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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 Diffusion wavelets ： ウィキペディア英語版 Diffusion wavelets Diffusion wavelets are a fast multiscale framework for the analysis of functions on discrete (or discretized continuous) structures like graphs, manifolds, and point clouds in Euclidean space. Diffusion wavelets are an extension of classical wavelet theory from harmonic analysis. Unlike classical wavelets whose basis functions are predetermined, diffusion wavelets are adapted to the geometry of a given diffusion operator $T$ (e.g., a heat kernel or a random walk). Moreover, the diffusion wavelet basis functions are constructed by dilation using the dyadic powers (powers of two) of $T$. These dyadic powers of $T$ diffusion over the space and propagate local relationships in the function throughout the space until they become global. And if the rank of higher powers of $T$ decrease (i.e., its spectrum decays), then these higher powers become compressible. From these decaying dyadic powers of $T$ comes a chain of decreasing subspaces. These subspaces are the scaling function approximation subspaces, and the differences in the subspace chain are the wavelet subspaces. Diffusion wavelets were first introduced in 2004 by Ronald Coifman and Mauro Maggioni at Yale University. == Algorithm == This algorithm constructs the scaling basis functions and the wavelet basis functions along with the representations of the diffusion operator $T$ at these scales. In the algorithm below, the subscript notation $\backslash Phi\_a$ and $\backslash Psi\_b$ represents the scaling basis functions at scale $a$ and the wavelet basis functions at scale $b$ respectively. The notation $()\_$ denotes the matrix representation of the scaling basis $\backslash Phi\_b$ represented with respect to the basis $\backslash Phi\_a$. Lastly, the notation $()\_^$ denotes the matrix represents of the operator $T$, where the row space of $T$ is represented with respect to the basis $\backslash Phi\_a$, and the column space of $T$ is represented with respect to the basis $\backslash Phi\_b$. Otherwise put, the domain of operator $T$ is represented with respect to the basis $\backslash Phi\_a$ and the range is represented with respect to the basis $\backslash Phi\_b$. The function $QR$ is a sparse QR decomposition with $\backslash epsilon$ precision. // Input: // $T$ is the matrix representation of the diffusion operator. // $\backslash epsilon$ is the precision of the QR decomposition, e.g., 1e-6. // $J$ is the maximum number of scale levels (note: this is an optional upper bound, it may converge sooner.) // Output: // $\backslash lbrace\backslash Phi\_j\backslash rbrace$ is the set of scaling basis functions indexed by scale $j$. // $\backslash lbrace\backslash Psi\_j\backslash rbrace$ is the set of wavelet basis functions indexed by scale $j$. $\backslash lbrace\backslash Phi\_j\backslash rbrace,\; \backslash lbrace\backslash Psi\_j\backslash rbrace\; \backslash leftarrow\; \backslash text\; (\; T\; ,\; \backslash epsilon\; ,\; J\; ):$ $\backslash textbf\; j\backslash leftarrow\; 0\; \backslash text\; J-1$: $()\_,\; ()\_^"\; TITLE="T^">)\_^\}"\; TITLE="T^\}^"\; TITLE="T^">)\_^"\; TITLE="\backslash Phi\_">)\_\backslash right)^2$ $()\_\; \backslash leftarrow\; QR\backslash left(I\_-()\_\backslash left(()\_\backslash right)^$ *, \epsilon\right) $\backslash textbf$ 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「Diffusion wavelets」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.