@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 [1]: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}
}

{"_id":"Sz5eDdTCsghjDLXRh","bibbaseid":"furht-discretewavelettransformdwt-2008","authorIDs":[],"bibdata":{"bibtype":"incollection","type":"incollection","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 [1]: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 ...","language":"en","urldate":"2016-05-03","booktitle":"Encyclopedia of Multimedia","publisher":"Springer US","editor":[{"propositions":[],"lastnames":["Furht"],"firstnames":["Borko"],"suffixes":[]}],"year":"2008","note":"00000","pages":"188–188","bibtex":"@incollection{furht_discrete_2008,\n\ttitle = {Discrete {Wavelet} {Transform} ({DWT})},\n\tcopyright = {©2008 Springer-Verlag},\n\tisbn = {978-0-387-74724-8 978-0-387-78414-4},\n\turl = {http://link.springer.com/referenceworkentry/10.1007/978-0-387-78414-4_305},\n\tabstract = {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 [1]: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 ...},\n\tlanguage = {en},\n\turldate = {2016-05-03},\n\tbooktitle = {Encyclopedia of {Multimedia}},\n\tpublisher = {Springer US},\n\teditor = {Furht, Borko},\n\tyear = {2008},\n\tnote = {00000},\n\tpages = {188--188}\n}\n\n","editor_short":["Furht, B."],"key":"furht_discrete_2008","id":"furht_discrete_2008","bibbaseid":"furht-discretewavelettransformdwt-2008","role":"editor","urls":{"Paper":"http://link.springer.com/referenceworkentry/10.1007/978-0-387-78414-4_305"},"downloads":0,"html":""},"bibtype":"incollection","biburl":"http://www.telemidia.puc-rio.br/~alan/files/all.bib","creationDate":"2020-03-03T14:08:15.045Z","downloads":0,"keywords":[],"search_terms":["discrete","wavelet","transform","dwt"],"title":"Discrete Wavelet Transform (DWT)","year":2008,"dataSources":["jAxurbvLP8q5LTdLa"]}