Fast Linear Approximations of Euclidean Distance in Higher Dimensions

Ohashi, Y.

doi:10.1016/b978-0-12-336156-1.50020-3

Cited by 4 publications

(4 citation statements)

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“…Note the striking similarity between ( 5) and ( 6). Interestingly, a similar but less rigorous approach had been published earlier by Ohashi [9]. It should also be noted that several authors approached the problem from a Euclidean distance transform perspective and derived similar approximations for the 2-and 3-dimensional cases, see for example [10] and [11].…”

Section: Introductionmentioning

confidence: 90%

Comments on “On approximating Euclidean metrics by weighted t-cost distances in arbitrary dimension”

Celebi

Kingravi

Celiker

2012

Pattern Recognition Letters

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Mukherjee (Pattern Recognition Letters, vol. 32, pp. 824-831, 2011) recently introduced a class of distance functions called weighted t-cost distances that generalize m-neighbor, octagonal, and t-cost distances. He proved that weighted t-cost distances form a family of metrics and derived an approximation for the Euclidean norm in Z n . In this note we compare this approximation to two previously proposed Euclidean norm approximations and demonstrate that the empirical average errors given by Mukherjee are significantly optimistic in R n . We also propose a simple normalization scheme that improves the accuracy of his approximation substantially with respect to both average and maximum relative errors.

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Section: Introductionmentioning

confidence: 90%

Comments on “On approximating Euclidean metrics by weighted t-cost distances in arbitrary dimension”

Celebi

Kingravi

Celiker

2012

Pattern Recognition Letters

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“…It should be noted that a similar but less rigorous approach had been published earlier by Ohashi [8].…”

Section: Barni Et Al's Approximationmentioning

confidence: 98%

“…where E(·) is the expectation operator. Note that the formulation of D a,b is similar to that of D µ,λ (8) in that they both approximate D 2 by a linear combination of D ∞ and D 1 . These approximations differ in their methodologies for finding the optimal parameters.…”

Section: Seol and Cheun's Approximationmentioning

confidence: 99%

On Euclidean norm approximations

Celebi

Celiker

Kingravi

2011

Pattern Recognition

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Euclidean norm calculations arise frequently in scientific and engineering applications. Several approximations for this norm with differing complexity and accuracy have been proposed in the literature. Earlier approaches [1,2,3] were based on minimizing the maximum error. Recently, Seol and Cheun [4] proposed an approximation based on minimizing the average error. In this paper, we first examine these approximations in detail, show that they fit into a single mathematical formulation, and compare their average and maximum errors. We then show that the maximum errors given by Seol and Cheun are significantly optimistic.

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“…The proposal is inspired by similar problems in computer graphics and image processing where approximations for the Euclidean distance are used [8,9] (not for the squared Euclidean distances though). As is the case of the approximations that modify the forward-backward MAP algorithm into a max-log MAP algorithm .…”

Section: Introductionmentioning

confidence: 99%

Euclidean Distances in Quantized Spaces with Pre-stored Components for MIMO Detection

Monteiro

Wassell

2007

2007 European Conference on Wireless Technologies

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This paper proposes a technique to reduce the complexity involved in the maximum likelihood detection of multiple input multiple output (MIMO) spatial multiplexing systems. Both the received lattice and the components of the received signal are quantized, corresponding to a mapping into a multidimensional space divided into hypercubes. In the new space the maximum likelihood detection criterion can be applied making use of a small look-up table storing all the exact possible distance components in each dimension of the quantized space. The number of pre-stored elements can be as small as the number of quantization levels per dimension. This procedure eliminates the multiplications involved in the calculation of the squared Euclidean distances. The impact of the quantization error is assessed by means of simulations over fast flat fading channels. Near optimum performance is achieved with only 5 bits representing each dimension of the received signal.

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Fast Linear Approximations of Euclidean Distance in Higher Dimensions

Cited by 4 publications

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Comments on “On approximating Euclidean metrics by weighted t-cost distances in arbitrary dimension”

Comments on “On approximating Euclidean metrics by weighted t-cost distances in arbitrary dimension”

On Euclidean norm approximations

Euclidean Distances in Quantized Spaces with Pre-stored Components for MIMO Detection

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