Inversion ? no iteration same ambiguities additional instabilities parameter & potential atomic displacements exit object wave image direct interpretation by data reduction: Fourier filtering QUANTITEM Fuzzy & Neuro-Net Srain analysis deviations from reference structures: displacement field (Head) algebraic discretization reference beam (holography) defocus series Gerchberg-Saxton (Jansson) multi-slice inversion (vanDyck,Griblyuk,Lentzen) Pade-inversion (Spence) local linearization

Data lost? Additional data? Imaging process Scattering process phases linearity 3d-2d projection atom positions reference beam defocus series lattices & bonds shape & orientation displacement field inelastic spectra

regularization physically motivated Assumption:complex amplitudes are regular Cauchy relations: a/ x = a. / y a/ y = -a. / x 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

Direct & Inverse: black box gedankenexperiment operator A f input g output wave image thickness local orientation structure & defects composition microscope theory, hypothesis, model of scattering and imaging 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 if unique & stable inverse A -1 exists ill-posed & insufficient data => least square

perfect crystal:  = e 2  iAt  o distorted object:  / z=  i(  A+  xy +  )   / z continuous at boundaries  gu)/ z displacement field    / z = e -  t energy conservation oo  gg i,j i,j-1i,j+1 i+1,ji,j-1i,j+1 i-1,j solve equations of perfect crystal, discretize wave equations and boundary conditions => algebraic equation system of  at all nodes (i,j,k) and  Q g e 2  igu(i,j,k) = 0 forward wave equation =>  (i+1,j) backward energy conservation =>  (i-1,j)