mathjournal94

# mathjournal94 - Mathematica Journal, 4(1), 1994, p. 81-88...

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Image Compression Using the Discrete Cosine Transform Andrew B. Watson NASA Ames Research Center Abstract The discrete cosine transform (DCT) is a technique for converting a signal into elementary frequency components. It is widely used in image compression. Here we develop some simple functions to compute the DCT and to compress images. These functions illustrate the power of Mathematica in the prototyping of image processing algorithms. The rapid growth of digital imaging applications, including desktop publishing, multimedia, teleconferencing, and high-definition television (HDTV) has increased the need for effective and standardized image compression techniques. Among the emerging standards are JPEG, for compression of still images [Wallace 1991]; MPEG, for compression of motion video [Puri 1992]; and CCITT H.261 (also known as Px64), for compression of video telephony and teleconferencing. All three of these standards employ a basic technique known as the discrete cosine transform (DCT). Developed by Ahmed, Natarajan, and Rao [1974], the DCT is a close relative of the discrete Fourier transform (DFT). Its application to image compression was pioneered by Chen and Pratt [1984]. In this article, I will develop some simple functions to compute the DCT and show how it is used for image compression. We have used these functions in our laboratory to explore methods of optimizing image compression for the human viewer, using information about the human visual system [Watson 1993]. The goal of this paper is to illustrate the use of Mathematica in image processing and to provide the reader with the basic tools for further exploration of this subject. The One-Dimensional Discrete Cosine Transform The discrete cosine transform of a list of n real numbers s ( x ), x = 0, . .., n -1, is the list of length n given by: Mathematica Journal, 4(1), 1994, p. 81-88

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( 29 = 2 n C u ( 29 s x ( 29 cos 2x + 1 ( 29 u π 2n x = 0 n - 1 ε u = 0,. ..,n where C u ( 29 = 2 - 1 2 for u = 0 = 1 otherwise Each element of the transformed list S(u) is the inner (dot) product of the input list s(x) and a basis vector . The constant factors are chosen so that the basis vectors are orthogonal and normalized. The eight basis vectors for n = 8 are shown in Figure 1. The DCT can be written as the product of a vector (the input list) and the n x n orthogonal matrix whose rows are the basis vectors. This matrix, for n = 8, can be computed as follows: DCTMatrix = Table[ If[ k==0, Sqrt[1/8], Sqrt[2/8] Cos[Pi (2j + 1) k/16] ], {k, 0, 7}, {j, 0, 7}] // N; We can check that the matrix is orthogonal: DCTMatrix . Transpose[DCTMatrix] // Chop // MatrixForm 1. 0 0 0 0 0 0 0 0 1. 0 0 0 0 0 0 0 0 1. 0 0 0 0 0 0 0 0 1. 0 0 0 0 0 0 0 0 1. 0 0 0 0 0 0 0 0 1. 0 0 0 0 0 0 0 0 1. 0 0
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mathjournal94 - Mathematica Journal, 4(1), 1994, p. 81-88...

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