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 ? ˜ a/O

a

O

a?

d

d=?/2N.A.

October 21, 2005 A.J. Devaney IMA Lecture 4

Fourier Analysis in 2D

x

y

Kx

KyFT

IFT? K?

October 21, 2005 A.J. Devaney IMA Lecture 5

Plane Waves

z?

k ?

Propagating waves

Evanescent waves

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

z?

k Max K?=k sin ?

Each diffracted plane wave component carriessample information at specific spatial frequency

?

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 carriessample information at specific spatial frequency

October 21, 2005 A.J. Devaney IMA Lecture 8

Coherent Imaging

Analog Imaging

ComputationalImaging

Nature

Computer

Lens

Measurement plane

Thin sample Image

Illuminating plane wave

October 21, 2005 A.J. Devaney IMA Lecture 9

Coherent Computational Imaging

ComputationalImaging

Computer

Measurement planeIlluminating plane wave

S

S0 S

Undo PropagationPropagation

October 21, 2005 A.J. Devaney IMA Lecture 10

Plane Wave Expansion of the Solution to the Boundary Value Problem

z

S

=S0

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

S

S0

S

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

S

S0

S

Propagation

z

S

S0

S

Backpropagation

October 21, 2005 A.J. Devaney IMA Lecture 13

Back propagation of Bandlimited Fields

z

SS0

Propagation

Backpropagation

October 21, 2005 A.J. Devaney IMA Lecture 14

Coherent Imaging Via Backpropagation

S

Backpropagation

S0

• 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

S

Backpropagation

S0 z

?a

Abbe’s theory of the microscope

PSF of microscope

October 21, 2005 A.J. Devaney IMA Lecture 16

Abbe Resolution Limit

SS0 z

?a

k sin ?

-k sin ?

-k

k

Maximum Nyquist resolution = 2p/BW=λ/2sin?

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 λ 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 λ

Mach-Zender configuration

October 21, 2005 A.J. Devaney IMA Lecture 20

Retrieving the Complex Field

=

−

−++=+=

+++=+=

−

−

−

2

1*

*22

*11

11

1,

1,

DD

ee

ii

ieieIIie

eeIIe

sikz

sikz

sikz

sikz

ssikz

sikz

sikz

ssikz

ψψ

ψψψψ

ψψψψ

¼ λ plate

Four measurements required

October 21, 2005 A.J. Devaney IMA Lecture 21

Limited Numerical Aperture

CCD

sample

SS0 z

?a

Measurement plane

Sin ?=a/z<<1

Fuzzy Images

N.A.=.13z=44 m.m.a=6 m.m.

October 21, 2005 A.J. Devaney IMA Lecture 22

Pengyi and Capstone Team

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)

200µm

Scattered intensity Hologram 1

Hologram 2

October 21, 2005 A.J. Devaney IMA Lecture 26

Reconstruction of Ronchi ruling

PSDH reconstruction

50 100150 200 250

50

100

150

200

250100

200

300

400

500

Conventional optical microscope

100

200

300

400

500

CDH reconstruction

50 100150 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 δ x=6.7µm

October 21, 2005 A.J. Devaney IMA Lecture 27

Conventional Versus Backpropagated

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 δ x=6.7µm

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 δ x=6.7µm

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

150200

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 δ x=6.7µm

October 21, 2005 A.J. Devaney IMA Lecture 31

Reconstruction of salt-water specimen

pixel size δ x=1.675µm

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 100150200 250

50

100

150

200

250

200

400

600

800

Hologram 2

50 100150200250

50

100

150

200

250

pixel size δ x=6.7µm

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 δ x=1.675µm

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 µm

-2

0

2

Conventional optical microscope

200

300

400

500

October 21, 2005 A.J. Devaney IMA Lecture 37

Thick Sample System¼ λ 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

S

S0

Determines 3D Fourier transform over an Ewald hemi-sphere

Born Approximation

October 21, 2005 A.J. Devaney IMA Lecture 39

Generalized Projection Slice Theorem

-kz Kz

K?

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

K?

Kz

K?

Ewald hemi-spheresk

k

v2 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 λ 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 45

Short Wavelength Limit

Classical Tomographic 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 ImagingVol. 18 ed. H. Lee and G. Wade, 1989

Rytov versus Hybrid Model

October 21, 2005 A.J. Devaney IMA Lecture 49

Potential Scattering

Lippmann Schwinger Equation

October 21, 2005 A.J. Devaney IMA Lecture 50

Mathematical Structure of Inverse Scattering

LO

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=Nβ x Nα

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 53

SVD Based Inversion

October 21, 2005 A.J. Devaney IMA Lecture 54

Filtered Backpropagation Algorithm“Propagation”

“Backpropagation”

Basis image fields

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

October 21, 2005 A.J. Devaney IMA Lecture 56

October 21, 2005 A.J. Devaney IMA Lecture 57

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

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

October 21, 2005 A.J. Devaney IMA Lecture 60