3.2 The 2-dimensional dct

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This module introduces the 2-dimensional DCT.

In the equation from our discussion of the Haar transform: $y=TxT^T$ and to invert: $x=T^TyT$ we saw how a 1-D transform could be extended to 2-D by pre- and post-multiplication of a square matrix $x$ to give a matrix result $y$ . Our example then used $2\times 2$ matrices, but this technique applies to square matrices of any size.

Hence the DCT may be extended into 2-D by this method.

E.g. the $8\times 8$ DCT transforms a subimage of $8\times 8$ pels into a matrix of $8\times 8$ DCT coefficients.

The 2-D basis functions, from which $x$ may be reconstructed, are given by the $n^{2}$ separate products of the columns of $T^T$ with the rows of $T$ . These are shown for $n=8$ in (a) of as 64 subimages of size $8\times 8$ pels.

The result of applying the $8\times 8$ DCT to the Lenna image is shown in (b) of . Here each $8\times 8$ block of pels $x$ is replaced by the $8\times 8$ block of DCT coefficients $y$ . This shows the $8\times 8$ block structure clearly but is not very meaningful otherwise.

Part(c) of shows the same data, reordered into 64 subimages of $32\times 32$ coefficients each so that each subimage contains all the coefficients of a given type - e.g: the top left subimagecontains all the coefficients for the top left basis function from (a) of . The other subimages and basis functions correspond in the same way.

We see the major energy concentration to the subimages in the top left corner. (d) of is an enlargement of the top left 4 subimages of (c) of and bears a strong similarity to the group of third level Haar subimages in (b) of this figure . To emphasise this thehistograms and entropies of these 4 subimages are shown in .

Comparing with this figure , the Haar transform equivalent, we see that the Lo-Lo bands have identicalenergies and entropies. This is because the basis functions are identical flat surfaces in both cases. Comparing the other 3bands, we see that the DCT bands contain more energy and entropy than their Haar equivalents, which means less energy (and so hopefully less entropy) in the higher DCT bands (not shown) because the total energy isfixed (the transforms all preserve total energy). The mean entropy for all 64 subimages is 1.3622 bit/pel, which comparesfavourably with the 1.6103 bit/pel for the 4-level Haar transformed subimages using the same ${Q}_{\mathrm{step}}=15$ .

What is the optimum dct size?

This is a similar question to: What is the optimum number of levels for the Haar transform?

We have analysed Lenna using DCT sizes from $2\times 2$ to $16\times 16$ to investigate this. shows the $4\times 4$ and $16\times 16$ sets of DCT subimages. The $2\times 2$ DCT is identical to the level 1 Haar transform (so see (b) of ) and the $8\times 8$ set is in (c) of .

and show the mesh plots of the entropies of the subimages in .

compares the total entropy per pel for the 4 DCT sizes with the equivalent 4 Haartransform sizes. We see that the DCT is significantly better than the rather simpler Haar transform.

As regards the optimum DCT size, from , the $16\times 16$ DCT seems to be marginally better than the $8\times 8$ DCT, but subjectively this is not the case since quantisation artefacts become more visible as the block sizeincreases. In practise, for a wide range of images and viewing conditions, $8\times 8$ has been found to be the optimum DCT block size and is specified in most current coding standards.

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
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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
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Alexandre
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Rafiq
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Damian
How we are making nano material?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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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
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
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