Calculation Scheme Based on a Weighted Primitive: Application to Image Processing Transforms
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Research Article Calculation Scheme Based on a Weighted Primitive: Application to Image Processing Transforms Mar´ıa Teresa Signes Pont, Juan Manuel Garc´ıa Chamizo, Higinio Mora Mora, and Gregorio de Miguel Casado Departamento de Tecnolog´ıa Inform´atica y Computaci´on, Universidad de Alicante, 03690 San Vicente del Raspeig, 03071 Alicante, Spain Received 29 September 2006; Accepted 6 March 2007 Recommended by Nicola Mastronardi This paper presents a method to improve the calculation of functions which specially demand a great amount of computing resources. The method is based on the choice of a weighted primitive which enables the calculation of function values under the scope of a recursive operation. When tackling the design level, the method shows suitable for developing a processor which achieves a satisfying trade-off between time delay, area costs, and stability. The method is particularly suitable for the mathematical transforms used in signal processing applications. A generic calculation scheme is developed for the discrete fast Fourier transform (DFT) and then applied to other integral transforms such as the discrete Hartley transform (DHT), the discrete cosine transform (DCT), and the discrete sine transform (DST). Some comparisons with other well-known proposals are also provided. Copyright © 2007 Mar´ıa Teresa Signes Pont et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1.
INTRODUCTION
Mathematical notation aside, the motivation behind integral transforms is easy to understand. There are many classes of problems that are extremely difficult to solve or, at least, quite unwieldy from the algebraic standpoint in their original domains. An integral transform maps an equation from its original domain (time or space domain) into another domain (frequency domain). Manipulating and solving the equation in the target domain is, ideally, easier than manipulating and solving it in the original domain. The solution is then mapped back into the original domain. Integral transforms work because they are based upon the concept of spectral factorization over orthonormal bases. Equation (1) shows the generic formulation of a discrete integral transform where f (x), 0 ≤ x < N, and F(u), 0 ≤ u < N, are the original and the transformed sequences, respectively. Both have N = 2n values, n ∈ N and T(x, u) is the kernel of the transform F(u) =
N −1 x=0
T(x, u) f (x).
(1)
The inverse transform can be defined in√a similar way. Table 1 shows some integral transforms ( j = −1 as usual). The Fourier transform (FT) is a reference tool in image filtering [1, 2] and reconstruction [3]. A fast Fourier transform (FFT) scheme has been used in OFDM modulation (orthogonal frequency division multiplexing) and has shown to be a valuable tool in the scope of communications [4, 5]. The most relevant algorithm for FFT calculation was developed in 1965 by Cooley and
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