Gaussian DCT Coefficient Models
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Gaussian DCT Coefficient Models Saralees Nadarajah
Received: 5 May 2008 / Accepted: 21 August 2008 / Published online: 11 September 2008 © Springer Science+Business Media B.V. 2008
Abstract It has been known that the distribution of the discrete cosine transform (DCT) coefficients of most natural images follow a Laplace distribution. However, recent work has shown that the Laplace distribution may not be a good fit for certain type of images and that the Gaussian distribution will be a realistic model in such cases. Assuming this alternative model, we derive a comprehensive collection of formulas for the distribution of the actual DCT coefficient. The corresponding estimation procedures are derived by the method of moments and the method of maximum likelihood. Finally, the superior performance of the derived distributions over the Gaussian model is illustrated. It is expected that this work could serve as a useful reference and lead to improved modeling with respect to image analysis and image coding. Keywords Discrete cosine transform (DCT) · Gaussian distribution · Generalized hypergeometric function · Image analysis · Image coding · Incomplete gamma function · Kummer function · Modified Bessel function
1 Introduction A discrete cosine transform (DCT) expresses a sequence of finitely many data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT was first introduced by Ahmed et al. [1]. Later Wang and Hunt [2] introduced a complete set of variants of the DCT. The DCT is included in many mathematical packages, such as Matlab, Mathematica and GNU Octave. DCTs are important to numerous applications in science and engineering, from lossy compression of audio and images (where small high-frequency components can be discarded), to spectral methods for the numerical solution of partial differential equations to Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials. The use of cosine rather than sine functions is critical in these applications: for compression, it S. Nadarajah () University of Manchester, Manchester M13 9PL, UK e-mail: [email protected]
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S. Nadarajah
turns out that cosine functions are much more efficient, whereas for differential equations the cosines express a particular choice of boundary conditions. We refer the readers to Jain [3] and Rao and Yip [4] for comprehensive accounts of the theory and applications of the DCT. For a tutorial account see Duhamel and Vetterli [5]. The DCT is widely used in image coding and processing systems, especially for lossy data compression, because it has a strong “energy compaction” property (Rao and Yip [4]). DCT coding relies on the premise that pixels in an image exhibit a certain level of correlation with their neighboring pixels. Similarly in a video transmission system, adjacent pixels in consecutive frames show very high correlation. (Frames usually consist of a representation of the original data to be transmitted, together with other bits which may be used for error d
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