Practical Entropy Coding Techniques2.22003/03/262003/05/02NickKingsburyngk10@cam.ac.ukLiqunWangliqun@rice.eduNickKingsburyngk10@cam.ac.ukArithmetic CodingEntropyHuffman CodingRun-length Coding (RLC)This module introduces practical entropy coding techniques, such as Huffman Coding, Run-length Coding (RLC) and Arithmetic Coding.
In the module of Use of
Laplacian PDFs in Image Compression we have assumed that
ideal entropy coding has been used in order to calculate the bit
rates for the coded data. In practise we must use real codes and
we shall now see how this affects the compression performance.
There are three main techniques for achieving entropy coding:
Huffman Coding - one of the simplest variable
length coding schemes.
Run-length Coding (RLC) - very useful for
binary data containing long runs of ones of zeros.
Arithmetic Coding - a relatively new variable
length coding scheme that can combine the best features of
Huffman and run-length coding, and also adapt to data with
non-stationary statistics.
We shall concentrate on the Huffman and RLC methods for
simplicity. Interested readers may find out more about
Arithmetic Coding in chapters 12 and 13 of the JPEG Book.
First we consider the change in compression performance if
simple Huffman Coding is used to code the subimages of the
4-level Haar transform.
The calculation of entropy in this equation from our discussion of
entropy assumed that each message with probability
pi could be represented by a word of length
i2pi bits. Huffman codes require the
i to be integers and assume that the
pi are adjusted to become:
pi^2i
where the
i are integers, chosen subject to the constraint that
ipi^1 (to guarantee that sufficient uniquely decodable code
words are available) and such that the mean Huffman word length
(Huffman entropy),
H^ipii, is minimised.
We can use the probability histograms which generated the
entropy plots in figures of level 1 energies, level 2
energies, level 3 energies and level 4 energies to
calculate the Huffman entropies
H^ for each subimage and compare these with the true
entropies to see the loss in performance caused by using real
Huffman codes.
An algorithm for finding the optimum codesizes
i is recommended in the JPEG specification [the JPEG
Book, Appendix A, Annex K.2, fig K.1]; and a Mathlab M-file to
implement it is given in M-file code.
Comparison of entropies (columns 1, 3, 5) and Huffman coded
bit rates (columns 2, 4, 6) for the original (columns 1 and
2) and transformed (columns 3 to 6) Lenna images. In columns
5 and 6, the zero amplitude state is run-length encoded to
produce many states with probabilities < 0.5.
shows the results of
applying this algorithm to the probability histograms and lists the same results
numerically for ease of analysis. Columns 1 and 2 compare the
ideal entropy with the mean word length or bit rate from using a
Huffman code (the Huffman entropy) for the case of the
untransformed image where the original pels are quantized with
Qstep15. We see that the increase in bit rate from using the
real code is:
3.76763.710611.5%But when we do the same for the 4-level
transformed subimages, we get columns 3 and 4. Here we see that
real Huffman codes require an increase in bit rate of:
1.84251.6103114.4%
Comparing the results for each subimage in columns 3 and 4, we
see that most of the increase in bit rate arises in the three
level-1 subimages at the bottom of the columns. This is because
each of the probability histograms for these subimages (see
figure) contain one probability that is
greater than 0.5. Huffman codes cannot allocate a word length of
less than 1 bit to a given event, and so they start to lose
efficiency rapidly when
2pi becomes less than 1, ie when
pi0.5.
Run-length codes (RLCs) are a simple and effective way of
improving the efficiency of Huffman coding when one event is
much more probable than all of the others combined. They operate
as follows:
The pels of the subimage are scanned sequentially (usually
in columns or rows) to form a long 1-dimensional vector.
Each run of consecutive zero samples (the most probable
events) in the vector is coded as a single event.
Each non-zero sample is coded as a single event in the
normal way.
The two types of event (runs-of-zeros and non-zero samples)
are allocated separate sets of codewords in the same Huffman
code, which may be designed from a histogram showing the
frequencies of all events.
To limit the number of run events, the maximum run length
may be limited to a certain value (we have used 128) and
runs longer than this may be represented by two or more run
codes in sequence, with negligible loss of efficiency.
Hence RLC may be added before Huffman coding as an extra
processing step, which converts the most probable event into
many separate events, each of which has
pi0.5 and may therefore be coded efficiently. shows the new probability
histograms and entropies for level 1 of the Haar transform when
RLC is applied to the zero event of the three bandpass
subimages. Comparing this with a previous figure, note the
absence of the high probability zero events and the new states
to the right of the original histograms corresponding to the run
lengths.
Probability histograms (dashed) and entropies (solid) of the
four subimage of the Level 1 Haar transform of Lenna (see
figure) after RLC.
The total entropy per event for an RLC
subimage is calculated as before from the entropy
histogram. However to get the entropy per
pel we scale the entropy by the ratio of the number of
events (runs and non-zero samples) in the subimage to the number
of pels in the subimage (note that with RLC this ratio will no
longer equal one - it will hopefully be much less).
gives the entropies per
pel after RLC for each subimage, which are now
less than the entropies in this figure. This is because RLC takes advantage
of spatial clustering of the zero samples in a subimage, rather
than just depending on the histogram of amplitudes.
Clearly if all the zeros were clustered into a single run, this
could be coded much more efficiently than if they are
distributed into many runs. The entropy of the zero event tells
us the mean number of bits to code each zero pel if
the zero pels are distributed randomly, ie if the
probability of a given pel being zero does not depend on the
amplitudes of any nearby pels.
In typical bandpass subimages, non-zero samples tend to be
clustered around key features such as object boundaries and
areas of high texture. Hence RLC usually reduces the entropy of
the data to be coded. There are many other ways to take
advantage of clustering (correlation) of the data - RLC is just
one of the simplest.
In , comparing column 5
with column 3, we see the modest (7%) reduction in entropy per
pel achieved by RLC, due clustering in the Lenna image. The main
advantage of RLC is apparent in column 6, which shows the mean
bit rate per pel when we use a real Huffman code on the RLC
histograms of . The
increase in bit rate over the RLC entropy is only
1.50711.497710.63%
compared with 14.4% when RLC is not used (columns 3 and 4).
Finally, comparing column 6 with column 3, we see that, relative
to the simple entropy measure, combined RLC and Huffman coding
can reduce the bit rate by
11.50711.61036.4%
The closeness of this ratio to unity justifies our use of simple
entropy as a tool for assessing the information compression
properties of the Haar transform - and of other energy
compression techniques as we meet them.
The following is the listing of the M-file to calculate the
Huffman entropy from a given histogram.
0);
while length(nz) > 1,
% Find v1 for least value of freq(v1) > 0.
[y,i] = min(freq(nz));
v1 = nz(i);
% Find v2 for next least value of freq(v2) > 0.
nz = nz([1:(i-1) (i+1):length(nz)]); % Remove v1 from nz.
[y,i] = min(freq(nz));
v2 = nz(i);
% Combine frequency values.
freq(v1) = freq(v1) + freq(v2);
freq(v2) = 0;
codesize(v1) = codesize(v1) + 1;
% Increment code sizes for all codewords in this tree branch.
while others(v1) > -1,
v1 = others(v1);
codesize(v1) = codesize(v1) + 1;
end
others(v1) = v2;
codesize(v2) = codesize(v2) + 1;
while others(v2) > -1,
v2 = others(v2);
codesize(v2) = codesize(v2) + 1;
end
nz = find(freq > 0);
end
% Generate Huffman entropies by multiplying probabilities by code sizes.
huffent = (huffhist(:)/sum(huffhist(:))) .* codesize;
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