Discrete Cosine Transform (DCT). Furht, B., editor In Encyclopedia of Multimedia, pages 186–188. Springer US, 2008. 00000Paper abstract bibtex DefinitionDiscrete Cosine Transform is a technique applied to image pixels in spatial domain in order to transform them into a frequency domain in which redundancy can be identified.In JPEG compression [1], image is divided into 8 × 8 blocks, then the two-dimensional Discrete Cosine Transform (DCT) is applied to each of these 8 × 8 blocks. In JPEG decompression, the Inverse Discrete Cosine Transform (IDCT) is applied to the 8 × 8 DCT coefficient blocks. DCT and IDCT are defined as follows:DCT:F(u,v)=14C(u)C(v)∑i=07∑j=07f(i,j)cos((2i+1)uπ/16)cos((2j+1)vπ/16)DCT:F(u,v)=14C(u)C(v)∑i=07∑j=07f(i,j)cos((2i+1)uπ/16)cos((2j+1)vπ/16)\textbackslashbegin\array\ \*\20\l\DCT: &\textbackslash\textbackslash & F(u,v) = \1\textbackslashover4\C(u)C(v)\textbackslashsum\textbackslashlimits_\i = 0\\textasciicircum7 \textbackslashsum\textbackslashlimits_\j = 0\\textasciicircum7 f(i,j) \textbackslash\textbackslash &\textbackslashcos ((2i + 1)u\textbackslashpi /16)\textbackslashcos ((2j + 1)v\textbackslashpi /16)\textbackslashend\array\IDCT:f(i,j)=14∑u=07∑v=07C(u)C(v)F(u,v)cos((2i+1)uπ/16)cos((2j+1)vπ/16)IDCT:f(i,j)=14∑u=07∑v=07C(u)C(v)F(u,v)cos((2i+1)uπ/16)cos((2j+1)vπ/16)\textbackslashbegin\array\ \*\20\l\IDCT: & \textbackslash\textbackslash & f(i,j) = \1\textbackslashover 4\\textbackslashsum\textbackslashlimits_\u = 0\\textasciicircum7 \textbackslashsum\textbackslashlimits_\v = 0\\textasciicircum7 C(u)C(v)F(u,v) \textbackslash\textbackslash &\textbackslashcos ((2i + 1)u \textbackslashpi /16)\textbackslashcos ((2j + 1)v \textbackslashpi /16) \textbackslashend\array\ where f(i, j) and F(u, ν) are respectively the pixel value and the DCT coefficient, and ...
@incollection{furht_discrete_2008-1,
title = {Discrete {Cosine} {Transform} ({DCT})},
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_304},
abstract = {DefinitionDiscrete Cosine Transform is a technique applied to image pixels in spatial domain in order to transform them into a frequency domain in which redundancy can be identified.In JPEG compression [1], image is divided into 8 × 8 blocks, then the two-dimensional Discrete Cosine Transform (DCT) is applied to each of these 8 × 8 blocks. In JPEG decompression, the Inverse Discrete Cosine Transform (IDCT) is applied to the 8 × 8 DCT coefficient blocks. DCT and IDCT are defined as follows:DCT:F(u,v)=14C(u)C(v)∑i=07∑j=07f(i,j)cos((2i+1)uπ/16)cos((2j+1)vπ/16)DCT:F(u,v)=14C(u)C(v)∑i=07∑j=07f(i,j)cos((2i+1)uπ/16)cos((2j+1)vπ/16){\textbackslash}begin\{array\} \{*\{20\}l\}DCT: \&{\textbackslash}{\textbackslash} \& F(u,v) = \{1{\textbackslash}over4\}C(u)C(v){\textbackslash}sum{\textbackslash}limits\_\{i = 0\}{\textasciicircum}7 {\textbackslash}sum{\textbackslash}limits\_\{j = 0\}{\textasciicircum}7 f(i,j) {\textbackslash}{\textbackslash} \&{\textbackslash}cos ((2i + 1)u{\textbackslash}pi /16){\textbackslash}cos ((2j + 1)v{\textbackslash}pi /16){\textbackslash}end\{array\}IDCT:f(i,j)=14∑u=07∑v=07C(u)C(v)F(u,v)cos((2i+1)uπ/16)cos((2j+1)vπ/16)IDCT:f(i,j)=14∑u=07∑v=07C(u)C(v)F(u,v)cos((2i+1)uπ/16)cos((2j+1)vπ/16){\textbackslash}begin\{array\} \{*\{20\}l\}IDCT: \& {\textbackslash}{\textbackslash} \& f(i,j) = \{1{\textbackslash}over 4\}{\textbackslash}sum{\textbackslash}limits\_\{u = 0\}{\textasciicircum}7 {\textbackslash}sum{\textbackslash}limits\_\{v = 0\}{\textasciicircum}7 C(u)C(v)F(u,v) {\textbackslash}{\textbackslash} \&{\textbackslash}cos ((2i + 1)u {\textbackslash}pi /16){\textbackslash}cos ((2j + 1)v {\textbackslash}pi /16) {\textbackslash}end\{array\} where f(i, j) and F(u, ν) are respectively the pixel value and the DCT coefficient, and ...},
language = {en},
urldate = {2016-05-03},
booktitle = {Encyclopedia of {Multimedia}},
publisher = {Springer US},
editor = {Furht, Borko},
year = {2008},
note = {00000},
pages = {186--188}
}
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DCT and IDCT are defined as follows:DCT:F(u,v)=14C(u)C(v)∑i=07∑j=07f(i,j)cos((2i+1)uπ/16)cos((2j+1)vπ/16)DCT:F(u,v)=14C(u)C(v)∑i=07∑j=07f(i,j)cos((2i+1)uπ/16)cos((2j+1)vπ/16)\\textbackslashbegin\\array\\ \\*\\20\\l\\DCT: &\\textbackslash\\textbackslash & F(u,v) = \\1\\textbackslashover4\\C(u)C(v)\\textbackslashsum\\textbackslashlimits_\\i = 0\\\\textasciicircum7 \\textbackslashsum\\textbackslashlimits_\\j = 0\\\\textasciicircum7 f(i,j) \\textbackslash\\textbackslash &\\textbackslashcos ((2i + 1)u\\textbackslashpi /16)\\textbackslashcos ((2j + 1)v\\textbackslashpi /16)\\textbackslashend\\array\\IDCT:f(i,j)=14∑u=07∑v=07C(u)C(v)F(u,v)cos((2i+1)uπ/16)cos((2j+1)vπ/16)IDCT:f(i,j)=14∑u=07∑v=07C(u)C(v)F(u,v)cos((2i+1)uπ/16)cos((2j+1)vπ/16)\\textbackslashbegin\\array\\ \\*\\20\\l\\IDCT: & \\textbackslash\\textbackslash & f(i,j) = \\1\\textbackslashover 4\\\\textbackslashsum\\textbackslashlimits_\\u = 0\\\\textasciicircum7 \\textbackslashsum\\textbackslashlimits_\\v = 0\\\\textasciicircum7 C(u)C(v)F(u,v) \\textbackslash\\textbackslash &\\textbackslashcos ((2i + 1)u \\textbackslashpi /16)\\textbackslashcos ((2j + 1)v \\textbackslashpi /16) \\textbackslashend\\array\\ where f(i, j) and F(u, ν) are respectively the pixel value and the DCT coefficient, and ...","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":"186–188","bibtex":"@incollection{furht_discrete_2008-1,\n\ttitle = {Discrete {Cosine} {Transform} ({DCT})},\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_304},\n\tabstract = {DefinitionDiscrete Cosine Transform is a technique applied to image pixels in spatial domain in order to transform them into a frequency domain in which redundancy can be identified.In JPEG compression [1], image is divided into 8 × 8 blocks, then the two-dimensional Discrete Cosine Transform (DCT) is applied to each of these 8 × 8 blocks. In JPEG decompression, the Inverse Discrete Cosine Transform (IDCT) is applied to the 8 × 8 DCT coefficient blocks. DCT and IDCT are defined as follows:DCT:F(u,v)=14C(u)C(v)∑i=07∑j=07f(i,j)cos((2i+1)uπ/16)cos((2j+1)vπ/16)DCT:F(u,v)=14C(u)C(v)∑i=07∑j=07f(i,j)cos((2i+1)uπ/16)cos((2j+1)vπ/16){\\textbackslash}begin\\{array\\} \\{*\\{20\\}l\\}DCT: \\&{\\textbackslash}{\\textbackslash} \\& F(u,v) = \\{1{\\textbackslash}over4\\}C(u)C(v){\\textbackslash}sum{\\textbackslash}limits\\_\\{i = 0\\}{\\textasciicircum}7 {\\textbackslash}sum{\\textbackslash}limits\\_\\{j = 0\\}{\\textasciicircum}7 f(i,j) {\\textbackslash}{\\textbackslash} \\&{\\textbackslash}cos ((2i + 1)u{\\textbackslash}pi /16){\\textbackslash}cos ((2j + 1)v{\\textbackslash}pi /16){\\textbackslash}end\\{array\\}IDCT:f(i,j)=14∑u=07∑v=07C(u)C(v)F(u,v)cos((2i+1)uπ/16)cos((2j+1)vπ/16)IDCT:f(i,j)=14∑u=07∑v=07C(u)C(v)F(u,v)cos((2i+1)uπ/16)cos((2j+1)vπ/16){\\textbackslash}begin\\{array\\} \\{*\\{20\\}l\\}IDCT: \\& {\\textbackslash}{\\textbackslash} \\& f(i,j) = \\{1{\\textbackslash}over 4\\}{\\textbackslash}sum{\\textbackslash}limits\\_\\{u = 0\\}{\\textasciicircum}7 {\\textbackslash}sum{\\textbackslash}limits\\_\\{v = 0\\}{\\textasciicircum}7 C(u)C(v)F(u,v) {\\textbackslash}{\\textbackslash} \\&{\\textbackslash}cos ((2i + 1)u {\\textbackslash}pi /16){\\textbackslash}cos ((2j + 1)v {\\textbackslash}pi /16) {\\textbackslash}end\\{array\\} where f(i, j) and F(u, ν) are respectively the pixel value and the DCT coefficient, and ...},\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 = {186--188}\n}\n\n","editor_short":["Furht, B."],"key":"furht_discrete_2008-1","id":"furht_discrete_2008-1","bibbaseid":"furht-discretecosinetransformdct-2008","role":"editor","urls":{"Paper":"http://link.springer.com/referenceworkentry/10.1007/978-0-387-78414-4_304"},"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","cosine","transform","dct"],"title":"Discrete Cosine Transform (DCT)","year":2008,"dataSources":["jAxurbvLP8q5LTdLa"]}