24 research outputs found

    A low multiplicative complexity fast recursive DCT-2 algorithm

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    A fast Discrete Cosine Transform (DCT) algorithm is introduced that can be of particular interest in image processing. The main features of the algorithm are regularity of the graph and very low arithmetic complexity. The 16-point version of the algorithm requires only 32 multiplications and 81 additions. The computational core of the algorithm consists of only 17 nontrivial multiplications, the rest 15 are scaling factors that can be compensated in the post-processing. The derivation of the algorithm is based on the algebraic signal processing theory (ASP).Comment: 4 pages, 2 figure

    Signal Flow Graph Approach to Efficient DST I-IV Algorithms

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    In this paper, fast and efficient discrete sine transformation (DST) algorithms are presented based on the factorization of sparse, scaled orthogonal, rotation, rotation-reflection, and butterfly matrices. These algorithms are completely recursive and solely based on DST I-IV. The presented algorithms have low arithmetic cost compared to the known fast DST algorithms. Furthermore, the language of signal flow graph representation of digital structures is used to describe these efficient and recursive DST algorithms having (n1)(n-1) points signal flow graph for DST-I and nn points signal flow graphs for DST II-IV

    Szybka dyskretna transformata sinusowa

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    The aim of this work is to present a fast calculation method for DST-IV and inverse transform, whose complexity isO(n*lgn) with regard to multiplication count. DST-IV was chosen due to lack of attractive dependencies in the matrix describing the transformation. Most works (both Polish and foreign [1,2,3]) elucidating effective methods of producing graphs describing the calculation process are based on DST-II/DST-III, whose analysis is by far less complicated. The proposed method will be presented in a mathematical form.Celem pracy jest zaproponowanie szybkiej metody obliczeniowej pozwalającej na wyznaczenie DST-IV (oraz transformaty odwrotnej) o złożoności O(n*lgn) pod względem liczby mnożeń. Wybór DST-IV podyktowany jest brakiem atrakcyjnych zależności w macierzy opisującej przekształcenie – większość prac polskich i zagranicznych [1,2,3] opisujących efektywne metody konstrukcji grafów przebiegu obliczeń opiera się o DST-II/DST-III, których analiza jest prostsza. Opracowana metoda zostanie przedstawiona w postaci matematycznej

    Fast cosine transform for FCC lattices

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    Voxel representation and processing is an important issue in a broad spectrum of applications. E.g., 3D imaging in biomedical engineering applications, video game development and volumetric displays are often based on data representation by voxels. By replacing the standard sampling lattice with a face-centered lattice one can obtain the same sampling density with less sampling points and reduce aliasing error, as well. We introduce an analog of the discrete cosine transform for the facecentered lattice relying on multivariate Chebyshev polynomials. A fast algorithm for this transform is deduced based on algebraic signal processing theory and the rich geometry of the special unitary Lie group of degree four.Comment: Presented at 13th APCA International Conference on Automatic Control and Soft Computing (CONTROLO 2018); 9 figure

    Type-II/III DCT/DST algorithms with reduced number of arithmetic operations

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    We present algorithms for the discrete cosine transform (DCT) and discrete sine transform (DST), of types II and III, that achieve a lower count of real multiplications and additions than previously published algorithms, without sacrificing numerical accuracy. Asymptotically, the operation count is reduced from ~ 2N log_2 N to ~ (17/9) N log_2 N for a power-of-two transform size N. Furthermore, we show that a further N multiplications may be saved by a certain rescaling of the inputs or outputs, generalizing a well-known technique for N=8 by Arai et al. These results are derived by considering the DCT to be a special case of a DFT of length 4N, with certain symmetries, and then pruning redundant operations from a recent improved fast Fourier transform algorithm (based on a recursive rescaling of the conjugate-pair split radix algorithm). The improved algorithms for DCT-III, DST-II, and DST-III follow immediately from the improved count for the DCT-II.Comment: 9 page

    Type-IV DCT, DST, and MDCT algorithms with reduced numbers of arithmetic operations

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    We present algorithms for the type-IV discrete cosine transform (DCT-IV) and discrete sine transform (DST-IV), as well as for the modified discrete cosine transform (MDCT) and its inverse, that achieve a lower count of real multiplications and additions than previously published algorithms, without sacrificing numerical accuracy. Asymptotically, the operation count is reduced from ~2NlogN to ~(17/9)NlogN for a power-of-two transform size N, and the exact count is strictly lowered for all N > 4. These results are derived by considering the DCT to be a special case of a DFT of length 8N, with certain symmetries, and then pruning redundant operations from a recent improved fast Fourier transform algorithm (based on a recursive rescaling of the conjugate-pair split radix algorithm). The improved algorithms for DST-IV and MDCT follow immediately from the improved count for the DCT-IV.Comment: 11 page

    Signal Flow Graph Approach to Efficient DST I-IV Algorithms

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    In this paper, fast and efficient discrete sine transformation (DST) algorithms are presented based on the factorization of sparse, scaled orthogonal, rotation, rotation-reflection, and butterfly matrices. These algorithms are completely recursive and solely based on DST I-IV. The presented algorithms have low arithmetic cost compared to the known fast DST algorithms. Furthermore, the language of signal flow graph representation of digital structures is used to describe these efficient and recursive DST algorithms having (n�1) points signal flow graph for DST-I and n points signal flow graphs for DST II-IV

    Harmonic analysis of finite lamplighter random walks

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    Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular when the underlying graph is the infinite path Z\mathbb{Z}. In the present paper, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the C2C_2-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph. In the case the graph has a transitive isometry group GG, we also describe the spectral analysis in terms of the representation theory of the wreath product C2GC_2\wr G. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples were already studied by Haggstrom and Jonasson by probabilistic methods.Comment: 29 page
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