Discrete Cosine Transform (DCT). Furht, B., editor In Encyclopedia of Multimedia, pages 186–188. Springer US, 2008. 00000
Discrete Cosine Transform (DCT) [link]Paper  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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