# 2.2 Entropy

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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.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
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