Direct Retrieval of Object Information using Inverse Solutions of Dynamical Electron Diffraction

Max Planck Institute of Microstructure PhysicsHalle/Saale, Germanyschee@mpi-halle.de

http://www.mpi-halle.de

Kurt Scheerschmidt

Quantitative Analysis: Trial-&-Error or Inverse Problems

Confidence: a priori Data versus Regularization

trial-and-errorimage analysis

direct objectreconstruction

1. objectmodeling

2. wave simulation

3. image process

4. likelihoodmeasure

repetition

parameter &potential

reconstruction

wavereconstruction

?

image

?

Inversion ? no iteration

same ambiguities

additional instabilities

parameter& potential

atomicdisplacementsexit object

wave

imagedirect interpretation by data reduction:Fourier filteringQUANTITEM

Fuzzy & Neuro-NetSrain analysis

deviations fromreference structures:

displacement field (Head)algebraic discretization

reference beam (holography)defocus series (Kirkland, van Dyck …)

Gerchberg-Saxton (Jansson)tilt-series, voltage variation

multi-slice inversion(van Dyck, Griblyuk, Lentzen,

Allen, Spargo, Koch)Pade-inversion (Spence) non-Convex sets (Spence)

local linearization

= M(X) 0

= M(X0) 0 + M(X0)(X-X0) 0

Assumptions:

- object: weakly distorted crystal

- described by unknown parameter set X={t, K,Vg, u}

- approximations of t0, K0 a priori known

M needs analytic solutions for inversion

Perturbation: eigensolution , C for K, V yields analytic solution of and its derivatives

for K+K, V+V with tr() + {1/(i-j)}

= C-1(1+)-1 {exp(2i(t+t)} (1+)C

The inversion needs generalized matrices due to different numbersof unknowns in X and measured reflexes in disturbed by noise

Generalized Inverse (Penrose-Moore):

X= X0+(MTM)-1MT.[exp- X]

A0 Ag1 Ag2 Ag3

P0 Pg1 Pg2 Pg3

...

...exp

X= X0+(MTM)-1MT.[exp- X]

i i i

j j jX X X...

t(i,j) Kx(i,j) Ky(i,j)

Regularized Generalized Inverse

X=(MTC1M + C2)-1 MT

as Maximum-Likelihood-Estimate of Gauss-distributed Errors

||ex-th||2 + ||X||2 = Min

with defect (ex-th)†C1(ex-th) (ex†C1ex)-1

and constraint XTC2Xwhich is physically interpretable as:

Weighting C1=W†

ghWgh

Smoothing C2=DTijDij

data itself: Dij=i-ip,j-jp

second derivative: Dij=-2i-ip,j-jp+i-ip±1,j-jp±1

-lg()

lg()Regularization

Kx(i,j)/a*

Ky(i,j)/a*

t(i,j)/Å

Retrieval with iterative fit of the confidence region

lg()

step

step

< t > / Å

relative beamincidence to zone axis [110]

[-1,1,0]

[002]

iii

iii

iiiiii

(i-iii increasing smoothing)

Ge-CdTe, 300kVSample: D. SmithHolo: H. Lichte,

M.Lehmann

10nm

object waveamplitude

object wavephase

FT

A000

P000

A1-11

P1-11

A1-1-1

A-111

P-111

P1-1-1

A-11-1

P-11-1

A-220

P-220

Kx(i,j)/a*

Ky(i,j)/a*

t(i,j)/Å

set 1: Ge set 2: CdTe dVo/Vo = 0.02% dV’o/V’o = 0.8%

Ky(i,j)/a*

Kx(i,j)/a*

K(i,j)/a*

t(i,j)/ Å

model/reco input 7 / 7 15 / 15 15 / 9 15 / 7beams used Influence of Modeling Errors

CONCLUSIONS & OUTLOOK

OBJECT RECONSTRUCTION:

Trial-&-Error Matching of Amplitudes & Phases

as well as

Inversion via local Linearization

ILL-POSEDNESS:

Ambiguity & Instability Generalization & RegularizationModeling Error & Confidence a priori Data

Thanks for your attention

Thanks for cooperation:

H.Lichte, M.Lehmann (Uni-Dresden)

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

0

1

-1

1

.6

.2

.5

-.5

Confidence range?

Kx(i,j)/a* Ky(i,j)/a* K(i,j)/a* t(i,j)/ Å

Properly posed problems (J. Hadamard 1902)Existence

UniquenessStability

if at least one solution But: exists which is unique and continuous with data

implies determinism (Laplaciandeamon, classical physics) ofintegrable systems for knowninitial/boundary conditions

suitable theory/model& a priori knowledge

inverse 1.kind

solution via construction

but small confidence(uniqueness/stability)

Direct & Inverse: black box gedankenexperiment

operator Af

input

g

output

waveimage

thicknesslocal orientation

structure & defectscompositionmicroscope 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 knowledgeintuition & induction

additional data

if unique & stable inverse A-1 exists

ill-posed & insufficient data => least square

restricted information channel (D. van Dyck)

a priori information: object & additional experiments

amorph1023coordinates

FT white noise

medium range orderPDF, ADF

FT densebut structured S(r)

crystalspace group with

basis / displacements

FT discreteconvolution withdefects and shape

Reference wave:Rexp(2ir)

Diffraction:

u(u-)

Aberrations:uuexp(-D-i

Interference: RF-1{uu}

Hologram:

h(R) = * = 1+(R) *(R) + 2|R| cos(2ir+R)

Reconstruction: F-1{h(R)} = (u) +d(u)

+[uexp(-D-i(u-)

+[ uexp(-D+i(u+)

Diffraction:

uF{(R,t)} = g(u)*(u-k-g)

Object wave:(R,t) g exp (2i(k+g)r)

Uniqueness (J. Spence):

Scattering Matrix: S = e2iAt

however: t ± n/Re[] multiplicity

SS-1

=1 for all t => S(A)=S(B) only if A=B

Uniqueness (D.M. Barnett):

/z ~ gu/z => series expansion of u => unique coefficient relations