Upload
hadieu
View
216
Download
2
Embed Size (px)
Citation preview
Bronnikov Algorithms
Phase problem and the Radon transform
Andrei V. Bronnikov
Bronnikov Algorithms The Netherlands
Bronnikov Algorithms
• The Radon transform and applications• Inverse problem of phase-contrast CT• Fundamental theorem• Image reconstruction algorithms• Phantom studies• Experimental data
Bronnikov Algorithms
The Radon transform (J. Radon, 1917)
Object
Projectionθ
dlffsLine
s ∫=),(
,ˆ
θθ
s
Bronnikov Algorithms
The inverse 3D (Radon) transform (H.A. Lorentz, 1905)
θ
ωs
ωθωθπ
ddsfs
f ),,(ˆ4
12
2
2 ∂∂−
= ∫∫
Bronnikov Algorithms
Applications
• Medical imaging• Laboratory animals
• Materials research• Non-destructive testing• Reverse engineering• Geophysics
Bronnikov Algorithms
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
Bronnikov Algorithms
1D Filtering
θθ
π
dgqf ∫ ∗=0
Backprojection
Image reconstruction
•• Filling up the Fourier spaceFilling up the Fourier space
•• Filtered backprojection:Filtered backprojection:
Bronnikov Algorithms
Wavelet-based contrast enhancement
Original image Processed imageOriginal image Processed image
Bronnikov and Duifhuis, 1998
Bronnikov Algorithms
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).
Bronnikov Algorithms
Phase-contrast imaging
•• Coherent xCoherent x--raysrays
•• Microfocus tubeMicrofocus tube
Bronnikov Algorithms
Insect knee E=18 KeVInsect knee E=18 KeVSnigirev et al, ESRFSnigirev et al, ESRF
Bronnikov Algorithms
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
Bronnikov Algorithms
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
Bronnikov Algorithms
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
Bronnikov Algorithms
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)
Bronnikov Algorithms
Wigner transform
∫∞
∞−
′− ′′
+′
−= xdexxVxxVxW xizzz
ξπξ 2* )2
()2
(),(
)()0()(),(),( 0*
00 xVxVxWzxIWignerInversetransformRadon
ϕξ ⇒⇒⇒
Bronnikov Algorithms
Radon-Wigner transform
),()(),(0 zxIdxdxzxxW =′′−−′∫ ∫∞
∞−
∞
∞−
ξξδξ
ξ
x'
zarctan =θ
),(0 ξxW ′
),( zxI
Bronnikov Algorithms
Implementation (Bronnikov et al 1991)
function Wigner discrete theof samples ofnumber -n
2 planesdetection ofNumber nπ≥
zDetector positions
Object
)()0()(),(),( 0*
00 xVxVxWzxIWignerInversetransformRadon
ϕξ ⇒⇒⇒
Bronnikov Algorithms
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):
Bronnikov Algorithms
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
Bronnikov Algorithms
Radon transforms2D transform (lines) 3D transform (planes)2D transform (lines) 3D transform (planes)
dlffsLine
s ∫=),(
,ˆ
ωω σ
ωθωθ dff
sPlanes ∫=
),,(,,
ˆ
θ
ωssω
Bronnikov Algorithms
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
Bronnikov Algorithms
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
Bronnikov Algorithms
FBP algorithm (Bronnikov 1999, 2002)
2D FilteringBackprojection
θπ θ
π
dgqd
f ∫ ∗∗=0
241
θ
Bronnikov Algorithms
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
Bronnikov Algorithms
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
Bronnikov Algorithms
Phase objectPhase object
θπ θ
π
dgqd
f ∫ ∗∗=0
2411/),( −= i
d IIyxg θθ
Object
NearNear--field intensityfield intensity ReconstructionReconstructiondIθ
DetectorDetector
zd
θ
Bronnikov Algorithms
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
Bronnikov Algorithms
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
Bronnikov Algorithms
Mixed phase and amplitude objectMixed phase and amplitude object
Absorption function Absorption function µµ/2/2 Phase function Phase function ϕϕ
),(),(21
),( yxiyxeyxT ϕµ +−=
Bronnikov Algorithms
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
Bronnikov Algorithms
Comparison with heuristic methodsComparison with heuristic methods(Anastasio(Anastasio et al, PMB 2003)et al, PMB 2003)
BPBackprojection “Absorption” FBP
“Absorption” LFBP Bronnikov method
Bronnikov Algorithms
Polypropylene foamPolypropylene foam((Schena et al; ElettraSchena et al; Elettra, 2004), 2004)
Conventional FPB Bronnikov method
Bronnikov Algorithms
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
Bronnikov Algorithms
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)