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This module introduces image processing, 2D convolution, 2D sampling and 2D FTs.

Image processing

Images are 2D functions f x y

Linear shift invariant systems

H is LSI if:

  • H 1 f 1 x y 2 f 2 x y H f 1 x y H f 2 x y for all images f 1 , f 2 and scalar.
  • H f x x 0 y y 0 g x x 0 y y 0
LSI systems are expressed mathematically as 2D convolutions: g x y h x y f where h x y is the 2D impulse response (also called the point spread function ).

2d fourier analysis

u v y x f x y u x v y where is the 2D FT and u and v are frequency variables in x u and y v .

2D complex exponentials are eigenfunctions for 2D LSI systems:

h x y u 0 v 0 h u 0 x v 0 y u 0 x v 0 y h u 0 v 0
where h u 0 v 0 H u 0 v 0 H u 0 v 0 is the 2D Fourier transform of h x y evaluated at frequencies u 0 and v 0 .

g x y h x y f x y h x y f
G u v H u v u v

Inverse 2d ft

g x y 1 2 2 v u G u v u x v y

2d sampling theory

Think of the image as a 2D surface.

We can sample the height of the surface using a 2D impulse array.

Impulses spaced x apart in the horizontal direction and y in the vertical

f s x y S x y f x y where f s x y is sampled image in frequency

2D FT of s x y is a 2D impulse array in frequency S u v

multiplication in timeconvolution in frequency F s u v S u v u v

u v is bandlimited in both the horizontal and vertical directions.
periodically replicated in ( u , v ) frequency plane

Nyquist theorem

Assume that f x y is bandlimited to B x , B y :

If we sample f x y at spacings of x B x and y B y , then f x y can be perfectly recovered from the samples by lowpass filtering:

ideal lowpass filter, 1 inside rectangle, 0 outside

Aliasing in 2d

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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
How can I make nanorobot?
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
how can I make nanorobot?
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.
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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