Evaluate and compare two uniform scalar quantizers for encoding
a source ranging over the values in the range (-4.0, +4.0).
The first is a midrise quantizer defining 8 equal sized intervals.
The second is a midtread quantizer defining 7 equal sized intervals.
Analyze their expected error, mean square error, and entropy.
Do this for each of the following two distributions.
The first distribution has source values uniform over the full range.
The second distribution has half the values uniform over the full
range and the other half expected to be at 0+ε,
where ε is a very small positive value.
Write programs to produce histograms of
pixel values (range [0,255]), and
pixel-to-pixel differences (range [-255,+255]).
Use these programs to determine the histograms for
images sinan.img and earth.img (located in directory
http://www.ics.uci.edu/~dan/class/267/datasets/images/ )
Note that histograms for the Sinan image are in Sayood's book
(Figures 10.2 and 10.3).
Which image would you expect to compress better using
simple delta modulation? Explain why.
Let f(x) be a density which is
everywhere positive on the interval [a,b].
Let DN be the minimum distortion among all
N-level quantizers for f(x).
It is known that there is a positive constant Cf
such that
limN→∞(N2DN )
= Cf .
Suppose we measure DN in decibels:
[DN ]dB = 10 log10
[σ2 / DN ]
What is the approximate difference
[D2N ]dB -
[DN ]dB
for large N?
Interpret this result.
Let f(x) be the density
f(x) = 2x, 0 ≤ x ≤ 1
Find the exact values for the two quantization interval endpoints
y1 and y2 for the optimum
3-level quantizer for the density f(x).
(Hint: y1/y2 = (√5 - 1)/2. )
Find the exact values for the three quantization levels
L1, L2, L3
for the optimum 3-level quantizer.
A large image is partitioned into 4×4 blocks.
Assume that the image is a 6 bit per pixel image,
meaning that the pixel values are in [0,63].
Suppose that one of the blocks is
20 37 49 60
11 20 36 57
26 39 35 45
0 22 27 42
The block truncation coding scheme is used on this block.
Compute the two quantization levels and the
4×4 binary matrix that are transmitted to the decoder.
Determine the 4×4 block that is reconstructed by the decoder.
Determine the compression rate in bits/pixel.
Compute the SQNR in decibels resulting from the block truncation
coding of the particular block above.
In an eight bit per pixel image,
a variable-rate quantization scheme is used in which
half of the pixels are fixed-rate scalar quantized
at a rate of two bits per pixel, and the remaining half of the pixels
are fixed-rate scalar quantized at a rate of three bits per pixel.
Compute the optimum SQNR that results from such a lossy compression system.
Model each pixel value as being continuously and uniformly
distributed over the interval [0,255].
(Of course, the pixel values are really integers,
but the calculations will be much easier and you will make
very little error by assuming them to be continuously distributed.)
[Sayood Chapter 13 problem #1 on page 421]
Prove that use of an orthogonal transformation matrix
maintains distance invariance.