# 2.2 Entropy

 Page 1 / 1
This module introduces entropy of source information.

Entropy of source information was discussed in the third-year E5 Information and Coding course. For an image $x$ , quantised to $M$ levels, the entropy ${H}_{x}$ is defined as:

${H}_{x}=\sum_{i=0}^{M-1} {p}_{i}\log_{2}\left(\frac{1}{{p}_{i}}\right)=-\sum_{i=0}^{M-1} {p}_{i}\log_{2}{p}_{i}$
where ${p}_{i}$ , $i=0$ to $M-1$ , is the probability of the ${i}^{\mathrm{th}}$ quantiser level being used (often obtained from a histogram of the pel intensities).

${H}_{x}$ represents the mean number of bits per pel with which the quantised image $x$ can be represented using an ideal variable-length entropy code. AHuffman code usually approximates this bit-rate quite closely.

To obtain the number of bits to code an image (or subimage) $x$ containing $N$ pels:

• A histogram of $x$ is measured using $M$ bins corresponding to the $M$ quantiser levels.
• The $M$ histogram counts are each divided by $N$ to give the probabilities ${p}_{i}$ , which are then converted into entropies ${h}_{i}=-({p}_{i}\log_{2}{p}_{i})$ . This conversion law is illustrated in and shows that probabilities close to zero or one produce low entropy and intermediatevalues produce entropies near 0.5.
• The entropies ${h}_{i}$ of the separate quantiser levels are summed to give the total entropy ${H}_{x}$ for the subimage.
• Multiplying ${H}_{x}$ by $N$ gives the estimated total number of bits needed to code $x$ , assuming an ideal entropy code is available which is matched to the histogram of $x$ .

shows the probabilities ${p}_{i}$ and entropies ${h}_{i}$ for the original Lenna image and shows these for each of the subimages in this previous figure , assuming a uniform quantiser with a step-size ${Q}_{\mathrm{step}}=15$ in each case. The original Lenna image contained pel values from 3 to 238 and a mean level of 120 was subtracted fromeach pel value before the image was analysed or transformed in order that all samples would be approximately evenly distributedabout zero (a natural feature of highpass subimages).

The Haar transform preserves energy and so the expected distortion energy from quantising the transformed image $y$ with a given step size ${Q}_{\mathrm{step}}$ will be approximately the same as that from quantising the input image $x$ with the same step size. This is because quantising errors can usually bemodeled as independent random processes with variance (energy) = $\frac{{Q}_{\mathrm{step}}^{2}}{12}$ and the total squared quantising error (distortion) will tend to the sum of the variances over all pels. Thisapplies whether the error energies are summed before or after the inverse transform (reconstruction) in the decoder.

Hence equal quantiser step sizes before and after an energy-preserving transformation should generate equivalentquantising distortions and provide a fair estimate of the compression achieved by the transformation.

The first two columns of (original and level 1) compare the entropy (mean bit rate) per pel for the original image (3.71 bit / pel) with that of theHaar transformed image of this previous figure (2.08 bit / pel), using ${Q}_{\mathrm{step}}=15$ . Notice that the entropy of the original image is almost as great as the 4 bit / pel that would be needed to codethe 16 levels using a simple fixed-length code, because the histogram is relatively uniform.

The level 1 column of shows the contribution of each of the subimages of this previous figure to the total entropy per pel (the entropies from have been divided by 4 since each subimage has one quarter of the total number of pels). the Lo-Lo subimagecontributes 56% to the total entropy (bit rate) and has similar spatial correlations to the original image. Hence it is alogical step to apply the Haar transform again to this subimage.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!