Phase problem and the Radon transform - European ...€¦ · Phase problem and the Radon transform ... X-ray source. Bronnikov Algorithms ... is the counterpart of the Fourier slice

Bronnikov Algorithms

Phase problem and the Radon transform

Andrei V. Bronnikov

Bronnikov Algorithms The Netherlands


• The Radon transform and applications• Inverse problem of phase-contrast CT• Fundamental theorem• Image reconstruction algorithms• Phantom studies• Experimental data


The Radon transform (J. Radon, 1917)

Object

Projectionθ

dlffsLine

s ∫=),(

,ˆ

θθ

s


The inverse 3D (Radon) transform (H.A. Lorentz, 1905)

θ

ωs

ωθωθπ

ddsfs

f ),,(ˆ4

12

2

2 ∂∂−

= ∫∫


Applications

• Medical imaging• Laboratory animals

• Materials research• Non-destructive testing• Reverse engineering• Geophysics


Principle of x-ray tomography

Detector plane

Object

θ

f

g

X-ray source


Fourier slice theorem

),(ˆ),(ˆ vugvuf θθθ =

Fourier transform in the central plane Fourier transform in the central plane is the Fourier transform of the is the Fourier transform of the projectionprojection

Detector plane

Object f

θg


1D Filtering

θθ

π

dgqf ∫ ∗=0

Backprojection

Image reconstruction

•• Filling up the Fourier spaceFilling up the Fourier space

•• Filtered backprojection:Filtered backprojection:


Density reconstruction; non-planar orbits(Bronnikov, 1995-2002)


Wavelet-based contrast enhancement

Original image Processed imageOriginal image Processed image

Bronnikov and Duifhuis, 1998


Contrast-enhanced imaging

X-ray images of the rat cerebellum

Absorption contrast Phase contrast

A. Momose, T. Takeda, Y. Itai and K. Hirao, Nature Medicine, 2, 473-475 (1996).


Phase-contrast imaging

•• Coherent xCoherent x--raysrays

•• Microfocus tubeMicrofocus tube


Phase-contrast CT vs. absorption-contrast CT ( ESRF )


Insect knee E=18 KeVInsect knee E=18 KeVSnigirev et al, ESRFSnigirev et al, ESRF


Problems

•• the lack of the the lack of the linear linear theory comparable to theory comparable to that of conventional absorptionthat of conventional absorption--based CTbased CT

•• no quantitative solution (only edges of the no quantitative solution (only edges of the objects are reconstructed)objects are reconstructed)

•• huge amount of highhuge amount of high--resolution dataresolution data


Goals

•• to develop an accurate and practical to develop an accurate and practical mathematical tool for quantitative phasemathematical tool for quantitative phase--contrast image reconstructioncontrast image reconstruction

•• to suggest the fastest image reconstruction to suggest the fastest image reconstruction algorithm for phasealgorithm for phase--contrast CTcontrast CT


Inverse problem of phase-contrast tomography

πθθ <≤0),,( from),,(find 321 yxIxxxf z

- find the phase from intensity- invert Radon transform to reconstruct the object

• Phase retrieval (holographic) approach(Cloetens et al 1999):

• Reconstruction formula (CT approach):- a straightforward linear relation between the intensity and the object function


Phase retrieval

• Nonlinear iterative methods (Gershberg, Saxton 1972; Fienup, 1982)

• Linear TIE method (Gureyev et al, 2004)

• Linear Wigner transform (Maier, Preobrazshenski 1990; Raymer et al 1994)


Wigner transform

∫∞

∞−

′− ′′

+′

−= xdexxVxxVxW xizzz

ξπξ 2* )2

()2

(),(

)()0()(),(),( 0*

00 xVxVxWzxIWignerInversetransformRadon

ϕξ ⇒⇒⇒


Radon-Wigner transform

),()(),(0 zxIdxdxzxxW =′′−−′∫ ∫∞

∞−

∞

∞−

ξξδξ

ξ

x'

zarctan =θ

),(0 ξxW ′

),( zxI


Implementation (Bronnikov et al 1991)

function Wigner discrete theof samples ofnumber -n

2 planesdetection ofNumber nπ≥

zDetector positions

Object

)()0()(),(),( 0*

00 xVxVxWzxIWignerInversetransformRadon

ϕξ ⇒⇒⇒


Inverse problem of phase-contrast tomography

πθθ <≤0),,( from),,(find 321 yxIxxxf z

- find the phase from intensity- invert Radon transform to reconstruct the object- multiple detector positions along the beam

•• Phase retrieval (holographic) approach:Phase retrieval (holographic) approach:

- a straightforward linear relation between the intensity and the object function- single detector position

•• Reconstruction formula (CT approach):Reconstruction formula (CT approach):


Mathematical model: the Fresnel propagator


Theoretical background),(),(2

1

),( yxiyxeyxT θθ ϕµθ

+−= Transmission functionTransmission function

iUyxTyxU ),(),( θθ = WavefieldWavefield

πθθθ <≤∗∗= 0,),( 2UhyxI zz FresnelFresnel propagationpropagation

0, →∂∂

∂∂

yxθθ µµ ConditionsConditions2Dd <λ

∇−= ),(

21),( 20 yxdIyxId

θθθ ϕπλ Intensity equationIntensity equation


Radon transforms2D transform (lines) 3D transform (planes)2D transform (lines) 3D transform (planes)

dlffsLine

s ∫=),(

,ˆ

ωω σ

ωθωθ dff

sPlanes ∫=

),,(,,

ˆ

θ

ωssω


Fundamental theorem (Bronnikov 2002)

Second derivative of 3D Radon transform Second derivative of 3D Radon transform of the object is proportional to 2D Radon of the object is proportional to 2D Radon transform of the phasetransform of the phase--contrast projectioncontrast projection

Detector plane

Object

),(ˆ1),,(ˆ2

2

ωωθ θ sgd

sfs

−=

∂∂

2Dd <λ

f

g

d


Inversion formula (Bronnikov 1999)

3D backprojection of the 2D Radon 3D backprojection of the 2D Radon transform of the phasetransform of the phase--contrast projectioncontrast projection

Object

ωθωπ

ωθωθπ θ ddsg

dddsf

sf ),(ˆ

41),,(ˆ

41

22

2

2 ∫∫∫∫ =∂∂−

=

f

g

d

θ

ω


FBP algorithm (Bronnikov 1999, 2002)

2D FilteringBackprojection

θπ θ

π

dgqd

f ∫ ∗∗=0

241

θ


Analytical filter function (Bronnikov 1999, 2002)

θπ θ

π

dgqd

f ∫ ∗∗=0

241

22 yxy

q+

=

MTF of the 2D filter:MTF of the 2D filter:22 ηξ

ξ+

=Q


Implementation

•• FFTFFT--based linear filteringbased linear filtering•• 2D parallel2D parallel--beam backpojectionbeam backpojection•• FBP structure (parallelization)FBP structure (parallelization)•• Hardware accelerationHardware acceleration

The fastest algorithm possible


Phase objectPhase object

θπ θ

π

dgqd

f ∫ ∗∗=0

2411/),( −= i

d IIyxg θθ

Object

NearNear--field intensityfield intensity ReconstructionReconstructiondIθ

DetectorDetector

zd

θ


Phantom studies: polystyrene spherePhantom studies: polystyrene sphere

3D computer phantom3D computer phantom

d d = 2.5 cm= 2.5 cm

λ = 0.1 nm, ∆ = 1 µm

3D reconstruction3D reconstruction

PhasePhase--contrast contrast projectionsprojections


Quantification and the limits of the linear Quantification and the limits of the linear approximationapproximation

d = d = 2.5 cm 1.51.5 < d < < d < 20 cm2.5 cm 20 cm


Stability to noiseStability to noise

0%0% 1%1% 2%2% 5%5%


Mixed phase and amplitude objectMixed phase and amplitude object

Absorption function Absorption function µµ/2/2 Phase function Phase function ϕϕ

),(),(21

),( yxiyxeyxT ϕµ +−=


Mixed phase and amplitude objectMixed phase and amplitude object

θπ θ

π

dgqd

f ∫ ∗∗=0

2411/),( 0 −= θθθ IIyxg d

θ

0θI

Object

dIθ

ReconstructionReconstructionz

d

Contact intensity NearContact intensity Near--field intensityfield intensity


Comparison with heuristic methodsComparison with heuristic methods(Anastasio(Anastasio et al, PMB 2003)et al, PMB 2003)

BPBackprojection “Absorption” FBP

“Absorption” LFBP Bronnikov method


Polypropylene foamPolypropylene foam((Schena et al; ElettraSchena et al; Elettra, 2004), 2004)

Conventional FPB Bronnikov method


Polyethylene tube with polymer fibers Polyethylene tube with polymer fibers (Groso and Stampanoni(Groso and Stampanoni; SLS, 2005); SLS, 2005)

721 projections; 15 cm distance sample detector; Energy=13.5 keVPixel size 1.75 microns


SummarySummary

•• A fundamental theorem has been proved. This A fundamental theorem has been proved. This is the counterpart of the Fourier slice theorem is the counterpart of the Fourier slice theorem used in absorptionused in absorption--based CTbased CT

•• A fast image reconstruction algorithm has been A fast image reconstruction algorithm has been implemented in the form of filtered implemented in the form of filtered backprojectionbackprojection (FBP)(FBP)

Documents

Phase problem and the Radon transform - European ...€¦ · Phase problem and the Radon transform ... X-ray source. Bronnikov Algorithms ... is the counterpart of the Fourier slice