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The concept of quantization originates in the field of electrical engineering.
The basic idea behind quantization is to describe a continuous function,
or one with a large number of samples, by a
few representative values.
Let denote the input signal and
denote quantized values, where
is the quantizer mapping function.
There will certainly be a distortion if we use
to represent
. In the least-square sense, the distortion can be measured by
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(3) |
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(4) |
A way to solve this coupled set of nonlinear equations is to first generate an initial set
,
then apply equations (6) and (7) alternately until convergence is obtained.
This iteration is well known as the Lloyd-Max quantization algorithm (LMQ). A common
modification is to form
The LMQ scheme is designed to find the best representation of a distribution,
which is not what I am trying to do in this instance. Instead I am trying
to the achieve the representation of with as few
points as possible.
The twist on the standard
LMQ scheme is the replacement of
in equation 5. Instead
of being the probability density function I construct an error
from a background piece-wise linear function. I first construct
by
linear interpolating between
samples. I then calculate
, the error from the piecewise linear background.
Figure 4 demonstrates
the methodology. Figure 4a shows a curve with `*' the initial
points and the resulting
function.
Figure 4b shows the
function constructed from
and
. We now have something that is approximating the shape of
a probability density function except that it can be positive or negative.
To get around this problem I first
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(9) |
lloyd1
Figure 4. Panel (a) shows the original curve (solid line); an initial set of ![]() ![]() ![]() |
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As a result
is always positive.
Flipping the signs does not violate the LMQ concept. What equation
7 is attempting to do is a local center of mass calculation.
By applying equation 10 or 11 we are transforming
our coordinate system to obtain an accurate center of mass calculation.
How accurate the curve is represented is determined
by the number of
terms. In practice it is best
to start with a dense representation of
to avoid local minima
and then use the fitting criteria of equation 8 to eliminate
points in regions with small deviations. Figure 5 demonstrates
this concept. The solid curve in Figure 5 is the original
function. The three dashed curves show different deviation criteria. With
increasing accuracy an increasing number of points are needed to represent
the curve. In this example 2, 9, 28 and points are used.
lloyd
Figure 5. The effect of modifying the deviation criteria. In panel (a) the solid curve in Figure 5 is the original function. The three dashed curves show three different deviation criteria. The closer the fit to the original curve the more points that are needed for an accurate representation. Panel (b) shows the error in the fitting functions. [ER] |
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![]() | Lloyd and Viterbi for QC and auto-picking | ![]() |
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