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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Multi-scale approaches ： ウィキペディア英語版 Multi-scale approaches The scale space representation of a signal obtained by Gaussian smoothing satisfies a number of special properties, scale-space axioms, which make it into a special form of multi-scale representation. There are, however, also other types of "multi-scale approaches" in the areas of computer vision, image processing and signal processing, in particular the notion of wavelets. The purpose of this article is to describe a few of these approaches: ==Scale-space theory for one-dimensional signals== For ''one-dimensional signals'', there exists quite a well-developed theory for continuous and discrete kernels that guarantee that new local extrema or zero-crossings cannot be created by a convolution operation.〔(Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234-254. )〕 For ''continuous signals'', it holds that all scale-space kernels can be decomposed into the following sets of primitive smoothing kernels: * the ''Gaussian kernel'' :$g(x,\; t)\; =\; \backslash frac)$ where $t\; >\; 0$, * ''truncated exponential'' kernels (filters with one real pole in the ''s''-plane): ::$h(x)=\; \backslash exp()$ if $x\; \backslash geq\; 0$ and 0 otherwise where $a\; >\; 0$ ::$h(x)=\; \backslash exp()$ if $x\; \backslash leq\; 0$ and 0 otherwise where $b\; >\; 0$, * translations, * rescalings. For ''discrete signals'', we can, up to trivial translations and rescalings, decompose any discrete scale-space kernel into the following primitive operations: * the ''discrete Gaussian kernel'' ::$T(n,\; t)\; =\; I\_n(\backslash alpha\; t)$ where $\backslash alpha,\; t\; >\; 0$ where $I\_n$ are the modified Bessel functions of integer order, * ''generalized binomial kernels'' corresponding to linear smoothing of the form :$f\_(x)\; =\; p\; f\_(x)\; +\; q\; f\_(x-1)$ where $p,\; q\; >\; 0$ :$f\_(x)\; =\; p\; f\_(x)\; +\; q\; f\_(x+1)$ where $p,\; q\; >\; 0$, * ''first-order recursive filters'' corresponding to linear smoothing of the form :$f\_(x)\; =\; f\_(x)\; +\; \backslash alpha\; f\_(x-1)$ where $\backslash alpha\; >\; 0$ :$f\_(x)\; =\; f\_(x)\; +\; \backslash beta\; f\_(x+1)$ where $\backslash beta\; >\; 0$, * the one-sided ''Poisson kernel'' :$p(n,\; t)\; =\; e^\; \backslash frac$ for $n\; \backslash geq\; 0$ where $t\backslash geq0$ :$p(n,\; t)\; =\; e^\; \backslash frac$ for $n\; \backslash leq\; 0$ where $t\backslash geq0$. From this classification, it is apparent that it we require a continuous semi-group structure, there are only three classes of scale-space kernels with a continuous scale parameter; the Gaussian kernel which forms the scale-space of continuous signals, the discrete Gaussian kernel which forms the scale-space of discrete signals and the time-causal Poisson kernel that forms a temporal scale-space over discrete time. If we on the other hand sacrifice the continuous semi-group structure, there are more options: For discrete signals, the use of generalized binomial kernels provides a formal basis for defining the smoothing operation in a pyramid. For temporal data, the one-sided truncated exponential kernels and the first-order recursive filters provide a way to define ''time-causal scale-spaces'' 〔(Richard F. Lyon. "Speech recognition in scale space," Proc. of 1987 ICASSP. San Diego, March, pp. 29.3.14, 1987. )〕〔(Lindeberg, T. and Fagerstrom, F.: Scale-space with causal time direction, Proc. 4th European Conference on Computer Vision, Cambridge, England, April 1996. Springer-Verlag LNCS Vol 1064, pages 229--240. )〕 that allow for efficient numerical implementation and respect causality over time without access to the future. The first-order recursive filters also provide a framework for defining recursive approximations to the Gaussian kernel that in a weaker sense preserve some of the scale-space properties.〔(Young, I.I., van Vliet, L.J.: Recursive implementation of the Gaussian filter, Signal Processing, vol. 44, no. 2, 1995, 139-151. )〕〔(Deriche, R: Recursively implementing the Gaussian and its derivatives, INRIA Research Report 1893, 1993. )〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Multi-scale approaches」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.