1. October 21, 2005A.J. Devaney IMA Lecture1 Introduction to Wavefield Imaging and Inverse Scattering Anthony J. Devaney Department of Electrical and Computer Engineering Northeastern University Boston, MA 02115 email: devaney@ece.neu.edu Digital Holographic Microscopy Review conventional optical microscopy Describe digital holographic microscopy Analyze imaging performance for thin samples Give experimental examples Outline classical DT operation for 3D samples Review DT in non-uniform background Computer simulations

2. October 21, 2005A.J. Devaney IMA Lecture2 Optical Microscopy Objective Lens Condenser Semi-transparent Object Image Illuminating light spatially coherent over small scale: Complicated non-linear relationship between sample and image Poor image quality for 3D objects Need to thin slice Cannot image phase only objects: Need to stain Need to use special phase contrast methods Require high quality optics Images generated by analog process Remove all image forming optics and do it digitally

3. October 21, 2005A.J. Devaney IMA Lecture3 Magnification and Resolution Pin hole Camera O I LILI LOLO Magnification : M=L I /L O =I/O Real Camera O I Resolution : N.A.=sin θ ≈ a/O a O a θ δ δ=λ/2N.A.

4. October 21, 2005A.J. Devaney IMA Lecture4 Fourier Analysis in 2D x y KxKx KyKy FT IFT ρ KρKρ

5. October 21, 2005A.J. Devaney IMA Lecture5 Plane Waves z γ k θ Propagating waves Evanescent waves

6. October 21, 2005A.J. Devaney IMA Lecture6 Abbe’s Theory of Microscopy Illuminating light Thin sample Diffracted light Plane waves Image of sample z γ k Max K ρ =k sin θ Each diffracted plane wave component carries sample information at specific spatial frequency θ Lens focuses each plane wave at image point

7. October 21, 2005A.J. Devaney IMA Lecture7 Basic Digital Microscope Diffracted light Plane waves Image of sample Diffracted light Plane waves Image of sample PC Illuminating light Coherent light Lens Detector system Issues: Speckle noise, phase retrieval, numerical aperture Each diffracted plane wave component carries sample information at specific spatial frequency

8. October 21, 2005A.J. Devaney IMA Lecture8 Coherent Imaging Analog Imaging Computational Imaging Nature Computer Lens Measurement plane Thin sample Image Illuminating plane wave

9. October 21, 2005A.J. Devaney IMA Lecture9 Coherent Computational Imaging Computational Imaging Computer Measurement plane Illuminating plane wave Σ Σ0Σ0 Σ Undo Propagation Propagation

10. October 21, 2005A.J. Devaney IMA Lecture10 Plane Wave Expansion of the Solution to the Boundary Value Problem z Σ ≡ Σ0Σ0

11. October 21, 2005A.J. Devaney IMA Lecture11 Propagation in Fourier Space Free space propagation (z> 0) corresponds to low pass filtering of the field data z propagating evanescent propagating evanescent z Σ Σ0Σ0 Σ Propagation

12. October 21, 2005A.J. Devaney IMA Lecture12 Undoing Propagation: Back propagation Back propagation requires high pass filtering and is unstable (not well posed) propagating evanescent z Σ Σ0Σ0 Σ Propagation z Σ Σ0Σ0 Σ Backpropagation

13. October 21, 2005A.J. Devaney IMA Lecture13 Back propagation of Bandlimited Fields z Σ Σ0Σ0 Propagation Backpropagation

14. October 21, 2005A.J. Devaney IMA Lecture14 Coherent Imaging Via Backpropagation Σ Backpropagation Σ0Σ0 Very fast and efficient using FFT algorithm Need to know amplitude and phase of field Plane wave Kirchoff approximation

15. October 21, 2005A.J. Devaney IMA Lecture15 Limited Numerical Aperture Σ Backpropagation Σ0Σ0 z θ a Abbe’s theory of the microscope PSF of microscope

16. October 21, 2005A.J. Devaney IMA Lecture16 Abbe Resolution Limit Σ Σ0Σ0 z θ a k sin θ -k sin θ -k k Maximum Nyquist resolution = 2π/BW= /2sinθ

17. October 21, 2005A.J. Devaney IMA Lecture17 Phase Problem Gerchberg Saxon, Gerchberg Papoulis Multiple measurement plane versions Holographic approaches

18. October 21, 2005A.J. Devaney IMA Lecture18 The Phase Problem Camera # 2 collimator HE-NE Laser Sample incident plane wave Magnifying Lens Camera # 1 Diffraction Plane # 1 Diffraction Plane # 2 Beam Splitter collimator HE-NE Laser incident plane wave Camera Sample Beam Splitter Mirror

19. October 21, 2005A.J. Devaney IMA Lecture19 Laser polarizer Spatial filter lens mirror ¼ plate Beam splitter CCD sample Beam splitter Digital Holographic Microscope Two holograms acquired which yield complex field over CCD Backpropagate to obtain image of sample 1024X1024 10 bits/pixel Pixel size=10 Mach-Zender configuration

20. October 21, 2005A.J. Devaney IMA Lecture20 Retrieving the Complex Field ¼ plate Four measurements required

21. October 21, 2005A.J. Devaney IMA Lecture21 Limited Numerical Aperture CCD sample Σ Σ0Σ0 z θ a Measurement plane Sin θ=a/z<<1 Fuzzy Images N.A.=.13 z=44 m.m. a=6 m.m.

22. October 21, 2005A.J. Devaney IMA Lecture22 Pengyi and Capstone Team

23. October 21, 2005A.J. Devaney IMA Lecture23 5 μm Slit

24. October 21, 2005A.J. Devaney IMA Lecture24 Reconstruction of slit

25. October 21, 2005A.J. Devaney IMA Lecture25 Ronchi ruling (10 lines/mm) Scattered intensity Hologram 1 Hologram 2

26. October 21, 2005A.J. Devaney IMA Lecture26 Reconstruction of Ronchi ruling

27. October 21, 2005A.J. Devaney IMA Lecture27 Conventional Versus Backpropagated

28. October 21, 2005A.J. Devaney IMA Lecture28 Phase grating

29. October 21, 2005A.J. Devaney IMA Lecture29 Reconstruction of phase grating

30. October 21, 2005A.J. Devaney IMA Lecture30 Salt-water specimen

31. October 21, 2005A.J. Devaney IMA Lecture31 Reconstruction of salt-water specimen pixel size  x=1.675  m

32. October 21, 2005A.J. Devaney IMA Lecture32 Biological samples: mouse embryo

33. October 21, 2005A.J. Devaney IMA Lecture33 Reconstruction of mouse embryo (a) Intensity image by PSDH 20406080100120 20 40 60 80 100 120 2 4 6 8 10 12 (b) Phase image by PSDH 20406080100120 20 40 60 80 100 120 0 0.5 1 1.5 2 (c) Conventional optical microscope 50 100 150 200

34. October 21, 2005A.J. Devaney IMA Lecture34 Cheek cell

35. October 21, 2005A.J. Devaney IMA Lecture35 Reconstruction of cheek cell

36. October 21, 2005A.J. Devaney IMA Lecture36 Onion cell

37. October 21, 2005A.J. Devaney IMA Lecture37 Thick Sample System ¼ plate Thick (3D) sample of gimbaled mount Many experiments performed with sample at various orientations relative to the optical axis of the system Paper with Jakob showed that only rotation needed to (approximately) generate planar slices Use cylindrically symmetric samples

38. October 21, 2005A.J. Devaney IMA Lecture38 Thick Samples: Born Model Thick sample Σ Σ0Σ0 Determines 3D Fourier transform over an Ewald hemi-sphere Born Approximation

39. October 21, 2005A.J. Devaney IMA Lecture39 Generalized Projection Slice Theorem -kz KzKz KρKρ The scattered field data for any given orientation of the sample relative to the optical axis yields 3D transform of sample over Ewald hemi-sphere

40. October 21, 2005A.J. Devaney IMA Lecture40 Multiple Experiments KzKz KρKρ KzKz KρKρ Ewald hemi-spheres k k √2 k

41. October 21, 2005A.J. Devaney IMA Lecture41 Born Inversion for Fixed Frequency Inversion Algorithms: Fourier interpolation (classical X-ray crystallography) Filtered backpropagation (diffraction tomography) Problem: How to generate inversion from Fourier data on spherical surfaces A.J.D. Opts Letts, 7, p.111 (1982) Filtering of data followed by backpropagation: Filtered Backpropagation Algorithm

42. October 21, 2005A.J. Devaney IMA Lecture42 Inverse Scattering Computer Illuminating plane waves 3D semi-transparent object Object Reconstruction Essentially combine multiple 3D coherent images generated for each scattering experiment Filtering followed by back propagation

43. October 21, 2005A.J. Devaney IMA Lecture43 Inadequacy of Born Model ¼ plate Thick (3D) sample of gimbaled mount 1. Sample is placed in test tube with index matching fluid: Multiple scattering 2. Samples are often times many wavelengths thick: Born model saturates Adequately addressed by Rytov model Addressed by DWBA model

44. October 21, 2005A.J. Devaney IMA Lecture44 Complex Phase Representation (Non-linear) Ricatti Equation Linearize Rytov Model

45. October 21, 2005A.J. Devaney IMA Lecture45 Short Wavelength Limit Classical Tomographic Model

46. October 21, 2005A.J. Devaney IMA Lecture46 Free Space Propagation of Rytov Phase Within Rytov approximation phase of field satisfies linear PDE Rytov transformation

47. October 21, 2005A.J. Devaney IMA Lecture47 Degradation of the Rytov Model with Propagation Distance Rytov and Born approximations become identical in far field (David Colton) Experiments and computer simulations have shown Rytov to be much superior to Born for large objects—Back propagate field then use Rytov--Hybrid Model

48. October 21, 2005A.J. Devaney IMA Lecture48 N. Sponheim, I. Johansen, A.J. Devaney, Acoustical Imaging Vol. 18 ed. H. Lee and G. Wade, 1989 Rytov versus Hybrid Model

49. October 21, 2005A.J. Devaney IMA Lecture49 Potential Scattering Lippmann Schwinger Equation

50. October 21, 2005A.J. Devaney IMA Lecture50 Mathematical Structure of Inverse Scattering Non-linear operator (Lippmann Schwinger equation) Object function Scattered field data Use physics to derive model and linearize mapping Linear operator (Born approximation) Form normal equations for least squares solution Wavefield Backpropagation Compute pseudo-inverse Filtered backpropagation algorithm Successful procedure require coupling of mathematics physics and signal processing

51. October 21, 2005A.J. Devaney IMA Lecture51 Multi static Data Matrix Multi-static Data Matrix=“Generalized Scattering Amplitude”

52. October 21, 2005A.J. Devaney IMA Lecture52 Distorted Wave Born Approximation Linear Mapping 1.Vector space to vector space 2. Hilbert Space H V to vector space C N, N=N  x N  1 yields standard time-reversal processing useful for small sets of discrete targets 2 yields inverse scattering useful for large sets of discrete targets and distributed targets

53. October 21, 2005A.J. Devaney IMA Lecture53 SVD Based Inversion

54. October 21, 2005A.J. Devaney IMA Lecture54 Filtered Backpropagation Algorithm “Propagation” “Backpropagation” Basis image fields

55. October 21, 2005A.J. Devaney IMA Lecture55

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58. October 21, 2005A.J. Devaney IMA Lecture58

59. October 21, 2005A.J. Devaney IMA Lecture59

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