Shifted Fourier transform-based tensor algorithms for the 2-D DCT
In this paper, tensor algorithms for calculating the two-dimensional (2-D) discrete cosine transform (DCT) are presented. The tensor approach is based on the concept of the covering revealing the transforms, which yields in particular the splitting of the shifted 2/sup r//spl times/2/sup r/-point Fo...
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| Published in | IEEE transactions on signal processing Vol. 49; no. 9; pp. 2113 - 2126 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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New York, NY
IEEE
01.09.2001
Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN | 1053-587X 1941-0476 |
| DOI | 10.1109/78.942639 |
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| Abstract | In this paper, tensor algorithms for calculating the two-dimensional (2-D) discrete cosine transform (DCT) are presented. The tensor approach is based on the concept of the covering revealing the transforms, which yields in particular the splitting of the shifted 2/sup r//spl times/2/sup r/-point Fourier and cosine transforms into 2/sup r-1/3 one-dimensional (1-D) incomplete 2/sup r/-point transforms. The multiplicative complexity of the 2-D 2/sup r//spl times/2/sup r/-point discrete cosine transforms in terms of the tensor representation is 4/sup r/3-2/sup r-2/(r/sup 2/+7r+14), which is reduced to 4/sup r/8/3-2/sup r/(r/sup 2/+7r+10)-20/3 when using the improved tensor algorithm. The multiplicative complexity in the general L/sup r//spl times/L/sup r/ case, with a prime L>2, as well as in the L/sub 1/L/sub 2//spl times/L/sub 1/L/sub 2/ case, with arbitrary co-prime L/sub 1/, L/sub 2/>1, is provided. The examples of the tensor algorithms for calculating the 8/spl times/8-point DCT through 104, 88, and 84 multiplications are given in detail. Based on the proposed concept, the fast algorithm for calculating the 1-D DCT-I is also developed. The multiplicative complexity of the 2/sup r/-point DCT-I is 2/sup r+1/-(r-2)(r+5)/2-8. The comparative estimates with the known algorithms are given. |
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| AbstractList | In this paper, tensor algorithms for calculating the two-dimensional (2-D) discrete cosine transform (DCT) are presented. The tensor approach is based on the concept of the covering revealing the transforms, which yields in particular the splitting of the shifted 2 super(r)2 super(r)-p oint Fourier and cosine transforms into 2 super(r-1)3 one-dimensional (1-D) incomplete 2 super(r)-point transforms. The multiplicative complexity of the 2-D 2 super(r)2 super(r)- point discrete cosine transforms in terms of the tensor representation is 4 super(r)3-2 super(r-2)(r super(2)+7r+14), which is reduced to 4 super(r/8)3-2 super(r)(r super(2)+7r+10)-20/3 when using the improved tensor algorithm. The multiplicative complexity in the general L super(r)L super(r) case, with a prime L>2, as well as in the L sub(1)L sub(2)L sub(1)L sub(2) case, with arbitrary co-prime L sub(1), L sub(2)>1, is provided. The examples of the tensor algorithms for calculating the 88-point DCT through 104, 88, and 84 multiplications are given in detail. Based on the proposed concept, the fast algorithm for calculating the 1-D DCT-I is also developed. The multiplicative complexity of the 2 super(r)-point DCT-I is 2 super(r+1)-(r-2)(r+5)/2-8. The comparative estimates with the known algorithms are given In this paper, tensor algorithms for calculating the two-dimensional (2-D) discrete cosine transform (DCT) are presented. The tensor approach is based on the concept of the covering revealing the transforms, which yields in particular the splitting of the shifted 2 super(r) x 2 super(r)-point Fourier and cosine transforms into 2 super(r-1) 3 one-dimensional (1-D) incomplete 2 super(r )-point transforms. The multiplicative complexity of the 2-D 2 super(r) x 2 super(r)-point discrete cosine transforms in terms of the tensor representation is 4 super(r)3 2 super(r-2) (r super(2) + 7r + 14), which is reduced to 4 super(r)8/3 - 2 super(r ) (r super(2) + 7r + 10) - 20/3 when using the improved tensor algorithm. The multiplicative complexity in the general L super(r) x L super(r) case, with a prime L > 2, as well as in the L sub(1)L sub(2) x L sub(1) L sub(2) case, with arbitrary co-prime L sub(1), L sub(2) > 1, is provided. The examples of the tensor algorithms for calculating the 8 x 8-point DCT through 104, 88, and 84 multiplications are given in detail. Based on the proposed concept, the fast algorithm for calculating the 1-D DCT-1 is also developed. The multiplicative complexity of the 2 super(r)-point DCT-I is 2 super(r+1) - (r - 2)(r + 5)/2 - 8. The comparative estimates with the known algorithms are given. In this paper, tensor algorithms for calculating the two-dimensional (2-D) discrete cosine transform (DCT) are presented. The tensor approach is based on the concept of the covering revealing the transforms, which yields in particular the splitting of the shifted 2/sup r//spl times/2/sup r/-point Fourier and cosine transforms into 2/sup r-1/3 one-dimensional (1-D) incomplete 2/sup r/-point transforms. The multiplicative complexity of the 2-D 2/sup r//spl times/2/sup r/-point discrete cosine transforms in terms of the tensor representation is 4/sup r/3-2/sup r-2/(r/sup 2/+7r+14), which is reduced to 4/sup r/8/3-2/sup r/(r/sup 2/+7r+10)-20/3 when using the improved tensor algorithm. The multiplicative complexity in the general L/sup r//spl times/L/sup r/ case, with a prime L>2, as well as in the L/sub 1/L/sub 2//spl times/L/sub 1/L/sub 2/ case, with arbitrary co-prime L/sub 1/, L/sub 2/>1, is provided. The examples of the tensor algorithms for calculating the 8/spl times/8-point DCT through 104, 88, and 84 multiplications are given in detail. Based on the proposed concept, the fast algorithm for calculating the 1-D DCT-I is also developed. The multiplicative complexity of the 2/sup r/-point DCT-I is 2/sup r+1/-(r-2)(r+5)/2-8. The comparative estimates with the known algorithms are given. In this paper, tensor algorithms for calculating the two-dimensional (2-D) discrete cosine transform (DCT) are presented. The tensor approach is based on the concept of the covering revealing the transforms, which yields in particular the splitting of the shifted 2 (r)x2(r)-point Fourier and cosine transforms into 2(r-1)3 one-dimensional (1-D) incomplete 2(r)-point transforms. The multiplicative complexity of the 2-D 2(r)x2(r)-point discrete cosine transforms in terms of the tensor representation is 4(r)3-2(r-2)(r (2) 7r 14), which is reduced to 4(r/8)3-2(r)(r(2) 7r 10)-20/3 when using the improved tensor algorithm. The multiplicative complexity in the general L(r)xL(r) case, with a prime L > 2, as well as in the L(1)L(2)xL(1)L(2) case, with arbitrary co-prime L(1), L(2) > 1, is provided. The examples of the tensor algorithms for calculating the 8x8-point DCT through 104, 88, and 84 multiplications are given in detail. Based on the proposed concept, the fast algorithm for calculating the 1-D DCT-I is also developed. The multiplicative complexity of the 2(r)-point DCT-I is 2(r 1)-(r-2)(r 5)/2-8. The comparative estimates with the known algorithms are given |
| Author | Grigoryan, A.M. Agaian, S.S. |
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| Cites_doi | 10.1109/83.650856 10.1109/ICASSP.1989.266596 10.1109/tassp.1987.1165060 10.1109/26.87153 10.1109/82.486464 10.1109/78.157218 10.1109/41.661303 10.1109/78.815487 10.1109/TASSP.1980.1163351 10.1109/ICASSP.1985.1168211 10.1109/78.80854 10.1109/ICSIGP.1996.567049 10.1109/78.157217 10.1109/76.664096 10.1109/78.205723 10.1007/978-3-642-45450-9 10.1109/TASSP.1984.1164443 10.1109/ICSIGP.1996.567066 10.1109/83.641417 10.1109/78.661324 10.1109/78.80833 10.1109/78.365284 10.1109/78.902116 10.1109/T-C.1974.223784 10.1006/jvci.1997.0323 10.1109/TASSP.1976.1162839 10.1016/0734-189X(88)90140-5 10.1016/0041-5553(86)90044-3 10.1109/76.585925 10.1002/(SICI)1520-6424(199708)80:8<81::AID-ECJA9>3.0.CO;2-7 10.1109/TCOM.1985.1096337 10.1007/978-1-4612-3912-3 10.1109/78.150010 10.1016/S0165-1684(96)00132-6 10.1109/TASSP.1985.1164737 |
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| Keywords | Fourier transformation Signal processing Mathematical method Discrete cosine transforms Algorithm Computational complexity Tensor method |
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| References_xml | – ident: ref10 doi: 10.1109/83.650856 – volume: 29 start-page: 20 issue: 12 year: 1986 ident: ref43 article-title: An optimal algorithm for computing the two-dimensional discrete Fourier transform publication-title: Izvestiya VUZov SSSR, Radioelectronica – ident: ref17 doi: 10.1109/ICASSP.1989.266596 – ident: ref15 doi: 10.1109/tassp.1987.1165060 – ident: ref49 doi: 10.1109/26.87153 – ident: ref28 doi: 10.1109/82.486464 – ident: ref40 doi: 10.1109/78.157218 – ident: ref8 doi: 10.1109/41.661303 – volume: 38 start-page: 397 year: 1991 ident: ref18 article-title: Fast algorithm for the discrete cosine transform publication-title: IEEE Trans. Circuits Syst. II – start-page: 21 issue: 1 year: 1986 ident: ref42 article-title: Tensor representation of the two-dimensional discrete Fourier transform and new orthogonal functions publication-title: Avtometria – ident: ref46 doi: 10.1109/78.815487 – ident: ref33 doi: 10.1109/TASSP.1980.1163351 – volume: 27 start-page: 52 issue: 10 year: 1984 ident: ref41 article-title: Two-dimensional Fourier transform algorithm publication-title: Izvestiya VUZov SSSR, Radioelectronica – start-page: 772 volume-title: Proc. ICASSP ident: ref26 article-title: Prime factor decomposition of the discrete cosine transform – volume: C-25 start-page: 1004 year: 1977 ident: ref14 article-title: A fast computational algorithm for discrete cosine transform publication-title: IEEE Trans. Comput. – ident: ref21 doi: 10.1109/ICASSP.1985.1168211 – ident: ref24 doi: 10.1109/78.80854 – ident: ref20 doi: 10.1109/ICSIGP.1996.567049 – start-page: 2025 volume-title: Proc. ICASSP ident: ref29 article-title: A method for fast approximate computation discrete cosine transforms – ident: ref38 doi: 10.1109/78.157217 – ident: ref39 doi: 10.1109/76.664096 – ident: ref47 doi: 10.1109/78.205723 – ident: ref1 doi: 10.1007/978-3-642-45450-9 – volume-title: JPEG Still Image Compression Standard year: 1993 ident: ref5 – ident: ref23 doi: 10.1109/TASSP.1984.1164443 – ident: ref30 doi: 10.1109/ICSIGP.1996.567066 – ident: ref32 doi: 10.1109/83.641417 – ident: ref25 doi: 10.1109/78.661324 – ident: ref34 doi: 10.1109/78.80833 – ident: ref31 doi: 10.1109/78.365284 – ident: ref45 doi: 10.1109/78.902116 – ident: ref13 doi: 10.1109/T-C.1974.223784 – volume-title: Fundamentals of Digital Image Processing year: 1989 ident: ref11 – volume: ASSP-37 start-page: 237 year: 1987 ident: ref27 article-title: Input and output mapping for a prime-factor-drcomposed computation of discrete cosine transform publication-title: IEEE Trans. Acoust. Speech, Signal Processing – ident: ref6 doi: 10.1006/jvci.1997.0323 – ident: ref9 doi: 10.1109/TASSP.1976.1162839 – ident: ref4 doi: 10.1016/0734-189X(88)90140-5 – volume-title: Discrete Cosine Transform—Algorithms, Advantages, Applications year: 1990 ident: ref12 – ident: ref44 doi: 10.1016/0041-5553(86)90044-3 – ident: ref37 doi: 10.1109/76.585925 – start-page: 1515 volume-title: Proc. ICASSP ident: ref22 article-title: Polynomial transform computation of the two-dimensional DCT – ident: ref7 doi: 10.1002/(SICI)1520-6424(199708)80:8<81::AID-ECJA9>3.0.CO;2-7 – ident: ref3 doi: 10.1109/TCOM.1985.1096337 – volume-title: Multiplicative Complexity, Convolution, and the DFT year: 1988 ident: ref48 doi: 10.1007/978-1-4612-3912-3 – ident: ref19 doi: 10.1109/78.150010 – volume: 1002 start-page: 541 volume-title: Proc. SPIE Int. Soc. Opt. Eng. ident: ref16 article-title: A fast recursive algorithm for computing discrete cosine transform – volume-title: Transform Coding Theory year: 1985 ident: ref2 – ident: ref36 doi: 10.1016/S0165-1684(96)00132-6 – ident: ref35 doi: 10.1109/TASSP.1985.1164737 |
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| SubjectTerms | Algorithms Applied sciences Complexity Discrete cosine transform Discrete cosine transforms Discrete Fourier transforms Discrete transforms Exact sciences and technology Fast Fourier transforms Fourier analysis Fourier transforms Image coding Information, signal and communications theory Mathematical analysis Mathematical methods Representations Signal processing algorithms Telecommunications and information theory Tensile stress Tensors Transform coding Transforms Two dimensional displays |
| Title | Shifted Fourier transform-based tensor algorithms for the 2-D DCT |
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