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General discussion

The space domain

In Part 1 of this series, I extended the concept of the Fourier transform from the time domain into the space domain. I pointed out that while the time domainis one-dimensional, the space domain is thee-dimensional. However, in order to keep the complexity of this module in check, we will assume that space is onlytwo-dimensional. This will serve us well later for such tasks as image processing.

(Three-dimensional Fourier transforms are beyond the scope of this module. I will write a module on using Fourier transforms inthree-dimensional space later if I have the time.)

A purely real space domain

Although the space domain can be (and often is) complex, many interesting problems, (such as photographic image processing) can be dealt with under the assumption that the space domain is purely real. We willmake that assumption in this module. This assumption will allow us to simplify our computations when performing the 2D Fourier transform to transform our datafrom the space domain into the wavenumber domain.

Preview

I will present and explain two complete Java programs in this module. The first program is a single class named ImgMod30 , which provides the capability to perform forward and inverse Fourier transforms onthree-dimensional surfaces. In addition, the class provides a method that can be used to reformat the wavenumber spectrum to make it more suitable for display.

The second program is named ImgMod31 . This is an executable program whose purpose is to exercise and to test the ImgMod30 class using several examples for which the results should already be known.

Sample programs

The class named ImgMod30

This class provides 2D Fourier transform capability that can be used for image processing and other purposes. The class provides three static methods:

  • xform2D : Performs a forward 2D Fourier transform on a purely real surface described by a 2D array of double values in the space domain to produce a complex spectrum in the wavenumberdomain. The method returns the real part, the imaginary part, and the amplitude spectrum, each in its own 2D array of double values.
  • inverseXform2D : Performs an inverse 2D Fourier transform from the complex wavenumber domain into the real space domainusing the real and imaginary parts of the wavenumber spectrum as input. Returns the surface in the space domain as a 2D array of double values. Assumes that the real and imaginary parts in the wavenumber domainare consistent with a purely real surface in the space domain, and does not return an imaginary surface for the space domain
  • shiftOrigin : The wavenumber spectrum produced by xform2D has its origin in the top-left corner with the Nyquist folding wave numbers near the center. This is not a very suitableformat for visual analysis. This method rearranges the data to place the origin at the center with the Nyquist folding wave numbers along the edges.

The class was tested using JDK 1.8 and Windows 7. The class uses the static import capability that was introduced in J2SE 5.0. Therefore, it should notcompile using earlier versions of the compiler.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
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