Oversampling method and its application to x-ray diffraction

- A novel x-ray microscopy in nano/atomic resolution -


An x-ray diffraction intensity is proportional to the square amplitude of the Fourier transform of an object image (an electron number density) in the kinematical theory. If the phase of the Fourier transform is also measured, the object image can be reconstructed by back Fourier transformation. However the phase is not directly obtained in x-ray diffraction measurement. Therefore the half of necessary information for image reconstruction is missing. The problem is known as the phase problem.

The oversampling method is a solution to the phase problem. In the method a zero density region around the sample object is assumed. If the zero density region is larger than the sample region, more than half of total information is known in real space. Therefore it is in principle possible to use the zero density information to make up for the missing phase information.

In the oversampling method the following iterative procedure is taken:
..... → Image of n-th iteration →
Fourier transformation →
Set Fourier amplitudes to known (measured) values →
Back Fourier transformation →
Set the density in the zero density region and at negative density pixels gradually to zero →
Image of (n+1)-th iteration → .....

[Simulation 1]

We consider a 2-dimensional sample image and a zero density region around it in discretized real space (Fig. 1), and its Fourier transform in the first Brillouin zone of discretized reciprocal space (Fig. 2).

Fig. 1 Sample image with zero density region around it
Fig 2. Oversampled Fourier transform (a simulated diffraction pattern)
SP8 logo initial (256 x 256 pixels)
SP8 logo f amp log scale (256 x 256 pixels)

The diffraction pattern is oversampled due to the zero density region in real space. It means that the diffraction pattern is sampled finer than the frequency determined by the sample image size.

In the simulation a random image in the sample region was taken as the initial image. After 80 iterations the original image was almost perfectly reconstructed. Total computation time was 2.83 sec on an Athlon 900 MHz PC.

Fig 3. Iterative improvement of image
SP8 logo animation

The resultant reconstructed image may be translationally shifted or centro-inverted depending on a given initial image, because they give the same diffraction amplitude.

[Simulation 2]

An example with a lager and more complex sample image

Y. Nishino