Average Profiling and Fourier

To limit the influence from noise or smooth the profiles SPIP offers three ways for calculating average profiles:

· from a region of lines parallel to the line marker

· from the entire image

· from a region selected by the zoom box.

Furthermore, it is possible to calculate the average Fourier amplitude spectra of profiles within a selected area.

Profile Averaging of Lines Parallel to the Line Marker:

Activate the Average Parallel Lines marker by its tool key $images\ebx_215181721.gif$ , or MarkersàAverage Parallel Lines. A region around the Line marker is now marked and the corresponding average profile is calculated and shown. To change the size of the average region keep the ‘A’ key and the left mouse button pressed while moving the mouse. The number of averaged lines is written in the caption of the profile window.

$images\waffleparlines.gif$

$images\profileaverage.gif$

To turn off the averaging pres the ‘L’ key or Markers®Line Average Parallel Lines once more. You can conveniently toggle between the average mode and the normal profile mode and observe the difference between the plain profile and the average profiles by pressing the ‘A’ and ‘L’ key. As for other profiles there are several options for further analysis available on the right-mouse key menu and you can for example activate the Fourier Auto Apply to get the Fourier spectrum calculated automatically while changing the size and orientation of the line marker.

Profile Averaging from Entire Image:

From the Processing®Average popup menu it is possible to calculate average profiles and average Fourier on profiles:

$images\averaging.gif$

Average X- and Y-Profiles:

It is possible to obtain the average X-profile zax and the average Y-profile zay calculated as:

$images\ebx_167033598.gif$ $images\ebx_1003813078.gif$

$images\averageprofile.gif$

This function is especially powerful when analyzing line profiles aligned parallel to the scan directions.

The functions are activated from the menu item ProcessingAverageAverage XProfile or item ProcessingAverageAverage Y_Profile.

Profile Averaging of Area Defined by Zoom Box:

By setting Processing®Average®Only Marked Area the calculation will be limited to the pixels inside the rectangle drawn by the rectangle marker tool. Note, that this setting can be used for the Average Fourier transforms as well.

Average Fourier and Power Spectrum Density(PSD)

By clicking ProcessingàAverage (Main Image) you will see a number of menu items for creating average Fourier transforms and Power Spectrum Density

$images\averagefouriermenu.gif$

Average X- and Y-Fourier:

To get a smooth 1D Fourier it can be an advantage to calculate the average Fourier amplitude Fau and Fav of the individual profiles:

$images\ebx_604160420.gif$ $images\ebx_-747634252.gif$ ,

where Fy is the Fourier transform of the profile having the row number equal to y and Fx is the Fourier transform of the profile having the column number equal to x.

Average X-Fourier x 8 and Y-Fourier x 8

Same as above but with 8 times higher resolution, which gives higher accuracy when measuring wavelength associated with Fourier peaks.

Average X-Fourier PSD and Y-Fourier PSD

Same as X-Fourier but with calculation of the power spectrum for each scan line instead of the amplitude spectrum. The power spectrum equals the amplitude spectrum squared. The resulting spectrum is called Power Spectrum Density (PSD). The spectrum is normalized by the number of pixels such that the result is relatively insensitive to the pixel resolution.

Average X-Fourier PSD x 8 and Y-Fourier PSD x 8

Same as above but with 8 times higher resolution.

Below is seen how an average Fourier transform may look when displayed on a dB scale. The cursors indicate the first and third harmonic components. Note also that the corresponding wavelengths are written in mm. It is therefore possible to estimate the pitch by positioning a cursor on the first harmonic. For accurate estimation of the pitch, a statistical mean value can be calculated based on the other harmonic components. However, if the profile is based on an image, you will find the pitch more easily and accurately by the unit cell detection algorithm.

$images\averagefourierdb.gif$

To learn more about 1D Fourier analysis, please consult section 1D Fourier Analysis.