[3dem] Williams and Carter

Terje Dokland dokland at uab.edu
Wed May 30 07:55:02 PDT 2012

Which part is it that you are having problems with? I agree it can be a bit hard to follow unless you also read Reimer and/or Spence to fill in the blanks!  

I'm no expert, but to me Equation 28.22 is just saying that the image function is the convolution of the point spread function h(x,y) with the 2D projection of the 3D specimen, which is [1- isV(x,y)] according to the WPOA in their lingo. 

Since the PSF is a phase shift it can be broken into sin and cos components (28.23). 

Then when you take the square of that  (28.24) you  get a lot of second order terms that are ignored because they are too small, yielding 28.25. 

The next part is a bit hard to follow, but basically you take the fourier transform of that, which turns the convolution into multiplication, which would give you

FTimage(u,v) = d(0) + 2s FTspecimentprojection(u,v) sin(Chi)

since they already showed previously the relationship between Chi and h (28.6)

and Chi is of course the phase shift due to spherical aberration and defocus. 

They then multiply that with the aperture function A and the envelope function E as well. 

On May 30, 2012, at 9:16 AM, Philip Köck wrote:

> I’ve been trying to read chapter 28 in the second edition of “Transmission Electron Microscopy” by
> Williams and Carter.
> Is there anybody out there who can make sense of the derivation in section 28.4?
> Philip
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> 3dem at ncmir.ucsd.edu
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