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The Maximally Embedded H(V)parallelogram is at Least 2/9 of the Convex Region
, 2005
"... The proportion of a maximally embedded h(v)parallelogram in a convex region is proved to be at least 2/9 in area. This study is motivated by a multiscale method of detecting the presence of a convex inhomogeneous region in a Gaussian random field. Such a constant (2/9) reveals the feasibility and e ..."
Abstract

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The proportion of a maximally embedded h(v)parallelogram in a convex region is proved to be at least 2/9 in area. This study is motivated by a multiscale method of detecting the presence of a convex inhomogeneous region in a Gaussian random field. Such a constant (2/9) reveals the feasibility and effectiveness of using a multiscale algorithm in detecting the presence of a convexshaped inhomogeneous region. This is a supporting document of a formal paper [5], and is a part of a thesis [6].
DETECTABILITY OF CONVEXSHAPED OBJECTS IN DIGITAL IMAGES, ITS FUNDAMENTAL LIMIT AND MULTISCALE ANALYSIS
"... Abstract: Given a convexshape inhomogeneous region embedded in a noisy image, we consider the conditions under which such an embedded region is detectable. The existence of low orderofcomplexity detection algorithms is also studied. The main results are (1) an analytical threshold (of a statistic ..."
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Abstract: Given a convexshape inhomogeneous region embedded in a noisy image, we consider the conditions under which such an embedded region is detectable. The existence of low orderofcomplexity detection algorithms is also studied. The main results are (1) an analytical threshold (of a statistic) that specifies what is detectable, and (2) the existence of a multiscale detection algorithm whose order of complexity is roughly the optimal O(n 2 log 2 (n)). Our analysis has two main components. We first show that in a discrete image, the number of convex sets increases faster than any finite degree polynomial of the image size n. Hence the idea of generalized likelihood ratio test cannot be directly adopted to derive the asymptotic detectability bound. Secondly, we show that the maximally embedded hvparallelogram is at least 2/9 of the convex region (in area). We then apply the results of hvparallelograms in AriasCastro, Donoho, and Huo (2005) on detecting convex sets. Numerical examples are provided. Our results have potential applications in several fields, which are described with corresponding references.