Compressive phase retrieval

Compressive phase retrieval

George Barbastathis

University of Michigan - Shanghai Jiao Tong University Joint Institute 密西根交大学院 (academic year 2013-14)

Massachusetts Institute of Technology Department of Mechanical Engineering

Singapore-MIT Alliance for Research and Technology (SMART) Centre

Acknowledgments

• Zhengyun Zhang, Adam Pan, Kelli Xu, Yunhui Zhou, 
 Yi Liu, Justin Lee, Shakil Rehman

• Lei Tian, Laura Waller UC Berkeley

• Jon Petruccelli SUNY Albany

• David Brady Duke University

• Colin J. R. Sheppard Italian Institute of Technology

• Rajiv Gupta Massachusetts General Hospital

• Haris Kudrolli, Vivek Nagarkar RMD Inc

• Singapore’s National Research Foundation

• US Department of Homeland Security

• Chevron Technology Company

This talk is about

• ^Phase

• Phase space

• Phase space tomography

• Compressive imaging

• Phase tomography

This talk is about

• ^Phase

• Phase space

• Phase space tomography

• Compressive imaging

• Phase tomography

The significance of phase

(F. Zernike, Science 121, 1955)

intensity image phase-contrast image

Visible X-ray

includes losses due to scattering) can thereby be pro- duced,⁶and because they rely on different contrast mech- anisms, they can provide complementary information.

The refraction image, being sensitive to the gradient of the refractive index, shows dramatic edge enhancement and provides a map of the boundaries between regions with differing refractive indices (see figure 3).

DEI requires an intense monochromatic x-ray beam for reasonable exposure times. Synchrotron sources have a definite advantage in this respect, but are not practical for clinical applications. Pisano and company have recent- ly demonstrated the improved cancer detail visualization that is possible with synchrotron-based DEI,⁷and the team has also shown that DEI can be used for CT,⁸which could increase the clinical potential for this technique. A Euro- pean collaboration led by Ralf Menk (Elettra) that is bring- ing together researchers at synchrotrons (Daresbury in the UK, DESY in Germany, Elettra, and ESRF), universities (Bremen and Siegen), and industry (Siemens AG) has also begun exploring the medical potential of DEI. Ingal and Beliaevskaya, meanwhile, have continued to explore the use of commercial x-ray tubes.⁹

Phase-contrast radiography

If just a detector, and no analyzer crystal, is in the beam path (figure 1c), the x rays emerging from the sample at their various angles will propagate through free space until they reach the detector. With the detector immedi- ately behind the sample, one will get a conventional absorption image. If the source is very highly coherent and the detector is placed very far behind the sample, one will observe a fringe pattern as different components of the beam, having been diffracted by the sample, interfere with each other on further propagation through space.

This regime corresponds to Fraunhofer or far-field dif- fraction. The interference pattern contains useful phase information, but extracting that information is an ongoing

computational and physics challenge.

With the detector placed at an intermediate distance, one gets Fresnel or near-field diffraction. Here, a combi- nation of imaging and diffraction effects is found, typical- ly involving interference fringes at the edges of features.

These fringes improve edge visibility (see figure 4). The optimum positioning of the detector for best enhancement effects varies from sample to sample, depending on the x- ray wavelength and the size of the features of interest.

This “in-line” phase-sensitive technique, exploiting Fresnel diffraction and dubbed phase-contrast imaging (although it is distinct from optical phase-contrast imag- ing), was first explored by Anatoly Snigirev and coworkers at ESRF¹⁰and by Wilkins and colleagues at CSIRO.¹¹It is very similar to the original techniques for holography developed by Dennis Gabor in 1948.

In the absence of absorption, the contrast depends on the Laplacian of the phase shift f in the sample. “Inter- pretation of the measured ¹²f or f in terms of object prop- erties becomes more difficult for thick objects due to the effects of multiple scattering within the sample,” says Wilkins. X rays that have been scattered through large angles will miss the detector altogether, which improves the signal-to-noise ratio of the image.

JULY2000 PHYSICSTODAY

25

FIGURE2. COMPUTED TOMOGRAMof a human kidney sample (5 mm in diameter) obtained with phase-contrast x-ray interfer- ometry. The image maps the difference in the refractive index between the sample and water. The darker region on the right is cancerous. The density difference between the normal and can- cerous tissues is calculated from the image to be 10 mg/cm³. (From A. Momose et al., SPIE Proc. 3659, 365 [1999].)

FIGURE3. HUMAN BREAST CANCER SPECIMEN, imaged in vitro with conventional absorption (top) and diffraction- enhanced techniques (bottom). The DEI technique is sensitive to variations in the refractive index, which are greatest at the boundaries between different regions. Because of the resulting edge enhancement, spiculations (thin lines of cancerous growth) that are barely discernable in the top image (arrow) are more clearly visible in the DEI refraction image. (From ref. 7.)

Downloaded 06 Apr 2013 to 18.111.12.56. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://www.physicstoday.org/about_us/terms

includes losses due to scattering) can thereby be pro- duced,⁶and because they rely on different contrast mech- anisms, they can provide complementary information.

The refraction image, being sensitive to the gradient of the refractive index, shows dramatic edge enhancement and provides a map of the boundaries between regions with differing refractive indices (see figure 3).

DEI requires an intense monochromatic x-ray beam for reasonable exposure times. Synchrotron sources have a definite advantage in this respect, but are not practical for clinical applications. Pisano and company have recent- ly demonstrated the improved cancer detail visualization that is possible with synchrotron-based DEI,⁷and the team has also shown that DEI can be used for CT,⁸which could increase the clinical potential for this technique. A Euro- pean collaboration led by Ralf Menk (Elettra) that is bring- ing together researchers at synchrotrons (Daresbury in the UK, DESY in Germany, Elettra, and ESRF), universities (Bremen and Siegen), and industry (Siemens AG) has also begun exploring the medical potential of DEI. Ingal and Beliaevskaya, meanwhile, have continued to explore the use of commercial x-ray tubes.⁹

Phase-contrast radiography

If just a detector, and no analyzer crystal, is in the beam path (figure 1c), the x rays emerging from the sample at their various angles will propagate through free space until they reach the detector. With the detector immedi- ately behind the sample, one will get a conventional absorption image. If the source is very highly coherent and the detector is placed very far behind the sample, one will observe a fringe pattern as different components of the beam, having been diffracted by the sample, interfere with each other on further propagation through space.

This regime corresponds to Fraunhofer or far-field dif- fraction. The interference pattern contains useful phase information, but extracting that information is an ongoing

computational and physics challenge.

With the detector placed at an intermediate distance, one gets Fresnel or near-field diffraction. Here, a combi- nation of imaging and diffraction effects is found, typical- ly involving interference fringes at the edges of features.

These fringes improve edge visibility (see figure 4). The optimum positioning of the detector for best enhancement effects varies from sample to sample, depending on the x- ray wavelength and the size of the features of interest.

This “in-line” phase-sensitive technique, exploiting Fresnel diffraction and dubbed phase-contrast imaging (although it is distinct from optical phase-contrast imag- ing), was first explored by Anatoly Snigirev and coworkers at ESRF¹⁰and by Wilkins and colleagues at CSIRO.¹¹It is very similar to the original techniques for holography developed by Dennis Gabor in 1948.

In the absence of absorption, the contrast depends on the Laplacian of the phase shift f in the sample. “Inter- pretation of the measured ¹²f or f in terms of object prop- erties becomes more difficult for thick objects due to the effects of multiple scattering within the sample,” says Wilkins. X rays that have been scattered through large angles will miss the detector altogether, which improves the signal-to-noise ratio of the image.

JULY2000 PHYSICSTODAY

25

FIGURE2. COMPUTED TOMOGRAMof a human kidney sample (5 mm in diameter) obtained with phase-contrast x-ray interfer- ometry. The image maps the difference in the refractive index between the sample and water. The darker region on the right is cancerous. The density difference between the normal and can- cerous tissues is calculated from the image to be 10 mg/cm³. (From A. Momose et al., SPIE Proc. 3659, 365 [1999].)

FIGURE3. HUMAN BREAST CANCER SPECIMEN, imaged in vitro with conventional absorption (top) and diffraction- enhanced techniques (bottom). The DEI technique is sensitive to variations in the refractive index, which are greatest at the boundaries between different regions. Because of the resulting edge enhancement, spiculations (thin lines of cancerous growth) that are barely discernable in the top image (arrow) are more clearly visible in the DEI refraction image. (From ref. 7.)

Downloaded 06 Apr 2013 to 18.111.12.56. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://www.physicstoday.org/about_us/terms

(E. D. Pisano et al., Radiology 214, 2000)

Density Refractive index

temperature pressure humidity

(x _o ) = k Z

n(r)dl ⇢ / n ² 1 n ² + 2

⇢ /

attenuation image phase-contrast image

(human breast cancer specimen)

Phase Imaging

• Non-quantitative 


(phase contrast) • Quantitative 


(brightness ∝ ^OPL)

OPL 0 L 0

no t q ua nt ifia ble

OPL = optical path length

Phase Retrieval

• Interferometric • Axial stack

camera

camera camera camera camera

External reference • Lippmann / Lateral stack

• Refractive

back focal plane sensor

Wavefront Sensing

lens array computed

wavefront slope (phase profile)

incoming coherent

wave

(Shack-Hartman)

Captured Image

• spot location → slope estimate

• spot shape → fine details

• wavefront sensing approach


throws away fringe info

• light field imaging approach


assumes fringes are extra rays

• can we do better?

+ + +

+ + +

+

+ +

Z. Zhang and G. Barbastathis, Focus on Microscopy 2014, paper MO-AF1-PAR-D

Sydney, Australia

coherence modes measured

intensity predicted intensity

Factored Form Descent (FFD)

lens array input field

x

back focal plane

output field

y = Kx coherence retrieval

output intensity

➔ input field

argmin _X X M

m=1

w _m ² y _m k _m XX ^T k ^T _m ²

X ⌘ (x ¹ x ₂ . . . x _M )

R = 1 = ) phase retrieval (purely coherent) M = 1 = )

Zhang et al Opt. Express 21:5759 (2013)

Single image not sufficient...

• each lens separately focuses

• low spatial frequency →
 no overlap (crosstalk)

• phase relationship between
 A and B can be obtained

• phase relationship between
 A and C cannot be obtained

lens array input field

x

back focal plane A

C

B

Fix with multiple images

lens array positions

(1D)

lens array positions

(2D)

Experimental Geometry (not to scale)

cover slip

cover slip specimen

Olympus 40X/0.75 NA

objective

LED illumination, 540 nm bandpass filter,

aperture and field 
 fully stopped down

micro- scope body

camera

port wavefront

sensor

XYZ translation stage Thorlabs WFS150-7AR 150 micron pitch 4.65 micron pixels

1280×1024 25 micron

pinhole

Experiment: 50μm bead

➡ Polysterene in 
 ethylene glycol

➡ Five images:

➡ four shifts

➡ one background for subtraction

➡ repeated 16 times for noise statistics

➡ Reconstruction using

rank-constrained FFD

Experiment: 50μm bead

amplitude phase

reconstruction (scalebar = 10 microns in specimen)

single lens element

Z. Zhang and G. Barbastathis, Focus on Microscopy 2014, paper MO-AF1-PAR-D

Sydney, Australia

Experiment: cheek cells

amplitude phase

reconstruction (scalebar = 10 microns in specimen)

single lens element

cell 1

cell 2

cell 1

cell 2

Z. Zhang and G. Barbastathis, Focus on Microscopy 2014, paper MO-AF1-PAR-D

Sydney, Australia

This talk is about

• ^Phase

• Phase space

• Phase space tomography

• Compressive imaging

• Phase tomography

back focal plane sensor

Wavefront Sensing

lens array

incoming coherent

wave

(Shack-Hartman)

x ₁

x ₂

x ₃

u u u _{3 2 1} u ₄ u ₅

u u u _{3 2 1} u ₄ u ₅

u u u _{3 2 1} u ₄ u ₅

sampling position momentum

Phase-space

partially coherent

u u u _{3 2 1} u ₄ u ₅ x ₁ x ₂ x ₃

position

mo mentum

L. Tian et al, Opt. Express

21:10511, 2013

Partially coherent light

U (x) Random field

J(x, x ⁰ ) ⌘ hU(x)U ^⇤ (x ⁰ ) i Correlation function (mutual intensity)

Young’s two-slit experiment x

x ⁰

B. J. Thompson and E. Wolf, J. Opt. Soc. Am., 47:895, 1957.

|J| / contrast

The mutual intensity

J(x, x ⁰ ) ⌘ hU(x)U ^⇤ (x ⁰ ) i

x

y

(x, y)

(x ⁰ , y ⁰ )

D. L. Marks, R. A. Stack, and D. Brady, Appl. Opt. 38:1332, 1999

The mutual intensity

J(x, x ⁰ ) ⌘ hU(x)U ^⇤ (x ⁰ ) i

• completely characterizes the (quasi-monochromatic) partially coherent field,

• in particular, the Optical Path Length (OPL);

• J. C. Petruccelli, L. Tian, and G. Barbastathis, Opt. Express 21:14430, 2013

• is analogous to the density matrix in quantum mechanics;

• is semi-positive definite (eigenvalues≥0);

• is a 4-dimensional quantity;

• but does it contain 4D information?

• J. Rosen and A. Yariv, Opt. Lett. 21:1011, 1996

• J. Rosen and A. Yariv, Opt. Lett. 21:1803, 1996

• D. L. Marks, R. A. Stack, and D. Brady, Appl. Opt. 38:1332, 1999

Eugene Paul Wigner

1902 Budapest, Hungary -!

1995 Princeton, New Jersey!

1927 symmetries in quantum mechanics!

1932 “On the quantum correction for !

thermodynamic equilibrium”!

1960 “The unreasonable effectiveness of!

mathematics in the natural sciences”!

1963 Nobel prize in Physics

http://en.wikipedia.org/wiki/E._P._Wigner

W (x, u) =

Z ✓

x + x ⁰ 2

◆

⇤

✓

x x ⁰ 2

◆

exp ( i2⇡ux ⁰ ) dx ⁰ W (x, u) =

Z ✓

x + x ⁰ 2

◆

⇤

✓

x x ⁰ 2

◆

exp ( i2⇡ux ⁰ ) dx ⁰

The Phase Space

• Wigner distribution function

W (x, u) =

Z ✓

x + x ⁰ 2

◆

⇤

✓

x x ⁰ 2

◆

exp ( i2⇡ux ⁰ ) dx ⁰

<.> < ... >

W (x, u) = Z

J

✓

x + x ⁰

2 , x x ⁰ 2

◆

exp ( i2⇡ux ⁰ ) dx ⁰

• Ambiguity function

A(u ⁰ , x ⁰ ) = Z

J

✓

x + x ⁰

2 , x x ⁰ 2

◆

exp ( i2⇡u ⁰ x) dx

F x $ u ⁰ u $ x ⁰

• By the way, W (x, u) is real.

Phase space (Wigner space)

time Temporal frequency

Chirp function Spherical wave

space variable x

Lo ca l s pa tia l fr equency u

Phase space description

space variable x

0 2 4 6 8 10

−1

−0.5 0 0.5 1

x (Wigner distribution function, WdF)

point source

x

z

x-z space

u

x

Wigner space (x-u space)

(x) = (x x ₀ )

axial!

position lateral!

position

lateral!

position momentum!

(spatial frequency)

spherical wave

x

z

x-z space

u

x

Wigner space (x-u space)

(x) = exp

⇢

i ⇡ (x x ₀ ) ² z

WDF shears/rotates upon propagation

boxcar (“rect”) function: 1D slit

A

1

u

x

x

integrate WDF along frequency axis

integ ra te WDF a lo ng s pa ce axi s

original function

FT of

original function

diffraction from a rectangular slit aperture

u

x _x

u

Example: waveguide (3rd mode) + lens

waveguide lens free space

position position position position

mo me n tu m

This talk is about

• ^Phase

• Phase space

• Phase space tomography

• Compressive imaging

• Phase tomography

Wavefunction evolution and the WDF

time evolution ^t

propagation distance z ( ) propagation ^Fresnel

Tomographic measurement

time evolution ^t

propagation distance z ( ) propagation ^Fresnel

measurement (quantum demolition)

intensity measurement

⇠

x

from evolution/propagation

Phase-space tomography

partially coherent

field (unknown)

camera (intensity measurement)

z 0 z 1 z 2 z 3 z 4

Phase-space tomography

x 1

x 2

x Wigner u

Mutual Intensity function

J (x 1 , x 2 ) W J (x, u)

Wigner Distribution function

z=z 0

z=z 1

z=z 2

z=z 3

z=z 4

x u

W J (x, u)

Wigner Distribution function

Phase-space tomography

Fourier

Δu Δx

Ambiguity function

z=z 0

z=z 1

z=z 2

z=z 3

z=z 4

A J (Δx, Δu)

Quantum phase space tomography

C. Kurtsiefer, and et al, Nature, 1997 J. Itatanl, and et al, Nature, 2000

Squeezed state recovery Matter wave interference

measurement

Optical Homodyne Tomography

Tomographic reconstruction

D. Smithey, and et al, Phys. Rev. Lett. 1993

measurement Tomographic

reconstruction

Optical phase space tomography

Axial intensity measurement Reconstructed WDF Reconstructed MI Spatial coherence measurements of a 1D soft x-ray beam

C.Q.Tran, and et al, JOSA A 22, 1691-1700(2005)

• Non-interferometric technique

The problem of limited data

u x

too close

too far z < 0

inaccessible ¹

2

• Assume intensity symmetric about z=0

measurement range

Make up for limited data?

☛ Compressive reconstruction

This talk is about

• ^Phase

• Phase space

• Phase space tomography

• Compressive imaging

• Phase tomography

0 10 20 30 40 50 60

−1

−0.5 0 0.5 1

Numerical example: 3 spikes

Original signal with 3 spikes (total length=64) DFT measurements (# of samples=12)

Compressive (L1) reconstruction Conventional (L2) reconstruction

DFT samples

0 10 20 30 40 50 60

−1.5

−1

−0.5 0 0.5 1 1.5

0 10 20 30 40 50 60

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15

0 10 20 30 40 50 60

−1

−0.5 0 0.5 1

f = argmin ˆ _f kfk ₂

s.t. F ¹ y _red = F ¹ f

f = argmin ˆ _f kfk ₁

s.t. F ¹ y _red = F ¹ f

Why L1?

Least squares solution (minimizes L2 on the line)

soluti on (und erdetermi

ned )

NOT Sparse

Why L1?

Compressive solution (minimizes L1 on the line)

soluti on (und erdetermi

ned )

Sparse

Generally, of the form

(0, . . . , 0, ⇠, 0, . . . , 0)

Reconstruction success is subject to sparsity

# N yqui st sa mpl es

(# non-zero samples) / (# Nyquist samples) (# non-zero samples) / (# Nyquist samples)

E. Candés, J. Romberg, and T. Tao, IEEE Trans. Info. Th. 52:489, 2006

Exprerimental compressive phase-space tomography

• Illumination central wavelength: 620nm; 


bandwith: 20nm

• Width of illumination slit: 300μm

• Coherence length: 93μm

• Width of object slit: 400μm

• 32 measurements (axial positions)

LED

1D object

Scanning z detector

f

Slit Lens

f = 75mm

Lei Tian et al, Opt. Expr. 20(8):8296, 2012

Limited data in our experiment

u x

too close

too far

• Total # of slices: 32

• Missing angle : 38 ^o

• Missing angle : 22 ₂ ^o

1

1

2

measurement

range

Ground Truth

LED

Illumination Slit

Imaging system

van Cittert-Zernike theorem

Global Degree of Coherence

μ=0.49

Filtered back-projection fails

Non-physical


correlation function

Underestimates the degree of

coherence μ=0.12

Compressive reconstruction

μ=0.46

Error around the edge due to resolution limit of

the imaging system

Error in compressive reconstruction

(compared to Van Cittert-Zernike)

Estimate of the coherent modes

Coherent modes eigenvalues

Validation: vCZ theorem

eigenvalues Difference between

CS and vCZ eigenvalue estimates

Compressive estimate

vCZ


estimate

4D phase space tomography:

astigmatic imaging

LED

diffuser

collimating lens

object

cylindrical
 lens (along x)

cylindrical
 lens (along y)

linear stage linear stage camera

“coherence”

aperture

100μm 300μm

640μm

L. Tian, S. Rehman, and G. Barbastathis, in Frontiers in Optics 2012, paper FM4C.4.

M. Raymer, M. Beck and D. McAlister, Phys.Rev.Lett. 1994 D. Marks, R. Stack and D. Brady, Opt. Lett. 2000

central wavelength: 620nm

bandwidth: 20nm

vʹ′

yʹ′

Inaccessible Inaccessible

Inaccessible

Inaccessible

xʹ′

• Dimension of unknown mutual intensity: 64 uʹ′ ⁴

• Total # of samples: 64 ² (# of samples in each image)×20(# of planes in a focal stack)×12(# of focal stack)

“Missing slices” in Ambiguity space

L. Tian, S. Rehman, and G. Barbastathis, in Frontiers in Optics 2012, paper FM4C.4.

x 1

x 2

Mutual intensity

Theoretical Prediction

300μm 100

μ m VCZ theorem

640μm

coherence

aperture object

y 2

y 1

0 5 10 15 20 25 30

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Modes

Eigenvalue

eigenvalues

horizontal direction has more modal structure due to lower coherence

coherent

modes

L. Tian, S. Rehman, and G. Barbastathis, Frontiers in Optics 2012, paper FM4C.4.

0 5 10 15 20 25 30

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Modes

Eigenvalue

Mutual intensity

x 2

x 1

y 1

y 2

Compressive Reconstruction

This talk is about

• ^Phase

• Phase space

• Phase space tomography

• Compressive imaging

• Phase tomography

Why X-ray phase imaging?

• X-ray absorption images do NOT provide good contrast for soft tissues

• Call for alternative contrast mechanism


☛ X-ray PHASE imaging

http://www.presstv.ir/detail/206921.html

“X-ray not good for lung cancer screening” refractive index of water (n=1-δ-iβ)

15 20 25

1E−10 1E−8 1E−6 1E−4

X−ray photon energy (keV)

TIE for x-rays?

http://science.howstuffworks.com/synchrotron.htm

Coherent?

➯ synchrotron

© 1996 Nature Publishing Group http://www.nature.com/naturemedicine

x-ray interferometer (Nat. Med. 2, 473-475) extremely sensitive to mech. stability & alignment

Talbot interferometer (Nat. Phys. 2, 256-261) requires 3 gratings & complicated measurement

LETTERS

d between G

and G

corresponding to the first Talbot distance

of d = p

/ 8l, a coherence length of ξ

= ^l l/γ

p

≥ p

is required, where l is the distance between G

and G

. With a typical value of a few micrometres for p

, the spatial coherence length required is of the order of ξ

≥ ¹⁰

m, similar to the requirements of existing methods (see above). It is important to note, however, that a setup with only two gratings (G

and G

) already requires, in principle, no spatial coherence in the direction parallel to the grating lines, in contrast with propagation-based methods

.

As the source mask G

can contain a large number of individual apertures, each creating a sufficiently coherent virtual line source, standard X-ray generators with source sizes of more than a square millimetre can be used efficiently. To ensure that each line source produced by G

contributes constructively to the image-formation process, the geometry of the setup should satisfy the condition (Fig. 1b)

p

= p

× l d .

It is important to note that the total source size w only determines the final imaging resolution, which is given by wd/l . The arrayed source thus decouples spatial resolution from spatial coherence, and allows the use of X-ray illumination with coherence lengths as small as ξ

= ^l l/w ∼ ¹⁰

^{m in both} directions, if the corresponding spatial resolution wd/l can be tolerated in the experiment. Finally, as a temporal coherence of ξ

≥ ¹⁰

^{m (} # E/E ≥ 10%) is sufficient

, we deduce that the method presented here requires the smallest minimum coherence volume ξ

× ξ

× ξ

for phase-sensitive imaging if compared with existing techniques.

The DPC image-formation process achieved by the two gratings G

and G

is similar to Schlieren imaging

and diffraction-enhanced imaging

. It essentially relies on the fact that a phase object placed in the X-ray beam path causes slight refraction of the beam transmitted through the object. The fundamental idea of DPC imaging depends on locally detecting these angular deviations (Fig. 1b). The angle, α is directly proportional to the local gradient of the object’s phase shift, and can be quantified by

α = ^l 2 π

∂ Φ (x, y)

∂ x ,

where x and y are the cartesian coordinates perpendicular to the optical axis, Φ ( x, y) represents the phase shift of the wavefront, and l is the wavelength of the radiation. For hard X-rays, with l < 0 . 1 nm, the angle is relatively small, typically of the order of a few microradians.

In our case, determination of the angle is achieved by the arrangement formed by G

and G

. Most simply, it can be thought of as a multi-collimator translating the angular deviations into changes of the locally transmitted intensity, which can be detected with a standard imaging detector. For weakly absorbing objects, the detected intensity is a direct measure of the object’s local phase gradient d Φ (x, y)/ d x . The total phase shift of the object can thus be retrieved by a simple one-dimensional integration along x . As described in more detail in refs 19,20, a higher precision of the measurement can be achieved by splitting a single exposure into a set of images taken for different positions of the grating G

. This approach also allows the separation of the DPC signal from other contributions, such as a non-negligible absorption of the object, or an already inhomogeneous wavefront phase profile before the object. Note that our method is fully compatible with conventional absorption radiography, because it simultaneously yields separate absorption and phase-contrast images, so that information is available from both.

w

p₀ p₁

p₂ α

x y z

G₀

G₀ G₁ G₂

G₁ G₂

Object

Imaging detector

Incoherent X-ray source

Au Si Si Au

Si

l d

p₀

p₂ p₁

a

b

200 µm 4 µm 2 µm

Figure 1

Talbot–Lau-type hard-X-ray imaging interferometer. a, Principle: the source grating (G

) creates an array of individually coherent, but mutually incoherent sources. A phase object in the beam path causes a slight refraction for each

coherent subset of X-rays, which is proportional to the local differential phase gradient of the object. This small angular deviation results in changes of the locally transmitted intensity through the combination of gratings G

and G

. A standard X-ray imaging detector is used to record the final images. b, Scanning electron micrographs of cross-sections through the gratings. The gratings are made from Si wafers using standard photolithography techniques, and subsequent electroplating to fill the grooves with gold (G

and G

).

Figure 2 shows processed absorption, DPC and reconstructed phase images of a reference sample containing spheres made of polytetrafluoroethylene (PTFE) and polymethylmethacrylate (PMMA). From the cross-section profiles (Fig. 2d), we deduce that 19 ± ⁰ . 6% (7 . 0 ± ⁰ . 6%) of the incoming radiation is absorbed in the centre of the PTFE (PMMA) sphere. By comparison with tabulated literature values

, a mean energy of the effective X-ray spectrum of E

≈ ²² . 4 ± ¹ . 2 keV (l ≈ ⁰ . 0553 nm) is deduced. To discuss the results for the object’s phase shift (Fig. 2c), we have to consider the real part of the refractive index, which, for X-rays, is typically expressed as n = ¹ − δ . For a homogenous sphere with radius r , the total phase shift through the centre of the sphere is Φ ₌ 4 πrδ/ l.

The experimentally observed maxima of the integrated phase shifts are Φ

= ⁵⁴ ± ² π and Φ

= ³² ± ² π (Fig. 2f), from which we obtain: δ

= ⁹ . 5 ± ⁰ . 8 × ¹⁰

^and δ

= ⁵ . 9 ± ⁰ . 6 × ¹⁰

^, which is in agreement with the values in the literature

. Note that because our method provides a quantitative measure of the absorption and integrated phase shift of the object in each pixel, the results can be used for further quantitative analysis, such as the reconstruction of a three-dimensional map of the real and

naturephysics VOL 2 APRIL 2006 www.nature.com/naturephysics 259

Untitled-3 2 3/22/06, 3:26:31 PM

Nature Publishing Group

©2006

Transport of Intensity (TIE)

I i (z)

no object

I(z)

object

• Images corrected for lateral magnification

• Attenuation neglected

• Long propagation distance
 (short wavelength) 


➦ two instead of four measurements
 k

z

✓ I(z)

I _i (z) 1

◆

⇡ r ² x

z

x-ray source

(with small spot size) x-ray detector

k @I

@z = r ^x · (Ir ^x )

I ⟹ ⇡ 1

Experimental geometry

• Experimental arrangement at RMD, Inc., Watertown, Mass.

• Three experiments:

• Polystyrene spheres on flat tape (Andor

camera)

• Polystyrene spheres taped onto drinking straw (Andor camera)

• Beetle (Andor camera)

Hamamatsu micro-focus source

∅5µm spot size operated at 20kVp

(λ=0.06nm) Rad-icon

CMOS camera 48µm pixel size 2000×2000 pixels

100″

object on rotation

stage

30″

Andor EM CCD with scintillator by RMD

16µm pixel size

512×512 pixels

X-ray phase tomography: beetle

y (mm)

x (mm)

2 4 6 8 10 12 14

2

4

6

8

10

12

14

30 ^o

y (mm)

x (mm)

2 4 6 8 10 12 14

2

4

6

8

10

12

14

60 ^o

y (mm)

x (mm)

2 4 6 8 10 12 14

2

4

6

8

10

12

14

0 ^o

intensity measurements

y (mm)

x (mm)

2 4 6 8 10 12 14

2

4

6

8

10

12

14

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01 0 0.01 0.02

90 ^o 0.03

1mm

30 ^o

60 ^o 0 ^o Phase

projections obtained by inverting the

intensity images using

TIE at each angle

90 ^o

1mm

Phase projections

Filtered backprojection:

reconstructed cross-sections

“phase CAT scan”

Total projections: 72 (every 5 degrees)

• Low frequencies

• TIE transfer function ➡ cloud-like artifacts


!

!

!

• ! High frequencies:

• missing Fourier slices ➡ streaking

• finite source size ➡ blurring

TIE tomography in the Fourier domain

u _y

u _x

(b)

(c) (a)

−4

−2 0 2 4 6 x 10

(a)

(b) (c)

illumi

nati on

Compressive reconstruction:

total variation

knk ^TV = X q

(r ^x n) ² + (r ^y n) ² + (r ^z n) ² Total variation (TV) function:

data fitting term sparsity constraint

• project the solution onto the gradient basis with few nonzero coefficients (which represents sharp boundaries)

• ^{look for} piecewise constant refractive index distribution minimize 1

2 g F ¹ H _TIE H _proj Fn ² + ⌧knk TV

−4

−2 0 2 4 6

x 10 ⁻⁶

−4

−2 0 2 4 6 x 10 ⁻⁶

−1 0 1 2 3 4 5 6 7 x 10⁻⁶

traditional reconstruction

TV reconstruction

refra cti ve ind ex

Compressive x-ray phase tomography

TV reconstruction

This talk was about

• ^Phase

• Axial stacks with partially coherent illumination 
 [Petruccelli Opt. Exp. 21:14430]

• TIE Phase from chromatic dispersion [Waller Opt. Exp. 18:2287]

• TIE Kalman filter [Waller Opt. Exp. 19:2805]

• Phase space

• Wigner distribution function from lenslet arrays 
 [Tian Opt. Exp. 21:10511]

• Phase from lens let arrays [Zhang FOM 2014, Sydney, Australia]

• Phase space tomography

• Compressive reconstruction of the mutual intensity 
 [Tian Opt. Exp. 20:8296]

• Factored Form Descent [Zhang Opt. Exp. 21:5756]

• Compressive phase tomography

• TIE Compressive x-ray tomography [Tian Opt. Lett. 38:3418]

• Nonlinear diffusion [Tian Opt. Lett. 37:4131]