Advances in visual data compression and communication by Wu F.

By Wu F.

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The matrix H is expressed in terms of its column as H = [h1 h2 h3 h4 h5 h6 h7 ] . 63) We can observe that the columns of H consist of the binary representations of the numbers 1 through 7, though not in numerical order. On the basis of this observation, we can generalize other Hamming codes. Hamming codes of length n = 2m 1 and dimension k = 2m m 1 exist for every m 2, having parity check matrices whose columns are binary representations of the numbers from 1 through n. Supposing that a codeword y is sent and the received vector is yˆ = y + z (addition modulo 2).

Source decoding recovers the source from the compressed bits. This chapter is organized according to T. M. Cover and J. A. Thomas’s book Elements of Information Theory [5], which is a comprehensive book about Shannon’s information theory. We do not pay much attention to either the completeness of the theory or the details of how to prove the theorems step by step. Instead, we try to highlight the core ideas behind the theory and make them more intuitively understood for further practical research on visual data compression and communication.

An intuitive idea is described here about why we can transmit C bits of information over a channel. The basic idea is that, for large block lengths, every channel looks like the noisy typewriter channel and the channel has a subset of inputs that produce essentially disjointed sequences at the output. For each input n-sequence, we wish to ensure that no two Y sequences produce the same Yˆ output sequence. Otherwise, we will not be able to decide which Y sequences was sent. The total ˆ number of possible Yˆ sequences is 2nH(Y ) .

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