The discrete cosine transform DCT

The DCT is defined as an inner product of cosine basis functions N-l 

The DCT was developed as an approximation to the KLT where, for large values of p, the DCT approximately diagonalises the above matrix Aj. In fact, the DCT is asymptotically equivalent to the KLT of a stationary process when the image block size tends to infinity (N oo). Even for small values of N (say, N = 64), the basis functions for the KLT and DCT for many natural images look remarkably similar. 

the DCT can be computed very efficiently with a time-complexity 0(N log N) as opposed to O(N ) for the case of the KLT. The computation can be made even more very efficient by blocking the image into K square blocks, each block with /lT x x/N pixels where N = K N (i.e. create disjoint blocks of sub-images of, say, 8 x 8 or 16 x 16 pixels), thereby reducing the DCT computations to each block. This makes the DCT the preferred algorithm in many standard eommercial compression algorithms such as JPEG. 

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Methods developed for traditional lossy image compression can be used in the compression of multispectral images. The discrete cosine transform (DCT),... 

The invertible transformation stage uses a different mathematical basis of features in an attempt to decorrelate the data. The resulting data will have a set of features that capture most of the independent features in the original data set. Typical features used include frequency and spatial location. The transformation is nearly loss-less as it is implemented using real arithmetic and is subject to (small) truncation errors. Examples of invertible transforms include the discrete cosine transform (DCT), the discrete wavelet transform (DWT) and the wavelet packet transform (WPT). We will investigate these transforms later. 

The MFCC can then be derived by taking the log of the band-passed frequency response and calculating the Discrete Cosine Transform (DCT) for each intermediate signal. 

Take an image of size MxM Consider the image in DCT domain, where the Discrete Cosine Transform has been done using block size N/N. There is a secret key K and a secret watermarking signal w. The watermarking signal w = [nq,..., wt is t = bits binary pattern. 

Frequency Analysis. The Discrete Fourier Transform (and its fast implementation, the Fast Fourier Transform [Brigham, 1974]) (FFT) as well as its cousin, the Discrete Cosine Transform [Rao and Yip, 1990] (DCT) require block operations, as opposed to single sample inputs. The DFT can be described recursively, with the basis being the 2 point DFT calculated as follows ... 

Before a frame is quantized, it is desirable to eliminate redundant data in the frame. There are many techniques for reducing the spatial redundancy, but by far, DPCM and discrete cosine transformation-(DCT-) based coding are the most widely used techniques for intraframe compression. (Rabbani and Jones (1991) discuss many image compression techniques. See Further Information.)... 

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