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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 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).
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.
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.
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.
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.
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.
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:
This section contains a variety of miscellaneous information.
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.
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.
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