INVERSE PROBLEMS IN X-RAY SCIENCE STEFANO MARCHESINI

Transcription

1 INVERSE PROBLEMS IN X-RAY SCIENCE STEFANO MARCHESINI overview of photon science inverse problems linear (under-determined): tomography non-linear: diffraction methods Inverse problems and the data deluge 1

2 When new materials are discovered we want to know... Structure (atomic resolution for crystals) Electronic properties Chemical structure Dynamical evolution 2

3 X-ray interaction with matter wavelength ~atom size penetration ~microns-m Energy: ~valence bands and up time: femtoseconds and longer scattering/diffraction (refraction) x-rays absorption/refraction fluorescence, electrons inelastic 3 scattering

4 APPLICATIONS OF X-RAY SCIENCE Solid state physics (spectroscopy: photoemission, inelastic scattering...) Biology (protein structures, cell imaging) Chemistry, Materials science (photovoltaics, batteries, catalysis, etc) Earth science Space science (stardust, metrology) Archeology Semiconductors (lithography, metrology)... 4

5 ADVANCED LIGHT SOURCE SYNCHROTRON Experimental Techniques Diffraction (~50%) spectro-microscopy (~25%) spectroscopy (~25%) - 45 beamlines around the ALS - covers a large range of science - structural biology - energy sciences - geo / environmental sciences - condensed matter physics - chemistry - materials science 5 - EUV lithography -.

6 X-ray interaction with matter wavelength ~atom size penetration ~microns-m Energy: ~valence bands and up time: femtoseconds and longer scattering/diffraction (refraction) x-rays absorption/refraction fluorescence, electrons inelastic 6 scattering

7 DISCOVERY OF MATERIALS ELECTRONIC BAND STRUCTURE Solid state physics Angle resolved photoemission Model A. Lanzara 7

8 X-ray interaction with matter wavelength ~atom size penetration ~microns-m Energy: ~valence bands and up time: femtoseconds and longer scattering/diffraction (refraction) x-rays absorption/refraction fluorescence, electrons inelastic 8 scattering

9 absorption Contrast mechanism I(x) =I 0 (x)t (x) absorption coefficient e (x,z 0) z 9

10 absorption Ray optics I 0 = I 0 e (z) z refraction E 0 = E 0 e n(z) z n =1 + i wavefield Ray optics, thick sample e (x,z 0) z e (x,z 1)... z Rythov approximation Born approximation e P i (x,z i) z log( I z I0 )= 10 I z I0 =1 X i (x, z i ) z X (x, z i ) i z

11 ray optics absorption coefficient e (x,z 0) z Tomography projection slice F 1D R (x, y)d = F2D (qˆ ) Fill out Fourier Slices IFT measure R (x, y)d 11

12 SHANNON WAS A PESSIMIST Sparsifying algorithms allow exact reconstruction from undersampled data Candes, Romber, Tao ( 06) Shannon theorem is worst case scenario if we know that the signal is sparse (and other conditions) see talk by Rebecca Willett Exact reconstruction using Total Variation norm (sparse gradient) 12

13 X-ray Microscopy the experiment ρ 2 Lens refractive index: Propagation Propagation 13

14 X-ray Diffraction: optical Fourier transform Simple geometry 14 Propagation

15 X-ray interaction with matter wavelength ~atom size penetration ~microns-m Energy: ~valence bands and up time: femtoseconds and longer scattering/diffraction (refraction) x-rays absorption/refraction fluorescence, electrons inelastic 15 scattering

16 NONLINEAR DATA ANALYSIS Diffraction & phase retrieval 16

17 TIMELINE 1905 Rontgen: x-ray imaging 1912 Bragg NaCl Diffraction pattern from a crystal 1951 Pauling et al Alpha helix stereochemistry+diffraction: 1 spot (5.1 A period) 1953 Watson & Crick DNA stereochemistry+diffraction 10 spots

18 Elastic scattering and Fourier space Each 2D measurement I(p) is a slice in Fourier space I(p)= F(q) 2 rotate for 3D detector pixels q i,j = k out k in, = 1 (p i,p j,z D ) p p 2 i +p 2 j +z2 D (0, 0, 1) 18 F (q) =F (x)

19 X-ray Diffraction: the phase problem F (q) =F (x) sample ρ Fρ 2 position (real space) FT momentum (Fourier space) ψ(-x)* ψ(x) Find ρ Measure Fρ 2 19 IFT Autocorrelation (real space) ρ(x)* ρ(-x)

20 SOLUTION IS NOT UNIQUE Find ρ Measure Fρ 2 Trivial ρ(x), ρ(-x), ρ(x+δx), e iφ identity flip translation phase homometric structures ρ(x) ρ(x)=ρ1(x)*ρ2(x), ρ(x)=ρ1(x)*ρ2(-x) ρ1(x) ρ2(x) Bruck, Sodin (1979) Bates (1982) Hayes (1982) homometric structures are rare (factorable polynomials are 20 rare in 2D and 3D)

21 X-ray Diffraction: the phase problem solution (1) (Gabor) Holography: introduce reference sample ρ ψ δ ρ(x)* δ ρ(x)* ρ(-x) ρ(-x)* δ position (real space) FT Fρ 2 IFT Find reference Measure 21 Use

22 X-ray Diffraction: the sampling problem F (q) =F (x) ρ(-x)* ρ(x) sample ρ Fρ 2 position (real space) FT momentum (Fourier space) IFT Autocorrelation (real space) Find Measure Sampling & aliasing ρ ρ(-x)* ρ(x) ρ Fρ 2 22

23 OPTIMIZATION PROBLEM Measure Find I(q) =c F (q) 2 F (q) =F (x) (q) Subject to: (x) finite, sparse or other prior (q) q Average over e.g. wavelength, rotation, crystallite sampling, e.g. Bragg condition, beamstop 23

24 ITERATIVE METHODS Start with some guess F -1 transform Real space constraint (q) O (q) F-transform Fit the data 24 projection onto a nonconvex set

25 BRANCHES OF DIFFRACTION Diffraction Small molecule crystal Small powder crystal protein crystal +strong contrast atom Solution scattering Direct methods seeking sparsity Direct methods, trial/error Direct methods, +maximum likelihood +model refinment Monte carlo 25

26 BRANCHES OF DIFFRACTION Diffraction Single particle diffraction nanocrystallography microcrystal scanning diffraction Non convex methods (seeking compact object) orientation determination (10^5 patterns) model fitting orientation determination model fitting L2 minimization unknowns 26

27 Sampling Schemes single particle Detector Mask I(q) =c F (q) 2 M(q) 27

28 Sampling Schemes Single molecule CDI Detector Mask Unknown rotation I(q) =c F (Rq) 2 M(q) 28

29 Sampling Schemes Protein crystal Detector Mask Bragg Sampling I(q) =c F (Rq) 2 S(Rq)M(q) 29

30 Sampling Schemes Protein nano-crystals Detector Mask Bragg Sampling Unknown rotation I(q) =c F (Rq) 2 S(Rq)M(q) 30

31 Sampling Schemes Powder diffraction Detector Mask Bragg Sampling average orientations I(q) =c F (q) 2 S(q) q M(q) 31

32 Sampling Schemes Solution scattering Detector Mask average orientations I(q) =c F (q) 2 q M(q) 32

33 Sampling Schemes Laue diffraction Thomas White CFEL, DESY Hamburg Detector Mask Bragg Sampling average wavelengths I(q) =c F (q) 2 S(q) (q) M(q) 33

34 Coherent diffractive imaging (1999) Coherent Diffraction Resolution extended by phasing alogirhtms + = J. Miao, P. Charalambous, J. Kirz & D Sayre,Nature 400, (1999) 34

35 Coherent Diffraction Ab-initio cohrent diffractive imaging + iteratively shrinking the support Set of all objects that have measured diffraction = support Find sparsest solution that fits the data S. Marchesini, H. He, H. N. Chapman et al. PRB 68, (R) (2003), 35

36 THREE DIMENSIONAL CDI Coherent Diffraction Diffraction data ab-initio Reconstruction Established billion-element phasing 15 nm resolution H. N. Chapman, A. Barty, S. Marchesini, et 36al. JOSAA 23, (2006)

37 IT DOESN T ALWAYS WORK Support determination algorithms (shrinkwrap) are not robust Even with simulated data, reconstruction is not guaranteed. Missing data is a main source of problem Table 1. Summary of various algorithms Algorithm Iteration (n+1) = ER SF HIO DM ASR HPR RAAR P s P m (n) R ( s P m (n) P m (n) (r) r S (I P m ) (n) (r) r / S {I + P s [(1 + s ) P m s I] P m [(1 + m ) P s m I]} (n) 1 [R 2 sr m + I] (n) 1 [R 2 s (R m +( 1)P m ) +I + (1 )P m ] (n) ˆ 1 (R 2 s R m + I) + (1 )P m (n) [Pm-I] ρ 2 / ρ0 2 target HIO conjugate gradient 37

38 WHAT FIX-POINT ITERATIONS WORK BEST? (q) O (q) Table 1. Summary of various algorithms Algorithm Iteration (n+1) = ER SF HIO DM ASR HPR RAAR P s P m (n) R ( s P m (n) P m (n) (r) r S (I P m ) (n) (r) r / S {I + P s [(1 + s ) P m s I] P m [(1 + m ) P s m I]} (n) 1 [R 2 sr m + I] (n) 1 [R 2 s (R m +( 1)P m ) +I + (1 )P m ] (n) ˆ 1 2 (R s R m + I) + (1 )P m (n) arxiv:physics/ arxiv:

39 DATA DELUGE OPPORTUNITIES Brighter x-rays, faster detectors More of the same more samples, more theory faster Statistical inference (Sparse modeling) recovering a small number of most relevant variables in high-dimensional data Reduction Reduce multi-frame data into higher resolution, higher contrast (SNR) images, volumes or movies. Examples: ptychography, (nanocrystallography, single molecule, A. Barty talk) 39

40 ptychography: combine scanning microscopy with diffraction x i a i 2 w ˆ w w z i m n z m m 40 z k. z 1 diffraction data a x (q) = Fw(r) ˆ(r + x), F FFT probe X unknown Karle, Hoppe, ~1970 Rodemburg ~ Chapman ~ Pfeiffer ~2007 Thibault ~

41 ( i amplitude a a i (q) Fourier transform )=( F Q )( )( ) a 1 a 2 F F F F FF m scanning illumination unknown w w w w w w Q 1 Q 2 i z i (r) w (r) FT Q x1 x 2 i r 2 m n q 2 r 1 q 1 r 2 41 x3 r 1 n

42 The Fourier magnitude projection P is used to ensure that F ix@r ˆ )g,=we Qxintroduce (r) = z x (r), Qx(3). (r) P= w(r)e. k sequences of various matrices as follows in Eq. be expressed as: F can Scanning Diffractive imaging Fz Q z F P z = F a. individual frames that we 1introduce for convenience. 1 F F z a1 cribing B matrices C B C w B C.. C,ofQvarious... (r) a (q) uences as follows i... where F = transform A, iz A.. A. A,Fourier the inverse operator. PF is a data unknown aqk 1 z k PF z = arg min kzi z i k, subject F to F z z Q1k F z x3 C.. C, z = B.. C, F = B. x 2. where A k k the Euclidian A x1 The overlap i projectio.. A norm. ) as a = F Q, or using the intermediate n variable z as: Qk z k the known set of illuminations F Q: q2 q1 a = F z, PQ z = Q intermediate singintroduce the intermediate variable z as: ˆ, z z = variable Q min, r2 where r1 min n = arg min kz where z, Q are the set of frames and set of illuminations re 2 Standard Projection ˆ is obtainedalgorithms of the unkown solution by solving the least squ F z, (3 a problems = feasibility called Fourier magnitude and overlapping i 1 = (Q Q) z. th ˆ The Fourier magnitude projection P is used to ensure min F = Q, (4 ely. Thez ptychographic reconstruction problem consists in Qfinding in Eq. (3). canoperator be expressed as: projections Pis F the where Q that multiplies conjugate o tive methods introduce an intermediate variablebyz,theand attem zi (r) alled Fourier imagnitude problems r frames zand the image. Qillumination is the operator which i ontooverlapping F z splits an in Eqs. (3,4) using projection algorithms, iterative transform Q) 1 ispaf normalization data = FMany iterativ a. ˆfitknowing each frame by a probe. (Q factor. T uction problem m consists in finding a, zq. F z on methods [10]. can be expressed as: iable z, and attempt to solve the two problems in Eqs. (3,4) 1 usin satisfy overlap where F is the inverse Fourier transform operator. PQ = Q(Q Q) QP,F is g section we will describe the standard operators commonly m methods, or alternating direction methods citezwen. 42 In the alternating projection algorithm, the approximation t on 3 we will introduce an intermediate variable PF z = arg min k, subjectpha toe i ci,z ireplacing is significant overlap among di erent areas of kz illumination,

43 ptychographic data ( ) ALS Ptychography super-resolution combining scanning with diffraction (future) Combine Tomography Blind deconvolution Phase retrieval multiple scattering Vibrations denoising 43 X 7 ALS Beamline 5.3.2, 2012 Scanning microscopy scanning diffraction d removed from web version resolution enhancement T Tyliszczak, R. Celestre, A D. Kilcoyne,, A. Schirotzek, T. Warwick(ALS),

44 CONCLUSIONS Experiments in photon science are very diverse Sparse modeling is a powerful method to extract information from noisy data High frame rate enables to achieve higher SNR or resolution: imaging of samples previously impossible Phase retrieval remains an open issue 44