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multi-slice inversion (vanDyck,Griblyuk,Lentzen) Pade-inversion (Spence) local linearization. deviations from reference structures: displacement field (Head) algebraic discretization. parameter & potential. Inversion ?. atomic displacements. exit object wave.

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Presentation Transcript
inversion

multi-slice inversion

(vanDyck,Griblyuk,Lentzen)

Pade-inversion (Spence)

local linearization

deviations from

reference structures:

displacement field (Head)

algebraic discretization

parameter

& potential

Inversion ?

atomic

displacements

exit object

wave

no iteration same ambiguities

additional instabilities

direct interpretation

by data reduction:

Fourier filtering

QUANTITEM

Fuzzy & Neuro-Net

Srain analysis

image

reference beam (holography)

defocus series

Gerchberg-Saxton (Jansson)

data lost additional data

Data lost?Additional data?

phases

linearity

reference beam

defocus series

Scattering process

Imaging process

lattices & bonds

shape & orientation

displacement field

inelastic spectra

3d-2d projection

atom positions

regularization physically motivated

regularization physically motivated

Assumption: complex amplitudes are regular

Cauchy relations: Ja/Jx = a.Jf/Jy Ja/Jy = -a.Jf/Jx

Linear inversion: t(x+1,y)-2t(x,y)+t(x-1,y)=0

t(x,y+1)-2t(x,y)+t(x,y-1)=0

slide4

Direct & Inverse: black box gedankenexperiment

thickness

local orientation

structure & defects

composition

microscope

g

output

f

input

operator A

wave

image

theory, hypothesis, model of

scattering and imaging

if unique & stable inverse A-1 exists

ill-posed & insufficient data => least square

direct: g=A<f experiment, measurement

invers 1.kind: f=A-1<g parameter determination

invers 2.kind: A=g$f -1 identification, interpretation

a priori knowledge

intuition & induction

additional data

slide5

perfect crystal: f = e2piAt fo

distorted object: Jf/Jz= pi(sA+Dxy+b) f

f, Jf/Jz continuous at boundaries

b = J(gu)/Jz displacement field

J(ff*)/Jz = e-pmt energy conservation

fo

fo

fg

i,j-1

i,j

i,j+1

i-1,j

forward

wave equation

=>f(i+1,j)

backward

energy conservation

=>b(i-1,j)

i+1,j

i,j-1

i,j

i,j+1

solve equations of perfect crystal, discretize wave equations and boundary conditions

=>algebraic equation system of f, b at all nodes (i,j,k) and SQge2pigu(i,j,k) = 0