Introduction to Wavefield Imaging and Inverse Scattering

1
Introduction to Wavefield Imaging and Inverse
Scattering
Anthony J. Devaney Department of Electrical and
Computer Engineering Northeastern
University Boston, MA 02115 email
devaney_at_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
Optical Microscopy

• 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
Magnification and Resolution
Pin hole Camera
Magnification
MLI/LOI/O
Real Camera
I
a
d
d?/2N.A.
O
a
?
Resolution N.A.sin ? a/O
O
4
Fourier Analysis in 2D
y
Ky
FT
IFT
?
K?
x
Kx
5
Plane Waves
k
?
z
?
6
Abbes Theory of Microscopy
Lens focuses each plane wave at image point
Plane waves
Thin sample
Image of sample
Illuminating light
Each diffracted plane wave component carries
sample information at specific spatial frequency
Diffracted light
Max K?k sin ?
k
?
z
?
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Basic Digital Microscope
Plane waves
Lens
Illuminating light
Image of sample
Diffracted light
Each diffracted plane wave component carries
sample information at specific spatial frequency
Plane waves
Detector system
Coherent light
PC
Image of sample
Diffracted light
Issues Speckle noise, phase retrieval, numerical
aperture
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Coherent Imaging
Lens
Image
Thin sample
Nature
Analog Imaging
Measurement plane
Illuminating plane wave
Computer
Computational Imaging
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Coherent Computational Imaging
Measurement plane
Illuminating plane wave
Computer
Computational Imaging
Propagation
Undo Propagation
S
S0
S
10
Plane Wave Expansion of the Solution to the
Boundary Value Problem
S

z
S0
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Propagation in Fourier Space
evanescent
z
propagating
Propagation
S
z
S0
propagating
evanescent
S

Free space propagation (zgt 0) corresponds to low
pass filtering of the field data
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Undoing Propagation Back propagation
Propagation
Backpropagation
S
S
z
z
S0
S0
S
S
propagating
evanescent
Back propagation requires high pass filtering and
is unstable (not well posed)
13
Back propagation of Bandlimited Fields
Propagation
z
Backpropagation
S0
S
Propagation
Backpropagation
14
Coherent Imaging Via Backpropagation
Kirchoff approximation
Backpropagation
Plane wave
S
S0

• Very fast and efficient using FFT algorithm
• Need to know amplitude and phase of field

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Limited Numerical Aperture
Backpropagation
a
?
S0
z
S
PSF of microscope
Abbes theory of the microscope
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Abbe Resolution Limit
-k
-k sin ?
a
?
S0
z
k sin ?
S
k
Maximum Nyquist resolution 2p/BW?/2sin?
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Phase Problem
Gerchberg Saxon, Gerchberg Papoulis
Multiple measurement plane versions
Holographic approaches
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The Phase Problem
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Digital Holographic Microscope
1024X1024 10 bits/pixel Pixel size10 ?
Mach-Zender configuration
Two holograms acquired which yield complex field
over CCD Backpropagate to obtain image of sample
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Retrieving the Complex Field
¼ ? plate
Four measurements required
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Limited Numerical Aperture
CCD
sample
Measurement plane
a
Sin ?a/zltlt1
?
z44 m.m. a6 m.m.
N.A..13
S0
Fuzzy Images
z
S
22
Pengyi and Capstone Team
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5 µm Slit
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Reconstruction of slit
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Ronchi ruling (10 lines/mm)
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Reconstruction of Ronchi ruling
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Conventional Versus Backpropagated
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Phase grating
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Reconstruction of phase grating
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Salt-water specimen
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Reconstruction of salt-water specimen
pixel size
d
x1.675
m
m
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Biological samples mouse embryo
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Reconstruction of mouse embryo
Intensity image by PSDH
Phase image by PSDH
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20
20
2
10
40
40
1.5
8
60
60
6
1
80
80
4
0.5
100
100
2
0
120
120
20
40
60
80
100
120
20
40
60
80
100
120
(a)
(b)
Conventional optical microscope
200
150
100
50
(c)
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Cheek cell
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Reconstruction of cheek cell
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Onion cell
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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
Thick Samples Born Model
Thick sample
S
S0
Born Approximation
Determines 3D Fourier transform over an Ewald
hemi-sphere
39
Generalized Projection Slice Theorem
K?
Kz
-kz
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
Multiple Experiments
K?
Ewald hemi-spheres
k
Kz
k
K?
v2 k
Kz
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Born Inversion for Fixed Frequency
Problem How to generate inversion from Fourier
data on spherical surfaces
Inversion Algorithms Fourier interpolation
(classical X-ray crystallography) Filtered
backpropagation (diffraction tomography)
A.J.D. Opts Letts, 7, p.111 (1982)
Filtering of data followed by backpropagation
Filtered Backpropagation Algorithm
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Inverse Scattering
Filtering followed by back propagation
3D semi-transparent object
Computer
Object Reconstruction
Illuminating plane waves
Essentially combine multiple 3D coherent images
generated for each scattering experiment
43
Inadequacy of Born Model
¼ ? plate
Thick (3D) sample of gimbaled mount
Addressed by DWBA model

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
44
Complex Phase Representation
(Non-linear) Ricatti Equation
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Short Wavelength Limit
Classical Tomographic Model
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Free Space Propagation of Rytov Phase
propagation
Within Rytov approximation phase of field
satisfies linear PDE
Rytov transformation
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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
objectsBack propagate field then use
Rytov--Hybrid Model
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Rytov versus Hybrid Model
N. Sponheim, I. Johansen, A.J. Devaney,
Acoustical Imaging Vol. 18 ed. H. Lee and G.
Wade, 1989
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Potential Scattering
Lippmann Schwinger Equation
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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
Multi static Data Matrix
Multi-static Data MatrixGeneralized Scattering
Amplitude
52
Distorted Wave Born Approximation
Linear Mapping
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
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SVD Based Inversion
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Filtered Backpropagation Algorithm
Propagation
Backpropagation
Basis image fields
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