First, the Distance transformation using mathematical morphology is developed. Then several approximations of the Euclidean Distance are discussed. The decomposition of the Euclidean Distance structuring element is presented. The decomposition technique employs a set of 3 by 3 gray scale morphological erosions with suitable weighted structuring elements and combines the outputs using the minimum operator. Real-valued Distance transformations are considered during the processes and the result is approximated to the closest integer in the final output image. >
Giorgio Ottaviani - One of the best experts on this subject based on the ideXlab platform.
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the Euclidean Distance degree of orthogonally invariant matrix varieties
Israel Journal of Mathematics, 2017Co-Authors: Dmitriy Drusvyatskiy, Honleung Lee, Giorgio Ottaviani, Rekha R ThomasAbstract:The Euclidean Distance degree of a real variety is an important invariant arising in Distance minimization problems. We show that the Euclidean Distance degree of an orthogonally invariant matrix variety equals the Euclidean Distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances.
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the Euclidean Distance degree of orthogonally invariant matrix varieties
arXiv: Optimization and Control, 2016Co-Authors: Dmitriy Drusvyatskiy, Honleung Lee, Giorgio Ottaviani, Rekha R ThomasAbstract:We show that the Euclidean Distance degree of a real orthogonally invariant matrix variety equals the Euclidean Distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances.
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The Euclidean Distance Degree of an Algebraic Variety
Foundations of Computational Mathematics, 2015Co-Authors: Jan Draisma, Giorgio Ottaviani, Emil Horobeţ, Bernd Sturmfels, Rekha R ThomasAbstract:The nearest point map of a real algebraic variety with respect to Euclidean Distance is an algebraic function. For instance, for varieties of low-rank matrices, the Eckart---Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest point maps from the perspective of computational algebraic geometry. The Euclidean Distance degree of a variety is the number of critical points of the squared Distance to a general point outside the variety. Focusing on varieties seen in applications, we present numerous tools for exact computations.
Henry Wolkowicz - One of the best experts on this subject based on the ideXlab platform.
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Coordinate Shadows of Semidefinite and Euclidean Distance Matrices
SIAM Journal on Optimization, 2015Co-Authors: Dmitriy Drusvyatskiy, Gábor Pataki, Henry WolkowiczAbstract:We consider the projected semidefinite and Euclidean Distance cones onto a subset of the matrix entries. These two sets are precisely the input data defining feasible semidefinite and Euclidean Distance completion problems. We classify when these sets are closed and use the boundary structure of these two sets to elucidate the Krislock--Wolkowicz facial reduction algorithm. In particular, we show that under a chordality assumption, the “minimal cones” of these problems admit combinatorial characterizations. As a by-product, we record a striking relationship between the complexity of the general facial reduction algorithm (singularity degree) and facial exposedness of conic images under a linear mapping.
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Euclidean Distance matrices and applications
2012Co-Authors: Nathan Krislock, Henry WolkowiczAbstract:Euclidean Distance matrices, or EDMs, have been receiving increased attention for two main reasons. The first reason is that the many applications of EDMs, such as molecular conformation in bioinformatics, dimensionality reduction in machine learning and statistics, and especially the problem of wireless sensor network localization, have all become very active areas of research. The second reason for this increased interest is the close connection between EDMs and semidefinite matrices. Our recent ability to solve semidefinite programs efficiently means we can now also solve many problems involving EDMs efficiently. This chapter connects the classical approaches for EDMs with the more recent tools from semidefinite programming. We emphasize the application to sensor network localization.
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Approximate and exact completion problems for Euclidean Distance matrices using semidefinite programming
Linear Algebra and its Applications, 2005Co-Authors: Suliman Al-homidan, Henry WolkowiczAbstract:AbstractA partial pre-Distance matrix A is a matrix with zero diagonal and with certain elements fixed to given nonnegative values; the other elements are considered free. The Euclidean Distance matrix completion problem chooses nonnegative values for the free elements in order to obtain a Euclidean Distance matrix, EDM. The nearest (or approximate) Euclidean Distance matrix problem is to find a Euclidean Distance matrix, EDM, that is nearest in the Frobenius norm to the matrix A, when the free variables are discounted.In this paper we introduce two algorithms: one for the exact completion problem and one for the approximate completion problem. Both use a reformulation of EDM into a semidefinite programming problem, SDP. The first algorithm is based on an implicit equation for the completion that for many instances provides an explicit solution. The other algorithm is based on primal–dual interior-point methods that exploit the structure and sparsity. Included are results on maps that arise that keep the EDM and SDP cones invariant.We briefly discuss numerical tests
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Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming
Computational Optimization and Applications, 1999Co-Authors: Abdo Y. Alfakih, Amir Khandani, Henry WolkowiczAbstract:Given a partial symmetric matrix A with only certain elements specified, the Euclidean Distance matrix completion problem (EDMCP) is to find the unspecified elements of A that make A a Euclidean Distance matrix (EDM). In this paper, we follow the successful approach in [20] and solve the EDMCP by generalizing the completion problem to allow for approximate completions. In particular, we introduce a primal-dual interior-point algorithm that solves an equivalent (quadratic objective function) semidefinite programming problem (SDP). Numerical results are included which illustrate the efficiency and robustness of our approach. Our randomly generated problems consistently resulted in low dimensional solutions when no completion existed.
