Wavenumber

Wavenumber, k, sometimes called the propagation number, is a quantification of spatial scale. It can be thought of as a spatial analog to the temporal frequency, and is often called spatial frequency. It is often defined as the number of wavelengths per unit distance, or in terms of wavelength, &lambda;:


 * $$k = \frac{1}{\lambda}\ $$

This is analogous to frequency f, which is the reciprocal of period T; that is, f = 1/T. In a sense, period can be thought of as a temporal 'wavelength—the length of an oscillation in time.

Angular wavenumber
If you've explored the applications of frequency in geophysics, you'll have noticed that we sometimes don't use ordinary frequency f, in Hertz. Because geophysics deals with oscillating waveforms, ones that vary around a central value (think of a wiggle trace of seismic data), we often use the circular or angular frequency. In this way, we can also express the close relationship between frequency and phase, which is an angle. So in many geophysical applications, we want the angular wavenumber. It is expressed in radians per metre:


 * $$k = \frac{2\pi}{\lambda}\ $$

The relationship between angular wavenumber and angular frequency is analogous to that between wavelength and ordinary frequency—they are related by the velocity V:


 * $$k = \frac{\omega}{V}\ $$

It's unfortunate that there are two definitions of wavenumber. Some people reserve the term spatial frequency for the ordinary wavenumber, or use &nu; (that's a Greek nu, not a vee — another potential source of confusion!), or even &sigma; for it. But just as many call it the wavenumber and use k, so the only sure way through the jargon is to specify what you mean by the terms you use. As usual!

Here are two spectra: the FFTs of an image of some waves, and a binary image of some particles:



Computing 2D Fourier transform in FIJI
You can use FIJI to compute the 2D FFT of an image. FIJI is a popular open-source, scriptable scientific image-processing tool. FIJI Is Just (a more awesome) ImageJ.

1 Download and install FIJI
Get the software from here.

2 Open your image
This image is a low-res version of a photomicrograph of some shale from Schieber et al (2010).



Please note: This image is not covered by the CC-BY license on the rest of this wiki. It is copyright of the authors. I'm calling the use of a low-res version 'fair use'. I will try to build an example with an open image at some point — feel free to change it!

3 Set the scale
No matter what you're doing in FIJI, it's often a good idea to tell FIJI the real-world units of your image, so you don't have to think in pixels. To do this, use the rectangle selection tool to measure a length in pixels (e.g. using a scale in the image), then select Analyze > Set scale...




 * 1) Select the rectangle select tool
 * 2) Draw a rectangle around your scale
 * 3) Note the length of the scale in pixels and select Analyze > Set Scale...
 * 4) Give the length, and set the units (mm in this case), and note the dimensions now appear for your image

4 Crop the image
You need to remove any decoration like scales, annotation, borders, etc. These would show up in the wavenumber domain. Select a good area, then Image > Crop



5 Run the FFT
There are two ways: Process > FFT > FFT or Plugins > Process > Fast FFT. The methods are basically the same, just slightly different implementations.



6 Play around with filters
To better understand what you're looking at, try running some spatial filters from Process > FFT > Bandpass Filter.... Be sure to check Display filter. This applies a mask in the 2D wavenumber domain, which results in removing some wavenumbers from the image. For example, to eliminate vertical stripes from an image, we want to eliminate all of the wavenumbers in the x direction that have a value of 0 in the y direction — this requires us to cut out a horizontal band in the 2D FFT display: