# 2.5 Practical entropy coding techniques

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 bitrates 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 ofHuffman 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 aboutArithmetic 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 the4-level Haar transform.

The calculation of entropy in this equation from our discussion of entropy assumed that each message with probability ${p}_{i}$ could be represented by a word of length ${}_{i}=-\log_{2}{p}_{i}$ bits. Huffman codes require the ${}_{i}$ to be integers and assume that the ${p}_{i}$ are adjusted to become:

$\stackrel{^}{{p}_{i}}=2^{-{}_{i}}$
where the ${}_{i}$ are integers, chosen subject to the constraint that $\sum \stackrel{^}{{p}_{i}}\le 1$ (to guarantee that sufficient uniquely decodable code words are available) and such that the mean Huffman word length(Huffman entropy), $\stackrel{^}{H}=\sum {p}_{i}{}_{i}$ , 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 $\stackrel{^}{H}$ for each subimage and compare these with the true entropies to see the loss in performance caused by using realHuffman 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 toimplement it is given in M-file code .

Numerical results used in the figure - entropies and bit rates of subimages for qstep=15
Column: 1 2 3 4 5 6 -
0.0264 0.0265 0.0264 0.0266
0.0220 0.0222 0.0221 0.0221 Level 4
0.0186 0.0187 0.0185 0.0186
0.0171 0.0172 0.0171 0.0173 -
0.0706 0.0713 0.0701 0.0705
0.0556 0.0561 0.0557 0.0560 Level 3
3.7106 3.7676 0.0476 0.0482 0.0466 0.0471 -
0.1872 0.1897 0.1785 0.1796
0.1389 0.1413 0.1340 0.1353 Level 2
0.1096 0.1170 0.1038 0.1048 -
0.4269 0.4566 0.3739 0.3762
0.2886 0.3634 0.2691 0.2702 Level 1
0.2012 0.3143 0.1819 0.1828 -
Totals: 3.7106 3.7676 1.6103 1.8425 1.4977 1.5071

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 theideal entropy with the mean word length or bit rate from using a Huffman code (the Huffman entropy) for the case of theuntransformed image where the original pels are quantized with ${Q}_{\mathrm{step}}=15$ . We see that the increase in bit rate from using the real code is: $\frac{3.7676}{3.7106}-1=1.5%$ But when we do the same for the 4-level transformed subimages, we get columns 3 and 4. Here we see thatreal Huffman codes require an increase in bit rate of: $\frac{1.8425}{1.6103}-1=14.4%$ Comparing the results for each subimage in columns 3 and 4, wesee that most of the increase in bit rate arises in the three level-1 subimages at the bottom of the columns. This is becauseeach 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 ofless than 1 bit to a given event, and so they start to lose efficiency rapidly when $-\log_{2}{p}_{i}$ becomes less than 1, ie when ${p}_{i}> 0.5$ .

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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