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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 Multidimensional sampling ： ウィキペディア英語版 Multidimensional sampling In digital signal processing, multidimensional sampling is the process of converting a function of a multidimensional variable into a discrete collection of values of the function measured on a discrete set of points. This article presents the basic result due to Petersen and Middleton〔D. P. Petersen and D. Middleton, "Sampling and Reconstruction of Wave-Number-Limited Functions in N-Dimensional Euclidean Spaces", Information and Control, vol. 5, pp. 279–323, 1962.〕 on conditions for perfectly reconstructing a wavenumber-limited function from its measurements on a discrete lattice of points. This result, also known as the Petersen–Middleton theorem, is a generalization of the Nyquist–Shannon sampling theorem for sampling one-dimensional band-limited functions to higher-dimensional Euclidean spaces. In essence, the Petersen–Middleton theorem shows that a wavenumber-limited function can be perfectly reconstructed from its values on an infinite lattice of points, provided the lattice is fine enough. The theorem provides conditions on the lattice under which perfect reconstruction is possible. As with the Nyquist–Shannon sampling theorem, this theorem also assumes an idealization of any real-world situation, as it only applies to functions that are sampled over an infinitude of points. Perfect reconstruction is mathematically possible for the idealized model but only an approximation for real-world functions and sampling techniques, albeit in practice often a very good one. ==Preliminaries== The concept of a bandlimited function in one dimension can be generalized to the notion of a wavenumber-limited function in higher dimensions. Recall that the Fourier transform of an integrable function $f(\backslash cdot)$ on ''n''-dimensional Euclidean space is defined as: :$\backslash hat(\backslash xi)\; =\; \backslash mathcal(f)(\backslash xi)\; =\; \backslash int\_\; f(x)\; e^\; \backslash ,\; dx$ where ''x'' and ''ξ'' are ''n''-dimensional vectors, and $\backslash langle\; x,\backslash xi\; \backslash rangle$ is the inner product of the vectors. The function $f(\backslash cdot)$ is said to be wavenumber-limited to a set $\backslash Omega$ if the Fourier transform satisfies $\backslash hat(\backslash xi)\; =\; 0$ for $\backslash xi\; \backslash notin\; \backslash Omega$. Similarly, the configuration of uniformly spaced sampling points in one-dimension can be generalized to a lattice in higher dimensions. A lattice is a collection of points $\backslash Lambda\; \backslash subset\; \backslash Re^n$ of the form $$ \Lambda = \left\ \right\} where is a basis for $\backslash Re^n$. The reciprocal lattice $\backslash Gamma$ corresponding to $\backslash Lambda$ is defined by :$$ \Gamma = \left\ \right\} where the vectors $u\_i$ are chosen to satisfy $\backslash langle\; u\_i,\; v\_j\; \backslash rangle\; =\; \backslash delta\_$. That is, if the vectors $u\_i$ form columns of a matrix $A$ and $v\_i$ the columns of a matrix $B$, then $A=B^$. An example of a sampling lattice is a hexagonal lattice depicted in Figure 1. The corresponding reciprocal lattice is shown in Figure 2. 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「Multidimensional sampling」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.