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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 Curvelet ： ウィキペディア英語版 Curvelet Curvelets are a non-adaptive technique for multi-scale object representation. Being an extension of the wavelet concept, they are becoming popular in similar fields, namely in image processing and scientific computing. Wavelets generalize the Fourier transform by using a basis that represents both location and spatial frequency. For 2D or 3D signals, directional wavelet transforms go further, by using basis functions that are also localized in ''orientation''. A curvelet transform differs from other directional wavelet transforms in that the degree of localisation in orientation varies with scale. In particular, fine-scale basis functions are long ridges; the shape of the basis functions at scale ''j'' is $2^$ by $2^$ so the fine-scale bases are skinny ridges with a precisely determined orientation. Curvelets are an appropriate basis for representing images (or other functions) which are smooth apart from singularities along smooth curves, ''where the curves have bounded curvature'', i.e. where objects in the image have a minimum length scale. This property holds for cartoons, geometrical diagrams, and text. As one zooms in on such images, the edges they contain appear increasingly straight. Curvelets take advantage of this property, by defining the higher resolution curvelets to be more elongated than the lower resolution curvelets. However, natural images (photographs) do not have this property; they have detail at every scale. Therefore, for natural images, it is preferable to use some sort of directional wavelet transform whose wavelets have the same aspect ratio at every scale. When the image is of the right type, curvelets provide a representation that is considerably sparser than other wavelet transforms. This can be quantified by considering the best approximation of a geometrical test image that can be represented using only $n$ wavelets, and analysing the approximation error as a function of $n$. For a Fourier transform, the squared error decreases only as $O(1/\backslash sqrt)$. For a wide variety of wavelet transforms, including both directional and non-directional variants, the squared error decreases as $O(1/n)$. The extra assumption underlying the curvelet transform allows it to achieve $O(^3/)$. Efficient numerical algorithms exist for computing the curvelet transform of discrete data. The computational cost of a curvelet transform is approximately 10–20 times that of an FFT, and has the same dependence of $O(n^2\; \backslash log\; n)$ for an image of size $n\; \backslash times\; n$. == Curvelet construction == To construct a basic curvelet $\backslash phi$ and provide a tiling of the 2-D frequency space, two main ideas should be followed: # Consider polar coordinates in frequency domain # Construct curvelet elements being locally supported near wedges The number of wedges is $N\_j\; =\; 4\; \backslash cdot\; 2^\; \backslash right\; \backslash rceil\}$ at the scale $2^$, i.e., it doubles in each second circular ring. Let $\backslash boldsymbol=(\backslash xi\_1,\backslash xi\_2)^T$ be the variable in frequency domain, and $r=\backslash sqrt,\; \backslash omega=\backslash arctan\; \backslash frac$ be the polar coordinates in the frequency domain. We use the ansatz for the ''dilated basic curvelets'' in polar coordinates: $\backslash hat\_:=2^\}W(2^r)\backslash tilde\_(\backslash omega),\; r\backslash ge\; 0,\; \backslash omega\; \backslash in\; [0,2\backslash pi),\; j\; \backslash in\; N\_0$ To construct a basic curvelet with compact support near a ″basic wedge″, the two windows $W$ and $\backslash tilde\_$need to have compact support. Here, we can simply take $W(r)$ to cover $(0,\backslash infty)$with dilated curvelets and $\backslash tilde\_$ such that each circular ring is covered by the translations of $\backslash tilde\_$ . Then the admissibility yields $\backslash sum\_^\backslash left|\; W(2^r)\; \backslash right|^2=1,\; r\; \backslash in\; (0,\backslash infty).$see ''Window Functions'' for more information For tiling a circular ring into $N$ wedges, where $N$ is an arbitrary positive integer, we need a $2\backslash pi$-periodic nonnegative window $\backslash tilde\_$ with support inside $\backslash left[\; \backslash frac,\; \backslash frac\; \backslash right]$ such that $\backslash sum\_^\backslash tilde^2\_N(\backslash omega-\backslash frac)=1$, for all $\backslash omega\; \backslash in\; \backslash left[\; 0,\; 2\backslash pi\; \backslash right)$ $\backslash tilde\_N$ can be simply constructed as $2\backslash pi$-periodizations of a scaled window $V(\backslash frac)$. Then, it follows that $\backslash sum\_^\backslash left|\; 2^\}\backslash hat\_(r,\; \backslash omega-\backslash frac)\; \backslash right|\; ^2=\backslash left|W(2^r)\; \backslash right|^2\backslash sum\_^\backslash tilde^2\_(\backslash omega-\backslash frac)=\backslash left|W(2^r)\; \backslash right|^2$ For a complete covering of the frequency plane including the region around zero, we need to define a low pass element $\backslash hat\_:=W\_0(\backslash left|\; \backslash xi\; \backslash right|)$ with $W\_0^2(r)^2:=1-\backslash sum\_^W(2^r)^2$ that is supported on the unit circle, and where we do not consider any rotation. 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「Curvelet」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.