Quantization (signal processing): Difference between revisions

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:<math> D=E[(x-Q(x))^2] = \int_{-\infty}^{\infty} (x-Q(x))^2f(x)dx = \sum_{k=1}^{M} \int_{b_{k-1}}^{b_k} (x-y_k)^2 f(x)dx =\sum_{k=1}^{M} d_k </math>.

Finding an optimal solution to the above problem results in a quantizer sometimes called a MMSQE (minimum mean-square quantization error) solution, and the resulting PDF-optimized (non-uniform) quantizer is referred to as a ''Lloyd–Max'' quantizer, named after two people who independently developed iterative methods<ref name=GrayNeuhoff/><ref>{{cite journal | last=Lloyd | first=S. | title=Least squares quantization in PCM | journal=IEEE Transactions on Information Theory | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=28 | issue=2 | year=1982 | issn=0018-9448 | doi=10.1109/tit.1982.1056489 | pages=129–137| s2cid=10833328 | citeseerx=10.1.1.131.1338 }} (work documented in a manuscript circulated for comments at [[Bell Laboratories]] with a department log date of 31 July 1957 and also presented at the 1957 meeting of the [[Institute of Mathematical Statistics]], although not formally published until 1982).</ref><ref>{{cite journal | last=Max | first=J. | title=Quantizing for minimum distortion | journal=IEEE Transactions on Information Theory | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=6 | issue=1 | year=1960 | issn=0018-9448 | doi=10.1109/tit.1960.1057548 | pages=7–12| bibcode=1960ITIT....6....7M }}</ref> to solve the two sets of simultaneous equations resulting from <math> {\partial D / \partial b_k} = 0 </math> and <math>{\partial D/ \partial y_k} = 0 </math>, as follows:

:<math> {\partial D \over\partial b_k} = 0 \Rightarrow b_k = {y_k + y_{k+1} \over 2} </math>,

which places each threshold at the midpoint between each pair of reconstruction values, and