Fourier transform

Quality metric that is computed from the Fourier transform. This is based on the method described in Krieger Lassen Ph.D. thesis [KriegerLassen1994]. It evaluates the noise present in the diffraction pattern by measuring the amount of low frequencies. The rational is that high quality patterns should have less noise therefore less high frequencies in the spectrum of the Fourier transform. Krieger Lassen used this quality index to identify recrystallized and non-recrystallized grains in an aluminum sample.

From the Fourier spectrum S(u,v) defined as the magnitude of the complex two dimensional Fourier transform,

\[S(u,v) = \left| F(u,v) \right| = \sqrt{\mathbb{R}(F(u,v))^2 + \mathbb{C}(F(u,v))^2}\]

Fourier spectrum

a measure of the high frequencies can be calculated by assigning a large weight to these high frequencies:

\[I = \frac{\sum\limits_{u=-n/2}^{n/2-1}{\sum\limits_{v=-n/2}^{n/2-1}{S(u,v)(u^2+v^2)}}} {\sum\limits_{u=-n/2}^{n/2-1}{\sum\limits_{v=-n/2}^{n/2-1}{S(u,v)}}}\]

In the latter equation, \(u^2+v^2\) term in the numerator is the weight factor. The high frequencies, located far from the center, are multiplied by a greater factor. It can be seen graphically as a radial mask where the intensity goes from 0 at the center to \((\text{width}/2)^2 + (\text{height}/2)^2\) at the corner of the mask.

Radial mask

This intensity I is then normalized by the maximum theoretical intensity \(I_\text{max}\)

\[I_\text{max} = \frac{1}{n^2} \sum\limits_{u=-n/2}^{n/2-1}{\sum\limits_{v=-n/2}^{n/2-1}{(u^2 + v^2)}}\]

which is the average of the radial mask.

The result is that the ratio between the intensity and maximum intensity varies between 0 for an uniform image and 1 for a white noise image. Since it is more intuitive to think of a quality index varying between 0 for a noisy pattern and 1 for a sharp pattern, the Fourier quality index is calculated as followed:

\[Q = 1 - \frac{I}{I_\text{max}}\]

Example on NiCoCrAlY sample