Phase contrast x ray imaging in two and three dimensions P. Cloetens

Transcription

1 Phase contrast x ray imaging in two and three dimensions P. Cloetens Propagation based phase contrast imaging the forward problem; Fresnel diffraction coherence conditions The inverse problem phase retrieval methods holotomography Recent developments Projection microscopy KB based imaging Slide: 1

2 Phase Contrast vs Absorption Simple transmission Absorption Phase Sample Dream 1: Zero Dose Increase the energy Absorption contrast Sample replaced by phase contrast Dream 2: Improve the Sensitivity Absorption contrast too low high spatial resolution light materials similar attenuation Slide: 2

3 Transmission through a sample refractive index n = 1 δ + iβ z e ik z medium =e i 2 n z =e i 2 z. e i 2 z 2. e z Transmission function projection of the refractive index distribution x y u inc (x,y) u 0 (x,y) = T(x,y). u inc (x,y) z Amplitude A x, y =e B x, y =e 2 x, y, z dz T(x,y) = A(x,y).e iϕ(x,y) Phase x, y = 0 2 x, y,z dz Slide: 3

4 Phase contrast vs Absorption Be C Al Fe Au Water window Hard X rays Energy (kev) Gain of up to a few 1000! Slide: 4

5 Phase Sensitive Techniques s Phase Retardation 2π ϕ = δ. s λ α ~ µrad Deflection Phase gradients = 2 x At zero distance: Intensity I 0 = u 0 2 =I inc.exp dz all phase information is lost Slide: 5

6 Phase Contrast Imaging methods Analyzer based PCI ID17 Medical Imaging Grating based PCI BM05/ID19 wavefront sensor Ch David, F Pfeiffer T Weitkamp Propagation based PCI ID19 / ID22NI Edge enhancement Holotomography Slide: 6

7 towards Fraunhofer diffraction edge detection versus holography (Fresnel diffraction) D = 0.15 m each edge imaged independently no access to phase, only to border D a λ = 0.7 Å 50 µm D = 3.1 m access to phase, if recorded at D s defocused image D a Slide: 7

8 Simulation Direct Space multiplication wave with transmission function OBJECT T = e B e iϕ Known Object Image Reciprocal Space convolution diffraction integral PROPAGATION multiplication FT wave with propagator P D = exp( iπλdf 2 ) I D coh ( x,y) = u(x,y) 2 INTENSITY auto correlation convolution intensity with COHERENCE / DETECTOR multiplication FT intensity with projected source degree of coherence µ 12 (λdf) PSF detector detector transfer function R(f) Slide: 8

9 Simulation Known Object Image Porosity in Quasi crystals Hole = dodecahedron seen along the 2 fold axis µm Simulations λ = 0.52 Å D = 0.4 m Slide: 9

10 Fresnel diffraction D u 0 (x,y) u D (x,y) In principle: complete object contributes to a point of the image In practice: only finite region: first Fresnel zone radius r F = D First Fresnel zone determines the sensed lengthscale Distance to be most sensitive to object with size a D opt = a2 2 For example at λ = 0.5Å (25 kev) a = 1 µm D = 10 mm a = 40 mm D = 16 m Slide: 10

11 Contrast Transfer Functions Fourier Transforms of the intensity and phase are linearly related I D (f) = δ Dirac (f) 2 cos(πλdf 2 ). B(f) + 2 sin(πλdf 2 ). ϕ(f) amplitude contrast factor phase contrast factor valid in case of a slowly varying phase, weak absorption 2 1 contrast factor amplitude phase frequency Slide: 11

12 Variable period Decreasing linewidth Increasing spatial frequency 2 µm period 720 nm period 530 nm period 610 nm Object invisible! Contrast depends strongly on period or spatial frequency Slide: 12

13 Spatial Coherence Hard X ray / neutron sources are ~ incoherent Wave is partially coherent when the source is small and far z 1 z 2 s α blurring = z 2.α = z 2 /z 1.s source object detector l Transverse coherence length coh Feature size a (1/f) λ λ. z = = 1 2α 2s D opt = a2 2 µ 12 (λdf) l coh > a / 2 Slide: 13

14 Temporal Coherence Incoherent sum of intensity for different wavelengths Approximation: second coherence envelope transfer function I D (f) = δ Dirac (f) + sinc( λdf 2 /2). 2 sin(πλ 0 Df 2 ). ϕ(f) = = Slide: 14

15 Edge Detection Essentially edge enhancement Weak defocusing (and weak contrast!) λd << a Radiography (2D) I D x, y I 0 x, y. [1 D 2 x, y ] xy Tomography (3D) absorption image phase term 2D Laplacian phase o x, y, z x, y, z D. xyz x, y, z absorption term Detection of cracks holes reinforcing fibres, particles refraction term Laplacian refractive index Slide: 15

16 Edge Detection U. Wegst Slide: 16

17 Insect Anatomy 100 µm Virtual slices through heads of tiny staphylinid beetles O. Betz (Univ. Tuebingen), U. Wegst, D. Weide et al, J. Microscopy, 227, 51 (2007) Slide: 17

18 Phase Retrieval How to retrieve the phase and amplitude in the object plane? Phase in image plane is lost Image(s) Object??? Inverse Problem Slide: 18

19 Phase retrieval and holography Consider weak object k inc k κ.k inc T x = x with x =0 I D f = 2 D * e i D f 2 f e i D f 2 * f NonLin. Back propagation e i D f 2 I D f 0 = * f e i 2D f 2 * f FT object FT conjugate object Multiple distances to avoid twin image problem e i D f 2 1 I 1 f 0 = * f e i 2D f 2 1 * f e i D f 2 N I N f 0 = * f e i 2D f 2 N * f Slide: 19

20 Phase Retrieval in practice Australian School: K. Nugent, T. Gureyev, D. Paganin TIE (transport of intensity) I z = 2 I Flemish School: D. Van Dyck, P. Cloetens, JP Guigay Transfer Function Approach / Focus variation Series of images recorded at different distances Each distance is most sensitive to a specific range of spatial frequencies D 1 D 2 D n Slide: 20

21 Phase Retrieval: multiple distances non absorbing foam 4 images recorded E = 18 kev D variable D D = 0.03 m D = 0.21 m D = 0.51 m D = 0.90 m Least squares minimization of cost functional K= m I exp m f I calc m [ f ] 2 50 µm Slide: 21

22 Holo tomography 1) phase retrieval with images at different distances D 2) tomography: repeated for 1000 angular positions Phase map 3D distribution of δ or the electron density improved resolution straightforward interpretation processing PS foam P.Cloetens et al., Appl. Phys. Lett. 75, 2912 (1999) Slide: 22

23 Phase retrieval: transfer function approach I m f =2sin D m f 2 f Linear least squares f = m sin D m f 2 I m f m 2 sin 2 D m f 2 sin( πλdf transfer function 2 ) Optimization of the choice of distances 1/ N m sin 2 D m f 2 4 distances Non iterative (fast!) Slide: 23

24 Phase retrieval: regularization Remains ill posed for low spatial frequencies Requires regularization: e.g. limit the energy of the solution K= m I m exp f I m calc [ f ] 2 f 2 Model error Regularization error f = m sin D m f 2.I m f m 2sin 2 D m f 2 Slide: 24

25 Phase Tomography of Arabidopsis seeds 3D structure of Arabidopsis seeds in their native state wet sample, no preparation no staining, no fixation, no cutting, no cryo cooling Holotomographic approach Contrast proportional to the electron density 4 distances, 800 angles E = 21 kev P Cloetens, R Mache, M Schlenker, S Lerbs Mache, PNAS (2006) 103, Slide: 25

26 Phase Tomography of Arabidopsis seeds Holotomographic approach Four distances E = 21 kev Seed of Arabidopsis Tomographic Slices 30 µm Cotyledon P Cloetens, R Mache, M Schlenker, S Lerbs Mache, PNAS (2006) 103, Slide: 26

27 Phase Tomography of Arabidopsis seeds Seed of Arabidopsis Tomographic Slice 10 µm tegumen protoderm intercellular spaces organites (protein stocks) 5 µm Slide: 27

28 Phase Tomography of Arabidopsis seeds Seed of Arabidopsis three dimensional network of intercellular air space gas exchange during germination Role? and/or rapid water uptake during imbibition P Cloetens, R Mache, M Schlenker, S Lerbs Mache, PNAS (2006) 103, Slide: 28

29 Phase retrieval: mixed approach (absorption case) Transfer function approach is robust, optimized solution but it is only valid for weak absorption Transport of Intensity Equation (TIE) is 'exact' ( D 0) but contrast is not optimized, less robust Merge and extend validity of both methods I D f =I D = 0 f 2sin D f 2 FT {I 0 } f cos D f 2 D 2 FT { I 0 } f Transfer function like Leads to a 'moderately iterative' solution (3 5 iterations) valid for slowly varying objects Perturbation term JP Guigay, M Langer, R Boistel, P Cloetens, Opt. Lett., 32, 1617 (2007) Slide: 29

30 Oldest terrestrial fossil plant from Rhynie (Scotland) 100 µm Tafforeau P., Kaminskyj S. Absorption D = 6 mm

31 Oldest terrestrial fossil plant from Rhynie (Scotland) 100 µm Tafforeau P., Kaminskyj S. Phase contrast D = 100 mm

32 Oldest terrestrial fossil plant from Rhynie (Scotland) 100 µm Tafforeau P., Kaminskyj S. Phase contrast D = 300 mm

33 Oldest terrestrial fossil plant from Rhynie (Scotland) 100 µm Tafforeau P., Kaminskyj S. Holotomographic reconstruction

34 Phase retrieval: regularization using a prior estimate Can the absorption provide the low spatial frequencies? Prior estimate (cf homogeneous object phase/attenuation duality) prior f = B f Use this as a prior estimate (different from a constraint) K= m I m exp f I m calc [ f ] 2 f prior f 2 Model error Regularization error f = m sin D m f 2. I m f prior f m 2 sin 2 D m f 2 M Langer, P Cloetens. F Peyrin, submitted to IEEE TIP Slide: 34

35 Phase retrieval: automatic parameter selection How to determine the regularization parameter ε? Automatic using L curve method! Follow curve log(regularization error) vs log(model error) Use point of maximum curvature M Langer, P Cloetens. F Peyrin, submitted to IEEE TIP Slide: 35

36 Phase retrieval: prior estimate Prior estimate from absorption and automatic parameter selection Without prior estimate With prior estimate Slide: 36

37 The Nano Imaging Project Projection Microscopy PXM Structure Dose efficient, fast Phase contrast Fluorescence mapping XFM Micro analysis Slow Trace elements Phase contrast Slide: 37

38 Projection Microscopy (PXM) object z 1 z 2 detector D equivalent to Limit cases z 1 >> z 2 D = z 2 ; M = 1 D= z 1. z 2 z 1 z 2 + M= z 1 z 2 z 1 z 1 << z 2 D = z 1 ; M = z 2 /z 1 Plane wave illumination Magnification Prop. by D Van Dyck, J Spence Slide: 38

39 Projection Microscopy Neuron cells Towards focus 10 µm D = 225 mm M = 17 D = 175 mm M = 22 D = 125 mm M = 31 D = 45 mm M = 87 E = 20.5 kev Exposure time = 1 s (16 bunch) 50 ms (ID22NI) No X ray optics behind the sample Dose efficient Slide: 39

40 Projection Microscopy: Phase Retrieval 10 µm 5 distances Neuron cell D = mm mm Rel. Phase Map Mass Quantitative Fluorescence Slide: 40

41 Zoom Tomography z1 = mm; mm;voxel voxel== nm nm µm Keep sample preparation straightforward Not too destructive Sample size >> field of view E = 17 kev 1801 projections; 0.1s/proj mm mm µm Slide: 41

42 Al / Si alloy tomographic slice Zoom Tomography Si Pore Al 5 FeSi 75 µm Inside φ = 1 mm sample local tomography! E = 20.5 kev X ray magnification ~ 80 (voxel size = 90 nm) R Mokso, P Cloetens, E Maire, W Ludwig, JY Buffière, APL, 2007, 90, Slide: 42

43 Nano XRF of a Siemens star E = 17.5 kev 600 x 600 pixels 30 ms / point 50 nm step (zap) 3 hours Detector limited 30 µm Slide: 43

44 Fresnel diffraction of a Siemens star E = 17.5 kev 500 ms exposure z 1 : mm 'Pixel' : nm Slide: 44

45 Fresnel diffraction of a Siemens star Simulation E = 17.5 kev D = 7 mm Slide: 45

46 Some references E. Hecht, Optics, 3th ed. (Addison Wesley, 1998). M. Born and E. Wolf, Principle of Optics, 6th ed. (Pergamon Press, Oxford, New York, 1980). J.W. Goodman, Introduction to Fourier optics, 2nd ed. (Mcgraw Hill, 1988). D. Paganin, Coherent X ray Optics (Oxford University Press, USA, 2006). P. Cloetens, R. Barrett, J. Baruchel, J.P. Guigay and M. Schlenker, J. Phys. D: Appl. Phys. 29, 133 (1996). K.A. Nugent, T.E. Gureyev, D.F. Cookson, D. Paganin, Z. Barnea, Phys. Rev. Lett. 77, 2961 (1996). P. Cloetens, M. Pateyron Salomé, J. Y. Buffière, G.Peix, J. Baruchel, F. Peyrin and M. Schlenker, J.Appl. Phys. 81, 9 (1997). P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Landuyt, J.P. Guigay, and M. Schlenker, Appl. Phys. Lett. 75, 2912 (1999). S. Zabler, P. Cloetens, J.P. Guigay, J. Baruchel, M. Schlenker, Rev. Sci. Instrum. 76, (2005). Slide: 46