Fourier analysis started even before Fourier, but in the context of Fourier’s work on the Analytic Theory of Heat, it began with a claim regarding expansions of functions as trigonometrical series, what we now call Fourier series. Such expansions form the framework for writing down solutions to the heat equation in terms of initial temperature distributions. [refer to Enrique A. Gonzalez-Velasc...

The discrete cosine transform (DCT) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. It is equivalent to a DFT 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), where in some variants the input and/or output data are shifted by half a sampl...

Each Discrete Cosine Transform uses N real basis vectors whose components are cosines. In the DCT-4, for example, the jth component of v k is cos(j + 1 2)(k + 1 2) N. These basis vectors are orthogonal and the transform is extremely useful in image processing. If the vector x gives the intensities along a row of pixels, its cosine series P c k v k has the coeecients c k = (x; v k)=N. They are q...

In this paper a reversible discrete cosine transform (RDCT) is presented. N-point reversible transform is firstly presented, then 8-point RDCT is obtained by substituting the 2 and 4point reversible transforms for 2 and 4-point transforms which compose 8-point discrete cosine transform (DCT), respectively. Integer input signal can be losslessly recovered, although the transform coefficients are...

Discrete frequency transforms provide a method to obtain a global view of data within a window. Discrete cosine transform is the frequency transform for practical image processing because of its excellent energy compaction property. Another reason for its popularity is the existence of a fast implementation for the algorithm. In this article, the basics of the discrete cosine transform, its use...

The discrete Fourier transform (DFT) maps a nite number of discrete time-domain samples to the same number of discrete Fourier-domain samples. Being practical to compute, it is the primary transform applied to real-world sampled data in digital signal processing. The DFT has special relationships with the discrete-time Fourier transform and the continuous-time Fourier transform that let it be u...

This paper presents a systematic investigation of the modified discrete cosine transform/inverse modified discrete cosine transform (MDCT/IMDCT) algorithm using a matrix representation. This approach results in new understanding of the MDCT/IMDCT, enables the development of new algorithms, and makes clear the connection between the algorithms. We represent in a matrix form the IMDCT as the prod...

A reversible data hiding scheme based on integer DWT and histogram modification is proposed. In the scheme, the cover media is firstly transformed by Integer DWT (Discrete Wavelet Transformation); then information is embedded through the modification of histograms of the middle and high frequency sub-bands of the DWT coefficients. In order to increase the embedding capacity, a multi-level schem...

The magneto-rheological fluid Dual Clutch Transmission (MDCT for short) adopts the magneto-rheological (MR for short) effect of magneto-rheological fluid (MRF for short) and the working characters of Dual-Clutch Transmission (DCT for short), and combines the characteristics with long service life, no wear, simple structure, accurate control of MR effect with the principles of DCT technology, it...

In this paper, a new scheme employing the time-recursive approach to compute the discrete cosine transform (DCT) and discrete sine transform (DST) is presented. Using such approach, parallel lattice structure that can dually generate the DCT and DST simultaneously is developed. The resulting architecture is regular, module, and without global communication and can be applied to any transform si...