26,318 research outputs found

    Generalized approach for the realization of discrete cosine transform using cyclic convolutions

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    Fast multi-dimensional scattered data approximation with Neumann boundary conditions

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    An important problem in applications is the approximation of a function ff from a finite set of randomly scattered data f(xj)f(x_j). A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials {e2πikx}\{e^{2\pi i kx}\}. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data, an assumption that is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials cos(πkx)\cos (\pi kx) as basis functions. We show that scattered data approximation using cosine polynomials leads to a least squares problem involving certain Toeplitz+Hankel matrices. We derive estimates on the condition number of these matrices. Unlike other Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context cannot be diagonalized by the discrete cosine transform, but they still allow a fast matrix-vector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. Finally we demonstrate the performance of the proposed method by applying it to a two-dimensional geophysical scattered data problem

    Analysis of the DCT coefficient distributions for document coding

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    It is known that the distribution of the discrete cosine transform (DCT) coefficients of most natural images follow a Laplacian distribution, and this knowledge has been employed to improve decoder design. However, such is not the case for text documents. In this letter, we present an analysis of their DCT coefficient distributions, and show that a Gaussian distribution can be a realistic model. Furthermore, we can use a generalized Gaussian model to incorporate the Laplacian distribution found for natural images.published_or_final_versio

    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
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