Discrete Cosine Transform Thesis

Discrete Cosine Transform Thesis-88
Multidimensional DCTs (MD DCTs) are developed to extend the concept of DCT on MD Signals. A new variety of fast algorithms are also developed to reduce the computational complexity of implementing DCT. DCTs are also widely employed in solving partial differential equations by spectral methods, where the different variants of the DCT correspond to slightly different even/odd boundary conditions at the two ends of the array.The discrete cosine transform (DCT) was first conceived by Nasir Ahmed while working at the University of Texas, and he proposed the concept to the National Science Foundation in 1972. DCTs are also closely related to Chebyshev polynomials, and fast DCT algorithms (below) are used in Chebyshev approximation of arbitrary functions by series of Chebyshev polynomials, for example in Clenshaw–Curtis quadrature.

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(A similar problem arises for the DST, in which the odd left boundary condition implies a discontinuity for any function that does not happen to be zero at that boundary.) In contrast, a DCT where both boundaries are even always yields a continuous extension at the boundaries (although the slope is generally discontinuous).

This is why DCTs, and in particular DCTs of types I, II, V, and VI (the types that have two even boundaries) generally perform better for signal compression than DFTs and DSTs.

These choices lead to all the standard variations of DCTs and also discrete sine transforms (DSTs).

Each boundary can be either even or odd (2 choices per boundary) and can be symmetric about a data point or the point halfway between two data points (2 choices per boundary), for a total of 2 × 2 × 2 × 2 = 16 possibilities.

element (top-left) is the DC (zero-frequency) component and entries with increasing vertical and horizontal index values represent higher vertical and horizontal spatial frequencies.

Multidimensional DCTs (MD DCTs) have several applications mainly 3-D DCT-II has several new applications like Hyperspectral Imaging coding systems, Due to enhancement in the hardware, software and introduction of several fast algorithms, the necessity of using M-D DCTs is rapidly increasing.In particular, a DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. These research papers, along with the original 1974 paper by Ahmed, Natarajan, and Rao, were cited by the Joint Photographic Experts Group as the basis for JPEG's lossy image compression algorithm in 1992.The DCTs are generally related to Fourier Series coefficients of a periodically and symmetrically extended sequence whereas DFTs are related to Fourier Series coefficients of a periodically extended sequence. DCT is also used in the more recent HEIF image format.DCTs are equivalent to DFTs of roughly twice the length, operating on real data with even symmetry (since the Fourier transform of a real and even function is real and even), whereas in some variants the input and/or output data are shifted by half a sample. In 1988, the first DCT-based video coding format, H.261, was introduced.There are eight standard DCT variants, of which four are common. DCT was later also used for the MPEG-1 format, introduced in 1992.Most directly, when using Fourier-related transforms to solve partial differential equations by spectral methods, the boundary conditions are directly specified as a part of the problem being solved.Or, for the MDCT (based on the type-IV DCT), the boundary conditions are intimately involved in the MDCT's critical property of time-domain aliasing cancellation.Half of these possibilities, those where the left boundary is even, correspond to the 8 types of DCT; the other half are the 8 types of DST.These different boundary conditions strongly affect the applications of the transform and lead to uniquely useful properties for the various DCT types.Its inverse, the type-III DCT, is correspondingly often called simply "the inverse DCT" or "the IDCT". For strongly correlated Markov processes, the DCT can approach the compaction efficiency of the Karhunen-Loève transform (which is optimal in the decorrelation sense).Two related transforms are the discrete sine transform (DST), which is equivalent to a DFT of real and odd functions, and the modified discrete cosine transform (MDCT), which is based on a DCT of overlapping data. As explained below, this stems from the boundary conditions implicit in the cosine functions.

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    Rights This volume was digitized and made accessible online due to deterioration of the original print copy. If you are the author of this work and would like to have online access removed, please contact the Library Administration Office, 785-532-7400, [email protected]

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