Direct Retrieval of Object Information using Inverse Solutions of Dynamical Electron Diffraction
Max Planck Institute of Microstructure PhysicsHalle/Saale, [email protected]
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