2,494 research outputs found
Comments on "On Approximating Euclidean Metrics by Weighted t-Cost Distances in Arbitrary Dimension"
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 . 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
. We also propose a simple normalization scheme that improves the
accuracy of his approximation substantially with respect to both average and
maximum relative errors.Comment: 7 pages, 1 figure, 3 tables. arXiv admin note: substantial text
overlap with arXiv:1008.487
On Euclidean Norm Approximations
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 were
based on minimizing the maximum error. Recently, Seol and Cheun 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.Comment: 9 pages, 1 figure, Pattern Recognitio
Distance Measures for Reduced Ordering Based Vector Filters
Reduced ordering based vector filters have proved successful in removing
long-tailed noise from color images while preserving edges and fine image
details. These filters commonly utilize variants of the Minkowski distance to
order the color vectors with the aim of distinguishing between noisy and
noise-free vectors. In this paper, we review various alternative distance
measures and evaluate their performance on a large and diverse set of images
using several effectiveness and efficiency criteria. The results demonstrate
that there are in fact strong alternatives to the popular Minkowski metrics
Greedy vector quantization
We investigate the greedy version of the -optimal vector quantization
problem for an -valued random vector . We show the
existence of a sequence such that minimizes
(-mean quantization error at level induced by
). We show that this sequence produces -rate
optimal -tuples ( the -mean
quantization error at level induced by goes to at rate
). Greedy optimal sequences also satisfy, under natural
additional assumptions, the distortion mismatch property: the -tuples
remain rate optimal with respect to the -norms, .
Finally, we propose optimization methods to compute greedy sequences, adapted
from usual Lloyd's I and Competitive Learning Vector Quantization procedures,
either in their deterministic (implementable when ) or stochastic
versions.Comment: 31 pages, 4 figures, few typos corrected (now an extended version of
an eponym paper to appear in Journal of Approximation
Median and related local filters for tensor-valued images
We develop a concept for the median filtering of tensor data. The main part of this concept is the definition of median for symmetric matrices. This definition is based on the minimisation of a geometrically motivated objective function which measures the sum of distances of a variable matrix to the given data matrices. This theoretically wellfounded concept fits into a context of similarly defined median filters for other multivariate data. Unlike some other approaches, we do not require by definition that the median has to be one of the given data values. Nevertheless, it happens so in many cases, equipping the matrix-valued median even with root signals similar to the scalar-valued situation. Like their scalar-valued counterparts, matrix-valued median filters show excellent capabilities for structure-preserving denoising. Experiments on diffusion tensor imaging, fluid dynamics and orientation estimation data are shown to demonstrate this. The orientation estimation examples give rise to a new variant of a robust adaptive structure tensor which can be compared to existing concepts. For the efficient computation of matrix medians, we present a convex programming framework. By generalising the idea of the matrix median filters, we design a variety of other local matrix filters. These include matrix-valued mid-range filters and, more generally, M-smoothers but also weighted medians and \alpha-quantiles. Mid-range filters and quantiles allow also interesting cross-links to fundamental concepts of matrix morphology
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
A New Algorithm for Exploratory Projection Pursuit
In this paper, we propose a new algorithm for exploratory projection pursuit.
The basis of the algorithm is the insight that previous approaches used fairly
narrow definitions of interestingness / non interestingness. We argue that
allowing these definitions to depend on the problem / data at hand is a more
natural approach in an exploratory technique. This also allows our technique
much greater applicability than the approaches extant in the literature.
Complementing this insight, we propose a class of projection indices based on
the spatial distribution function that can make use of such information.
Finally, with the help of real datasets, we demonstrate how a range of
multivariate exploratory tasks can be addressed with our algorithm. The
examples further demonstrate that the proposed indices are quite capable of
focussing on the interesting structure in the data, even when this structure is
otherwise hard to detect or arises from very subtle patterns.Comment: 29 pages, 8 figure
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