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The deepest part of the trough in the center is relatively narrow with respect to the folding wave numbers at the edges. However, it is somewhatbroader than the peak in the spectrum at the lower left.

The result of convolution

The image in the upper right shows the result of convolving the space domain surface in the upper left with the convolution operator in the upper center.This output space domain surface has a green square area in the center that is at the same level as the green background. In this case, green represents anelevation of 0, which is about midway between the lowest elevation (black) and the highest elevation (white).

Positive and negative fences

Surrounding the green square is a yellow and white fence representing very high elevations. Surrounding that fence is a black and blue fence, representingvery low elevations consisting of large negative values.

Thus, as you move from the outside to the inside of the square in the output surface, the elevation goes from a background level of zero, to a large negativevalue, followed immediately by a large positive value, followed by zero.

Edge detection

This is one form of edge detection. The edges of the square in the input surface have been emphasized and the flat portion of the inputsurface has been deemphasized in the convolution output.

Wavenumber spectrum of the convolution output

The wavenumber spectrum of the output from the convolution operation is shown in the lower right. The spectrum indicates that this surface is made up mostlyof wavenumber components having mid range to high values.

If you are familiar with digital signal processing, you will know that inorder for a space (or time) function to contain very rapid changes in value (such as the elevation changes at the fences described above) the function must contain significant high wavenumber (or frequency) components. That appears to be the case here indicated by the red areas on the four sides of the wavenumberspectrum.

Although this spectrum was produced by convolution in the space domain followed by a 2D Fourier transform on the convolution output, you should be ableto see that the shape of the spectrum on the bottom right approximates the product of the spectrum of the original surface on the bottom left and thespectral response of the convolution operator in the bottom center.

Thus, the same results could have been produced using multiplication in the wavenumber domain followed by an inverse Fourier transform to produce the spacedomain result. Convolution in the space domain is equivalent to multiplication in the wavenumber domain and vice versa.

Hidden watermarks and trademarks

Another interesting application that I can demonstrate online is using 2D Fourier transforms to hide secret trademarks and watermarks in images. Thepurpose of a hidden trademark or watermark is for the owner of the image to be able to demonstrate that the image may have been used inappropriately by someoneelse. Once again, this application can be satisfied by treating the space domain data as purely real. I plan to demonstrate how this is done in a future module.

Summary

I began by explaining how the space domain and the wavenumber domain in two-dimensional analysis are analogous to the time domain and the frequencydomain in one-dimensional analysis.

Then I introduced you to some practical examples showing how 2D Fourier transforms and wavenumber spectra can be useful in solving engineering problemsinvolving antenna arrays.

What's next?

In Part 2 of this two-part series, I will provide and explain a Java class that can beused to perform forward and inverse 2D Fourier transforms, and can also be used to shift the wavenumber origin from the upper left to the center for a morepleasing plot of the wavenumber spectrum.

In addition, I will provide and explain a program that is used to:

  • Test the forward and inverse 2D Fourier transforms to confirm that the code is correct and that the transformations behave as they should
  • Produce wavenumber spectra for simple surfaces to help the student gain a feel for the relationship that exists between the space domain and thewavenumber domain

Miscellaneous

This section contains a variety of miscellaneous information.

Housekeeping material
  • Module name: Java1490-2D Fourier Transforms using Java
  • File: Java1490.htm
  • Published: 07/12/05

Learn how the space domain and the wavenumber domain in two-dimensional analysis are analogous to the time domain and the frequency domain in one-dimensional analysis. Learn about some practical examples showing how 2D Fourier transforms and wavenumber spectra can be useful in solving engineering problems involving antenna arrays.

Disclaimers:

Financial : Although the Connexions site makes it possible for you to download a PDF file for thismodule at no charge, and also makes it possible for you to purchase a pre-printed version of the PDF file, you should beaware that some of the HTML elements in this module may not translate well into PDF.

I also want you to know that, I receive no financial compensation from the Connexions website even if you purchase the PDF version of the module.

In the past, unknown individuals have copied my modules from cnx.org, converted them to Kindle books, and placed them for sale on Amazon.com showing me as the author. Ineither receive compensation for those sales nor do I know who does receive compensation. If you purchase such a book, please beaware that it is a copy of a module that is freely available on cnx.org and that it was made and published withoutmy prior knowledge.

Affiliation : I am a professor of Computer Information Technology at Austin Community College in Austin, TX.

-end-

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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