Discrete Wavelet Transform (DWT). Furht, B., editor In Encyclopedia of Multimedia, pages 188–188. Springer US, 2008. 00000
DefinitionDiscrete Wavelet Transform is a technique to transform image pixels into wavelets, which are then used for wavelet-based compression and coding.The DWT is defined as :Wφ(j0,k)=1M‾‾√∑xf(x)φj0,k(x)Wφ(j0,k)=1M∑xf(x)φj0,k(x)W_\textbackslashvarphi (j_0 ,k) = \1\textbackslashover\textbackslashsqrt M \\textbackslashsum\textbackslashlimits_\x\ \f(x)\textbackslashvarphi _\j_0 ,k\ \ (x) (1)Wψ(j,k)=1M‾‾√∑kf(x)ψj,k(x)Wψ(j,k)=1M∑kf(x)ψj,k(x)W_\textbackslashpsi (j,k) = \1\textbackslashover\textbackslashsqrt M \\textbackslashsum\textbackslashlimits_k \f(x)\textbackslashpsi _\j,k\ (x)\ (2)for j≥j0 and the Inverse DWT (IDWT) is defined as:f(x)=1M√∑kWφ(j0,k)φj0,k(x)+1M√∑j=j0∞∑kWψ(j,k)ψj,k(x).f(x)=1M∑kWφ(j0,k)φj0,k(x)+1M∑j=j0∞∑kWψ(j,k)ψj,k(x).\textbackslashbegin\array\\*\20\l\f(x) = & \1\textbackslashover\\textbackslashsqrt \M\ \\\textbackslashsum\textbackslashlimits_\k\ \W_\textbackslashvarphi (j_0 ,k)\textbackslashvarphi_\j_0 ,k\ (x)\ \textbackslash\textbackslash &+ \1\textbackslashover\\textbackslashsqrt M \\\textbackslashsum\textbackslashlimits_\j = j_0 \\textasciicircum\textbackslashinfty\\textbackslashsum\textbackslashlimits_\k\ \W_\textbackslashpsi (\textbackslash,j,k)\textbackslashpsi_\j,k\\ (x).\\textbackslashend\array\ (3)where f(x), φj0,k(x)φj0,k(x)\textbackslashvarphi _\j_0 ,k\ (x), and ψj,k (x) are functions of the discrete variable x = 0,1,2,…,M−1. Normally we let j0 = 0 and select M to be a power of 2 (i.e., M = 2J) so that the summations in Equations (1), (2) and (3) are performed over x = 0,1,2,…,M−1, j = 0,1,2,…, J−1, and k = 0,1,2,…,2 j − 1. The coefficients defined in Equations (1) and (2) are usually ...
@incollection{furht_discrete_2008,
title = {Discrete {Wavelet} {Transform} ({DWT})},
copyright = {©2008 Springer-Verlag},
isbn = {978-0-387-74724-8 978-0-387-78414-4},
url = {http://link.springer.com/referenceworkentry/10.1007/978-0-387-78414-4_305},
abstract = {DefinitionDiscrete Wavelet Transform is a technique to transform image pixels into wavelets, which are then used for wavelet-based compression and coding.The DWT is defined as :Wφ(j0,k)=1M‾‾√∑xf(x)φj0,k(x)Wφ(j0,k)=1M∑xf(x)φj0,k(x)W\_{\textbackslash}varphi (j\_0 ,k) = \{1{\textbackslash}over{\textbackslash}sqrt M \}{\textbackslash}sum{\textbackslash}limits\_\{x\} \{f(x){\textbackslash}varphi \_\{j\_0 ,k\} \} (x) (1)Wψ(j,k)=1M‾‾√∑kf(x)ψj,k(x)Wψ(j,k)=1M∑kf(x)ψj,k(x)W\_{\textbackslash}psi (j,k) = \{1{\textbackslash}over{\textbackslash}sqrt M \}{\textbackslash}sum{\textbackslash}limits\_k \{f(x){\textbackslash}psi \_\{j,k\} (x)\} (2)for j≥j0 and the Inverse DWT (IDWT) is defined as:f(x)=1M√∑kWφ(j0,k)φj0,k(x)+1M√∑j=j0∞∑kWψ(j,k)ψj,k(x).f(x)=1M∑kWφ(j0,k)φj0,k(x)+1M∑j=j0∞∑kWψ(j,k)ψj,k(x).{\textbackslash}begin\{array\}\{*\{20\}l\}f(x) = \& \{1{\textbackslash}over\{{\textbackslash}sqrt \{M\} \}\}{\textbackslash}sum{\textbackslash}limits\_\{k\} \{W\_{\textbackslash}varphi (j\_0 ,k){\textbackslash}varphi\_\{j\_0 ,k\} (x)\} {\textbackslash}{\textbackslash} \&+ \{1{\textbackslash}over\{{\textbackslash}sqrt M \}\}{\textbackslash}sum{\textbackslash}limits\_\{j = j\_0 \}{\textasciicircum}{\textbackslash}infty\{{\textbackslash}sum{\textbackslash}limits\_\{k\} \{W\_{\textbackslash}psi ({\textbackslash},j,k){\textbackslash}psi\_\{j,k\}\} (x).\}{\textbackslash}end\{array\} (3)where f(x), φj0,k(x)φj0,k(x){\textbackslash}varphi \_\{j\_0 ,k\} (x), and ψj,k (x) are functions of the discrete variable x = 0,1,2,…,M−1. Normally we let j0 = 0 and select M to be a power of 2 (i.e., M = 2J) so that the summations in Equations (1), (2) and (3) are performed over x = 0,1,2,…,M−1, j = 0,1,2,…, J−1, and k = 0,1,2,…,2 j − 1. The coefficients defined in Equations (1) and (2) are usually ...},
language = {en},
urldate = {2016-05-03},
booktitle = {Encyclopedia of {Multimedia}},
publisher = {Springer US},
editor = {Furht, Borko},
year = {2008},
note = {00000},
pages = {188--188}
}
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