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

Bayesian Nonparametric Adaptive Control using Gaussian Processes

This technical report is a preprint of an article submitted to a journal.Most current Model Reference Adaptive Control (MRAC) methods rely on parametric adaptive elements, in which the number of parameters of the adaptive element are fixed a priori, often through expert judgment. An example of such an adaptive element are Radial Basis Function Networks (RBFNs), with RBF centers pre-allocated based on the expected operating domain. If the system operates outside of the expected operating domain, this adaptive element can become non-effective in capturing and canceling the uncertainty, thus rendering the adaptive controller only semi-global in nature. This paper investigates a Gaussian Process (GP) based Bayesian MRAC architecture (GP-MRAC), which leverages the power and flexibility of GP Bayesian nonparametric models of uncertainty. GP-MRAC does not require the centers to be preallocated, can inherently handle measurement noise, and enables MRAC to handle a broader set of uncertainties, including those that are defined as distributions over functions. We use stochastic stability arguments to show that GP-MRAC guarantees good closed loop performance with no prior domain knowledge of the uncertainty. Online implementable GP inference methods are compared in numerical simulations against RBFN-MRAC with preallocated centers and are shown to provide better tracking and improved long-term learning.This research was supported in part by ONR MURI Grant N000141110688 and NSF grant ECS #0846750

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 $\mathbb{Z}^n$. 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 $\mathbb{R}^n$. We also propose a simple normalization scheme that improves the accuracy of his approximation substantially with respect to both average and maximum relative errors.Comment: 7 pages, 1 figure, 3 tables. arXiv admin note: substantial text overlap with arXiv:1008.487

Fast and Accurate Border Detection in Dermoscopy Images Using Statistical Region Merging

Copyright 2007 Society of Photo-Optical Instrumentation Engineers. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited.As a result of advances in skin imaging technology and the development of suitable image processing techniques during the last decade, there has been a significant increase of interest in the computer-aided diagnosis of melanoma. Automated border detection is one of the most important steps in this procedure, since the accuracy of the subsequent steps crucially depends on it. In this paper, a fast and unsupervised approach to border detection in dermoscopy images of pigmented skin lesions based on the Statistical Region Merging algorithm is presented. The method is tested on a set of 90 dermoscopy images. The border detection error is quantified by a metric in which a set of dermatologist-determined borders is used as the ground-truth. The proposed method is compared to six state-of-the-art automated methods (optimized histogram thresholding, orientation-sensitive fuzzy c-means, gradient vector flow snakes, dermatologist-like tumor extraction algorithm, meanshift clustering, and the modified JSEG method) and borders determined by a second dermatologist. The results demonstrate that the presented method achieves both fast and accurate border detection in dermoscopy images.http://dx.doi.org/10.1117/12.70907