In the case of uniform quantizers, the
pdf of the analog sample was assumed to beuniform, and therefore, we obtained
the closed form solutions for optimal decision regions and output levels.
Moreover, the intervals between any two consecutive decision regions as well as
the intervals between any two consecutive output levels were constant. When the
pdf of the input analog samples is not uniform, then the quantization steps are
not constant and the optimal solutions are obtained by solving the
transcendental equations (2.31). This results in a nonuniformquantizer and is referred
to as pdf optimized quantizer.
Using
Lloyd’s algorithm, we design the quantizers for 3 and 5 bpp. The requantized
images at 3 and 5 bpp are shown in Figures 1.a,b, respectively. There are some
improvements in the flat areas compared with that of the corresponding uniform
quantizer. The SNR values are 20.1 and 33.68 dB for 3 and 5 bpp, respectively,
for the nonuniformquantizer, whereas they are 17.3 and 29.04 dB, respectively,
for the uniform quantizer. Figure 1.c shows a plot of the decision regions
versus output levels of the nonuniformquantizer for
L = 32 levels. The number of pixels
belonging to the different output levels is plotted against output levels and
is shown in Figure 1.d (top figure), while the bottom Figure 1.d shows the
histogram of the input image for the same number of bins—32 in the example. The
two are nearly identical and the slight difference is due to small number of
bits of quantization. Thus, we have designed the pdf-optimized quantizer for
the input image. Overall, we find the nonuniform quantizer
to perform much better than the uniform quantizer for the same number of bits
of quantization. Another important point of observation is that the iterations converge
much faster if we use a difference distortion relative to the previous
distortion rather than absolute difference in distortions. In fact, the average
number of iterations is 12 when the relative distortion measure is used to
check convergence,
Figure
1. An example of a nonuniformquantizer: (a) image requantized to 3
bpp, (b) image
requantized to 5 bpp, (c) plot of input decision regions versus
output levels of the nonuniform
quantizer, (d) the top figure is the count of quantized pixels
belonging to the different output levels versus output levels, and the bottom
figure is histogram of the input image for the same number of levels as in the
top figure.
while
it is 82 when the absolute difference measure is used for the same ε value of
0.1.
The MATLAB code for the Lloyd algorithm is listed below.
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