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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 i 0 M 1 p i 2 logbase --> 1 p i i 0 M 1 p i 2 logbase --> p i
where p i , i 0 to M 1 , is the probability of the i 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 2 logbase --> 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 .

Conversion from probability p i to entropy h i p i 2 logbase --> p i .

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 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).

Probability histogram (dashed) and entropies (solid) of the Lenna image in ( original image ).
Probability histogram (dashed) and entropies (solid) of the four subimages of the Level 1 Haar transform of Lenna (see previous figure ).

The Haar transform preserves energy and so the expected distortion energy from quantising the transformed image y with a given step size Q 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) = Q 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 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.

Mean bit rate for the original Lenna image and for the Haar transforms of the image after 1 to 4 levels, using a quantiserstep size Q step 15 .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Image coding. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10206/1.3
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