Focal series reconstruction

Long focal series of high resolution images contain enough information to determine most of the wavefunction at the exit surface of the specimen. Two methods are used to perform this reconstruction, linear restoration, which assumes a weakly scattering specimen and iterative methods, which in principle work for strongly scattering specimens.

Linear restoration provides a very accurate way of determining the defocus and measuring the displacement between the members of a focal series (RR Meyer, AI Kirkland, WO Saxton, Ultramicroscopy 92 (2002) 89-109). These are essential stages in comparing high resolution images with simulations.

Iterative restoration is still under development. A quick demonstration of a simple iterative focal series reconstruction program is shown below. In this case the "exit surface wavefunction" consists of uniform amplitude and a phase containing an image of the Taj Mahal.

Original wavefunction:

Top: real and imaginary parts (min 0.0703, max 0.998)

Bottom: amplitude and phase (amplitude min 0, max 1; phase min 0.704, max 0.626)



Calculated focal series from above wavefunction

300 kV, Cs 1.2 mm
obj ap 9.8433 mrad, step 0.0391 nm/pix, focal spread 5 steps 1 nm/step gaussian rad 1
Focal series from 0 nm to -1000 nm in steps of -100 nm

min 0.563, max 1.576



Reconstructed amplitude and phase from all 11 images

Iterations start at 0 nm and go to -1000 nm finishing with 0 nm again at the end.

Left amplitude, right phase
From top: 1, 2, 5, 10 iterations

amplitude min 0.970, max 1.035; phase min -0.304, max 0.354 rad



Reconstructed amplitude and phase as a function of number of images, all for 10 iterations

Iterations start at 0 nm and go to -1000 nm finishing with 0 nm again at the end.

Left amplitude, right phase
From top: 2, 3, 5, 11 images, always starting at 0 nm defocus

amplitude min 0.913, max 1.089; phase min -0.305, max 0.345 rad

General comments

Not much happends after about 5 iterations, except that the ripple in the amplitude increases.

The ripple in the amplitude appears because the focal spread means that the larger defoci have lost some of their high frequency information. It gets worse with more iterations.

There is no information in the focal series about the average phase and little information about the low frequencies in the phase. Higher defoci are required to get more low frequency phase information.

Dr Chris Boothroyd | Research