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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

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar 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, 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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