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Image reconstruction algorithmsfor microtomography
Andrei V. Bronnikov
Bronnikov Algorithms, The Netherlands
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BronnikovAlgorithms
• Introduction• Fundamentals of the algorithms
• State-of-the-art in 3D image reconstruction
• Phase-contrast image reconstruction• Summary
Contents
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BronnikovAlgorithms
Microtomography systems
I m a g e I n t e n s i f i e r
C a m e r a
M a n i p u l a t i o n s ys t e m
R a i l s ys t e m
O b j e c t
X - R a y t u b ePC
M o t o r c o n t r o l
C a m e r a c o n t r o l
NDT systems
Desktop systems
Synchrotron setup
Dental CBCTSmall animal CT
Nano x-ray microscopy
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Micro CT images
Stampanoni et al Sterling et al Dental CBCT
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• Object preparation, fixation, irradiation, etc
• Polychromatic source, miscalibrations, etc
• Small object size: insufficient absorption contrast
• Limited field-of-view, limited data, incomplete
geometry
• Large amount of digital data
Problems
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• Region-of-interest reconstruction
• Fully 3D cone-beam scanning and
reconstruction
• The use of phase contrast
• Software/hardware acceleration
Solutions
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Geometry
Parallel beamThe source is faraway from the object
Cone beamThe source is closeto the object:
- Increased flux- Magnification- Fully 3D
Synchrotron
Microfocus tube,microscopy
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`I nverse problem
g Af
Radon transform
Object: f
Projection
data: dl f g
s Line
s
),(
,
s To find f from g ?
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`Backprojection
Integration of the projection data
over the whole range of
0
* 1d gg A
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Algorithms: classification
gF F g AA A
g A A A f
g Af
nn 11
1**
*1*
Radon transform
Imagingequation
BPF
FBP
Fourier
• Fourier algorithm
• Filtered backprojection(FBP)
• Backprojection andfiltering (BPF)
• Iterative
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BronnikovAlgorithms
Parallel-beam geometry(Synchrotron)
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BronnikovAlgorithms
Fourier slice theorem
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BronnikovAlgorithms
I mage reconstruction with NFFT
gF F f 11
2
Interpolation from the polar gridto the Cartesian is required
Nonequispaced Fast Fourier transform (NFFT) can be used
Potts et al, 2001
Linogram (“pseudo-polar”) grid
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BronnikovAlgorithms
FBP and BPF algorithms
“ramp filter”
d gF f 0
11
1
1*
1
221*
2 AAF A AF
d gF f 0
2212
1
g AA Ag A A A f 1***1*
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BronnikovAlgorithms
FBP algorithm
1D Filtering
d gq f 0
Backprojection
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BronnikovAlgorithms
Local (“Lambda”) tomography
gs
H A f gs
H gF *1
1ˆ
dt t s
t gs Hg
gs
)(1)(
Local operator gs
A f *
Hilbert transform is non-local:
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BronnikovAlgorithms
Cone-beam geometry(Microfocus x-ray tube)
F ldk l ith ith
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BronnikovAlgorithms
Feldkamp algorithm with
a circular orbit
X
ZFiltering
Backprojection
d gq f
0
Feldkamp, Davis, Kress, 1984
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BronnikovAlgorithms
Kirillov-Tuy condition
dssu f ug ))(()),((0
a a
a ( )
s
detector
x-ray source trajectory; parametrized as a( )
Exact 3D reconstructionis possible if every plane throughthe object intersects the sourcetrajectory at least once
f
u
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Circular source orbit: artifacts
Slices of 3D reconstructionof a phantom (cone angle 30 deg):
Bronnikov 1995, 2000
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BronnikovAlgorithms
ROI reconstruction
Two-step data acquisition
Detector
Source
Sample
ROI
Position 2
Sample
Position 1
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BronnikovAlgorithms
Non-planar source orbits
- two orthogonal circles- two circles and line- helix (most feasible mechanically)- saddle
Non-planar3D reconstructionsof a phantom:
Non-planar orbits satisfy the Kirillov-Tuy condition,but special reconstruction algorithms are required
Katsevich algorithm for
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BronnikovAlgorithms
Katsevich algorithm fora non-planar source orbit
g H A f *
2
1
PI-line (“segment”, “chord”) between a( 1 ) and a( 2 ): 2 1=2
1. Differentiation of data2. Hilbert transform along
the filtration lines insidethe Tam-Danielson window
3. Backprojection
a ( 1)
a ( 2)
T h
R R2
,sin,cosaHelix:
Katsevich, 2002
BPF algorithms for ROI
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BronnikovAlgorithms
BPF algorithms for ROIreconstruction
g HHg
g A H f *
2
1
g A H f *1
21,2
1aa
1. Differentiation of data
2. Backprojection onto the PI chord (locality!)3. Hilbert transform along the PI chord
a ( )
Using that f has the finite support and
Zou, Pan, Sidky, 2005 derived:
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BronnikovAlgorithms
Ultra-fast implementation
Reconstruction of a 512x512x512 image from 360 projections:
~10 sec~20 sec~40 sec~80 secTime :
Twin
quad-core
Quad
core
Dual
core
SingleCPU
(~2 GHz) :
Reconstruction of a 1024x1024x1024 image from 800 projections:
~120sec~480 secTime :
Twin quad-coreDual coreCPU (~2 GHz) :
• Graphic card (GPU)• CPU
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BronnikovAlgorithms
Phase-contrast microtomography(Free propagation mode)
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BronnikovAlgorithms
Phase contrast
Interference of the phase-shiftedwave with the unrefracted waves
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BronnikovAlgorithms
I nline phase-contrast imaging
Snigirev et al , 1995
Polychromatic x-ray phase
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BronnikovAlgorithms
Polychromatic x ray phase
contrast
Wilkins et al , 1996
Phase-contrast tomography
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BronnikovAlgorithms
Phase contrast tomography
with Radon inversion: edges
I nverse problem of phase-
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BronnikovAlgorithms
e se p ob e o p asecontrast microtomography
0),,(from),,(find 321 y x I x x x f z
• CTF (Cloetens et al, 1999)
• TIE (Paganin and Nugent, 1998)
• Weak-absorption TIE (Bronnikov, 1999)
Object function: f = n – 1
Phase retrieval,more than one detection plane
FBP, single detection plane
R d t f l ti f TI E
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BronnikovAlgorithms
Radon transform solution of TI E
Bronnikov, 1999
Object
d d sgd
f ),(ˆ4
12
f
g
d
1 / ),( id
I I y xg
),(21),(20
y x
d
I y x I d
Phase-contrast reconstruction
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BronnikovAlgorithms
in the form of the FBP algorithm
2D Filtering
Backprojection
d gqd
f 0
222
y x
yq
22Q
Bronnikov, 1999, 2002, 2006
22Q
Gureyev et al , 2004: choice of for
linearly dependent absorption and refraction
I mplementation at SLS
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BronnikovAlgorithms
I mplementation at SLS
Phase tomography reconstruction (a) and the 3D
rendering (b) of a 350 microns thin wood sampleusing modified filter given in the Eq. (8). The length ofthe scale bar is 50 µm.
Validation of the MBA method: (a) Phase tomographic reconstruction of sampleconsisting of polyacrylate, starch and cross-linked rubber matrix obtained using DPCand (b) using MBA. The length of the scale bar is 100 µm.
22Q
“MBA: Modified Bronnikov Algorithm”Groso, Abela, Stampanoni, 2006
I mplementation at Ghent
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BronnikovAlgorithms
pUniversity
De Witte, Boone, Vlassenbroeck,Dierick, and Van Hoorebeke, 2009
Radon inversion “Modified Bronnikov Algorithm” “Bronnikov-Aided Correction”
Polychromatic source, mixed
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BronnikovAlgorithms
phase and amplitude object
Data provided by Xradia
Air bubbles in epoxy,
relatively strong absorption:
Reconstruction by the “Bronnikov Filter” with correction
6 mm
Summary
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BronnikovAlgorithms
• New developments in theory:- parallel-beam CT (synchrotron): the use of NFFT
- cone-beam CT (microfocus): exact reconstruction with non-planar orbits; exact ROI reconstruction
(Katsevich formula, PI line, Hilbert transform on chords)
• New developments in implementation:
– ultra-fast 3D reconstruction on multicore processors
– (10243 voxels within one-two minutes on a PC)
• New developments in coherent methods:
- robust algorithms for 3D phase reconstruction
- correction for the phase component
Summary