Andrei.bronnikov Image Reconstruction

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Image reconstruction algorithmsfor microtomography

Andrei V. Bronnikov

Bronnikov Algorithms, The Netherlands



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• Introduction• Fundamentals of the algorithms

• State-of-the-art in 3D image reconstruction

• Phase-contrast image reconstruction• Summary

Contents



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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



• 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



• Region-of-interest reconstruction

• Fully 3D cone-beam scanning and

reconstruction

• The use of phase contrast

• Software/hardware acceleration

Solutions



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



`I nverse problem

g Af

Radon transform

Object: f

Projection

data: dl f g

s Line

s

),(

,

s To find f from g ?



`Backprojection

Integration of the projection data

over the whole range of

0

* 1d gg A



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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Parallel-beam geometry(Synchrotron)



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Fourier slice theorem



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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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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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FBP algorithm

1D Filtering

d gq f 0

Backprojection



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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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Cone-beam geometry(Microfocus x-ray tube)

F ldk l ith ith



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Feldkamp algorithm with

a circular orbit

X

ZFiltering

Backprojection

d gq f

0

Feldkamp, Davis, Kress, 1984



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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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ROI reconstruction

Two-step data acquisition

Detector

Source

Sample

ROI

Position 2

Sample

Position 1



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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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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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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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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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Phase-contrast microtomography(Free propagation mode)



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Phase contrast

Interference of the phase-shiftedwave with the unrefracted waves



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I nline phase-contrast imaging

Snigirev et al , 1995

Polychromatic x-ray phase



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Polychromatic x ray phase

contrast

Wilkins et al , 1996

Phase-contrast tomography



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Phase contrast tomography

with Radon inversion: edges

I nverse problem of phase-



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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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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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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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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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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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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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• 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

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Andrei.bronnikov Image Reconstruction