october 21, 2005a.j. devaney ima lecture1 introduction to wavefield imaging and inverse scattering...
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October 21, 2005 A.J. Devaney IMA Lecture 1
Introduction to Wavefield Imaging and Inverse Scattering
Anthony J. DevaneyDepartment of Electrical and Computer Engineering
Northeastern UniversityBoston, 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
October 21, 2005 A.J. Devaney IMA Lecture 2
Optical Microscopy
Objective LensCondenser
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
October 21, 2005 A.J. Devaney IMA Lecture 3
Magnification and ResolutionPin hole Camera
O
I
LI
LO
Magnification: M=LI/LO=I/O
Real Camera
O
I
Resolution: N.A.=sin θ ≈ a/O
a
O
aθ
δ
δ=λ/2N.A.
October 21, 2005 A.J. Devaney IMA Lecture 6
Abbe’s Theory of Microscopy
Illuminating light
Thin sample
Diffracted light
Plane waves
Image of sample
zγ
k Max Kρ=k sin θ
Each diffracted plane wave component carries sample information at specific spatial frequency
θ
Lens focuses each plane wave at image point
October 21, 2005 A.J. Devaney IMA Lecture 7
Basic Digital Microscope
Diffracted light
Plane waves
Image of sample
Diffracted light
Plane waves
Image of samplePC
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
October 21, 2005 A.J. Devaney IMA Lecture 8
Coherent Imaging
Analog Imaging
Computational Imaging
Nature
Computer
Lens
Measurement plane
Thin sample Image
Illuminating plane wave
October 21, 2005 A.J. Devaney IMA Lecture 9
Coherent Computational Imaging
Computational Imaging
Computer
Measurement planeIlluminating plane wave
Σ
Σ0 Σ
Undo PropagationPropagation
October 21, 2005 A.J. Devaney IMA Lecture 10
Plane Wave Expansion of the Solution to the Boundary Value Problem
z
Σ
≡Σ0
October 21, 2005 A.J. Devaney IMA Lecture 11
Propagation in Fourier Space
Free space propagation (z> 0) corresponds to low pass filtering of the field data
zpropagating
evanescent
propagating
evanescent
z
Σ
Σ0
Σ
Propagation
October 21, 2005 A.J. Devaney IMA Lecture 12
Undoing Propagation: Back propagation
Back propagation requires high pass filtering and is unstable (not well posed)
propagating
evanescent
z
Σ
Σ0
Σ
Propagation
z
Σ
Σ0
Σ
Backpropagation
October 21, 2005 A.J. Devaney IMA Lecture 13
Back propagation of Bandlimited Fields
z
ΣΣ0
Propagation
Backpropagation
October 21, 2005 A.J. Devaney IMA Lecture 14
Coherent Imaging Via Backpropagation
Σ
Backpropagation
Σ0
• Very fast and efficient using FFT algorithm• Need to know amplitude and phase of field
Plane wave
Kirchoff approximation
October 21, 2005 A.J. Devaney IMA Lecture 15
Limited Numerical Aperture
Σ
Backpropagation
Σ0 z
θa
Abbe’s theory of the microscope
PSF of microscope
October 21, 2005 A.J. Devaney IMA Lecture 16
Abbe Resolution Limit
ΣΣ0 z
θa
k sin θ
-k sin θ
-k
k
Maximum Nyquist resolution = 2π/BW=/2sinθ
October 21, 2005 A.J. Devaney IMA Lecture 17
Phase Problem
Gerchberg Saxon, Gerchberg Papoulis
Multiple measurement plane versions
Holographic approaches
October 21, 2005 A.J. Devaney IMA Lecture 18
The Phase ProblemCamera # 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
October 21, 2005 A.J. Devaney IMA Lecture 19
Laser
polarizer
Spatial filter
lens
mirror
mirror¼ 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
1024X102410 bits/pixelPixel size=10
Mach-Zender configuration
October 21, 2005 A.J. Devaney IMA Lecture 20
Retrieving the Complex Field
2
1*
*22
*11
11
1,
1,
D
D
e
e
ii
ieieIIie
eeIIe
sikz
sikz
sikz
sikz
ssikz
sikz
sikz
ssikz
¼ plate
Four measurements required
October 21, 2005 A.J. Devaney IMA Lecture 21
Limited Numerical Aperture
CCD
sample
ΣΣ0 z
θa
Measurement plane
Sin θ=a/z<<1
Fuzzy Images
N.A.=.13z=44 m.m.a=6 m.m.
October 21, 2005 A.J. Devaney IMA Lecture 23
5 μm Slit
0
50
100
150
(a)
Scattered intensity
200 400 600
100
200
300
400
500
600
700
200
400
600
800
(b)
Hologram 1
200 400 600
100
200
300
400
500
600
700
100
200
300
400
500
600
700
(c)
Hologram 2
200 400 600
100
200
300
400
500
600
700
October 21, 2005 A.J. Devaney IMA Lecture 24
Reconstruction of slit
Reconstructed intensity image from CDH
(a)
100 200 300 400 500 600
50
100
150
200
(b)
Reconstructed intensity image from PSDH
100 200 300 400 500 600
50
100
150
2005
10
15
20
25
30
100
200
300
400
500
600
Pixel size=6.7¹ mPixel size=6.7¹ m
October 21, 2005 A.J. Devaney IMA Lecture 25
Ronchi ruling (10 lines/mm)
0
200
400
600
100
200
300
400
500
600
700
200
400
600
800
(a) (b)
(c)
200m
Scattered intensity Hologram 1
Hologram 2
October 21, 2005 A.J. Devaney IMA Lecture 26
Reconstruction of Ronchi ruling
PSDH reconstruction
50 100 150 200 250
50
100
150
200
250100
200
300
400
500
Conventional optical microscope
100
200
300
400
500
CDH reconstruction
50 100 150 200 250
50
100
150
200
250
5
10
15Reconstruction by "two-intensity" algorithm
50 100 150 200 250
50
100
150
200
250
5
10
15
20
25
pixel size x=6.7m
October 21, 2005 A.J. Devaney IMA Lecture 28
Phase grating
0
100
200
300
400
500
(a)
Scattered intensity
100 200 300
50
100
150
200
250
300 0
50
100
150
200
250
300
350
(b)
Hologram 1
100 200 300
50
100
150
200
250
300
50
100
150
200
250
300
(c)
Hologram 2
100 200 300
50
100
150
200
250
300
pixel size x=6.7m
October 21, 2005 A.J. Devaney IMA Lecture 29
Reconstruction of phase grating
-2
0
2
Phase image reconstructed by PSDH
20 40 60 80 100
20
40
60
80
100
Intensity image from an optical microscope
-2
-1
0
1
2
Phase image reconstructed by CDH
20 40 60 80 100
20
40
60
80
100
50
100
150
200
250 pixel size x=6.7m
October 21, 2005 A.J. Devaney IMA Lecture 30
Salt-water specimen
100
200
300
400
500
(a)
Scattered intensity
50 100 150
50
100
150
200
400
600
800
(b)
Hologram 1
50 100 150
50
100
150
200
400
600
800
(c)
Hologram 2
50 100 150
50
100
150
pixel size x=6.7m
October 21, 2005 A.J. Devaney IMA Lecture 31
Reconstruction of salt-water specimen
pixel size x=1.675m
Intensity
200 400
100
200
300
400
Phase
200 400
100
200
300
400-2
0
2
5
10
15Intensity
200 400
100
200
300
400
Phase
200 400
100
200
300
400-2
0
2
5
10
15 Conventional optical microscopy
200
400
600
800
October 21, 2005 A.J. Devaney IMA Lecture 32
Biological samples: mouse embryo
0
100
200
300
400
500
600
(a)
Scattered intensity
50 100 150 200 250
50
100
150
200
250
200
400
600
800
(b)
Hologram 1
50 100 150 200 250
50
100
150
200
250
200
400
600
800
(c)
Hologram 2
50 100 150 200 250
50
100
150
200
250
Pixel size=6.7¹ mPixel size=6.7¹ m
October 21, 2005 A.J. Devaney IMA Lecture 33
Reconstruction of mouse embryo
(a)
Intensity image by PSDH
20 40 60 80 100 120
20
40
60
80
100
1202
4
6
8
10
12
(b)
Phase image by PSDH
20 40 60 80 100 120
20
40
60
80
100
120 0
0.5
1
1.5
2
(c)
Conventional optical microscope
50
100
150
200
Pixel size ±x = 1:675¹ mPixel size ±x = 1:675¹ m
October 21, 2005 A.J. Devaney IMA Lecture 34
Cheek cell
0
200
400
600
Scattered intensity
50 100150200250
50
100
150
200
250
200
400
600
800
Hologram 1
50 100150200250
50
100
150
200
250
200
400
600
800
Hologram 2
50 100150200250
50
100
150
200
250
pixel size x=6.7m
October 21, 2005 A.J. Devaney IMA Lecture 35
Reconstruction of cheek cell
5
10
15
Intensity image by PSDH
100 200 300
100
200
300
400
-2
0
2
Phase image by PSDH
100 200 300
100
200
300
400
200
400
600
800
Intensity image from an optical microscopepixel size x=1.675m
October 21, 2005 A.J. Devaney IMA Lecture 36
Onion cell
Intensity (PSDH)
Phase (PSDH)
-2
0
2
10
20
30
5
10
15
20
100 m
-2
0
2
Conventional optical microscope
200
300
400
500
October 21, 2005 A.J. Devaney IMA Lecture 37
Thick Sample System¼ plate
Thick (3D) sample of gimbaled mount
Many experiments performed with sample at variousorientations 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
October 21, 2005 A.J. Devaney IMA Lecture 38
Thick Samples: Born Model
Thick sample
Σ
Σ0
Determines 3D Fourier transform over an Ewald hemi-sphere
Born Approximation
October 21, 2005 A.J. Devaney IMA Lecture 39
Generalized Projection Slice Theorem
-kzKz
Kρ
The scattered field data for any given orientationof the sample relative to the optical axis yields
3D transform of sample over Ewald hemi-sphere
October 21, 2005 A.J. Devaney IMA Lecture 40
Multiple Experiments
Kz
Kρ
Kz
Kρ
Ewald hemi-spheres
k
k
√2 k
October 21, 2005 A.J. Devaney IMA Lecture 41
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
October 21, 2005 A.J. Devaney IMA Lecture 42
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
October 21, 2005 A.J. Devaney IMA Lecture 43
Inadequacy of Born Model¼ plate
Thick (3D) sample of gimbaled mount
1. Sample is placed in test tube with index matching fluid: Multiple scattering2. Samples are often times many wavelengths thick: Born model saturates
Adequately addressed by Rytov model
Addressed by DWBA model
October 21, 2005 A.J. Devaney IMA Lecture 44
Complex Phase Representation
(Non-linear) Ricatti Equation
Linearize Rytov Model
October 21, 2005 A.J. Devaney IMA Lecture 46
Free Space Propagation of Rytov Phase
Within Rytov approximation phase of field satisfies linear PDE
Rytov transformation
October 21, 2005 A.J. Devaney IMA Lecture 47
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 muchsuperior to Born for large objects—Back propagate field then use Rytov--Hybrid Model
October 21, 2005 A.J. Devaney IMA Lecture 48
N. Sponheim, I. Johansen, A.J. Devaney, Acoustical Imaging Vol. 18 ed. H. Lee and G. Wade, 1989
Rytov versus Hybrid Model
October 21, 2005 A.J. Devaney IMA Lecture 50
Mathematical Structure of Inverse Scattering
j=Zdrj±LO=d
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 mathematicsphysics and signal processing
October 21, 2005 A.J. Devaney IMA Lecture 51
Multi static Data Matrix
Multi-static Data Matrix=“Generalized Scattering Amplitude”
October 21, 2005 A.J. Devaney IMA Lecture 52
Distorted Wave Born Approximation
Linear Mapping
1. Vector space to vector space2. Hilbert Space HV to vector space CN, N=N x N
1 yields standard time-reversal processing useful for small sets of discrete targets2 yields inverse scattering useful for large sets of discrete targets
and distributed targets
October 21, 2005 A.J. Devaney IMA Lecture 54
Filtered Backpropagation Algorithm
“Propagation”
“Backpropagation”
Basis image fields
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