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# A Basic Image Compression Example

Module by: Nick Kingsbury. E-mail the author

Summary: This module gives a basic image compression example.

We shall represent a monochrome (luminance) image by a matrix xx whose elements are xn x n , where n=( n1 n2 ) n n1 n2 is the integer vector of row and column indexes. The energy of xx is defined as

Energy of x=nx2n Energy of x n x n 2
(1)
where the sum is performed over all nn in xx.

Figure 1 shows the main blocks in any image coding system. The decoder is the inverse of the encoder. The three encoder blocks perform the following tasks:

• Energy compression - This is usually a transformation or filtering process which aims to concentrate a high proportion of the energy of the image xx into as few samples (coefficients) of yy as possible while preserving the total energy of xx in yy. This minimises the number of non-zero samples of yy which need to be transmitted for a given level of distortion in the reconstructed image x ^ x ^ .
• Quantisation - This represents the samples of yy to a given level of accuracy in the integer matrix qq. The quantiser step size controls the tradeoff between distortion and bit rate and may be adapted to take account of human visual sensitivities. The inverse quantiser reconstructs y ^ y ^ , the best estimate of yy from qq.
• Entropy coding - This encodes the integers in qq into a serial bit stream dd, using variable-length entropy codes which attempt to minimise the total number of bits in dd, based on the statistics (PDFs) of various classes of samples in qq.
The energy compression / reconstruction and the entropy coding / decoding processes are normally all lossless. Only the quantiser introduces loss and distortion: y ^ y ^ is a distorted version of yy, and hence x ^ x ^ is a distorted version of xx. In the absence of quantisation, if y ^ =y y ^ y , then x ^ =x x ^ x .

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