26.2 Multistage Pulse Code Modulation
In this section, we present a new concept called multistage pulse coding modulation (MPCM) that can be used for encoding spectral signatures in a progressive manner.
MPCM was originally developed for applications in image progressive transmission and reconstruction (Tzou, 1987; Wang and Goldberg, 1988, 1989). It can be viewed as a progressive version of a commonly used coding scheme in communications, pulse code modulation (PCM). It expands the hard-decision PCM-based quantizer to a soft-decision quantizer in such a fashion that it allows PCM to have a nondecision region that passes on its decisions to next stage progressively. As a result, a decision can be refined stage-by-stage so as to improve quantization results. The detailed idea of MPCM can be described as follows.
PCM is a quantizer, denoted by Q(x) that requires a set of quantization levels 
and a corresponding set of quantization thresholds 
. It quantizes a signal function x(n) according to
where τ0 and τM are initial conditions determined by the domain of the signal function x(n). It is a hard decision-based quantizer, referred to as a hard quantization because Q(x(n)) must make ...
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