Image reconstruction algorithms for microtomography - Bronnikov ...

Image reconstruction algorithms for microtomography Andrei V. Bronnikov

Bronnikov Algorithms, The Netherlands

Contents • • • • •

Introduction Fundamentals of the algorithms State-of-the-art in 3D image reconstruction Phase-contrast image reconstruction Summary

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Microtomography systems Desktop systems Image Intensifier Object

Camera control

PC

X-Ray tube

Motor control Camera Manipulation system

Rail system

NDT systems

Small animal CT

Nano x-ray microscopy

Dental CBCT

Synchrotron setup

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Micro CT images

Stampanoni et al Sterling et al

Dental CBCT

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Problems • 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

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Solutions • Region-of-interest reconstruction • Fully 3D cone-beam scanning and reconstruction • The use of phase contrast • Software/hardware acceleration

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Geometry

Parallel beam Synchrotron

The source is far away from the object

Cone beam Microfocus tube, microscopy

The source is close to the object: - Increased flux - Magnification - Fully 3D

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`

I nverse problem

Projection data: g , s

f dl Line ( , s )

s

Radon transform

Af

g

To find f from g ?

Object: f BronnikovAlgorithms

`

Backprojection Integration of the projection data over the whole range of * A g

1

g d 0

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Algorithms: classification Radon transform • Fourier algorithm • Filtered backprojection (FBP) • Backprojection and filtering (BPF) • Iterative

Af

f

Imaging equation

g *

1

*

*

A A

A AA

*

A g 1

BPF

g

FBP

1

Fn Fn 1 g

Fourier

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Parallel-beam geometry (Synchrotron)

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Fourier slice theorem

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I mage reconstruction w ith NFFT

f

1

F2 F1 g

Interpolation from the polar grid to the Cartesian is required

Linogram (“pseudo-polar”) grid

Nonequispaced Fast Fourier transform (NFFT) can be used Potts et al, 2001 BronnikovAlgorithms

FBP and BPF algorithms

f *

F2 A A

1

*

A A 1

2

*

A g 2

*

1

*

A AA

*

F1 AA

g

1

“ramp filter”

f

1

F2

1

2

2

g d 0

f

1

F1 1

g d

0

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FBP algorithm

1D Filtering

Backprojection

f

q g d 0

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Local ( “Lambda”) tomography F1 1

s

g

gˆ

H

s

g

Local operator

A* H

f

f

s

g

*

A

s

g

Hilbert transform is non-local:

Hg ( s )

1

g (t ) dt s t BronnikovAlgorithms

Cone-beam geometry (Microfocus x-ray tube)

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Feldkamp algorithm w ith a circular orbit Feldkamp, Davis, Kress, 1984

Filtering

Z

X

Backprojection

f

q g d 0 BronnikovAlgorithms

Kirillov-Tuy condition detector Exact 3D reconstruction is possible if every plane through the object intersects the source trajectory at least once u

s a( )

f

g (a( ), u )

f (a( ) su )ds 0

x-ray source trajectory; parametrized as a( )

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Circular source orbit: artifacts Slices of 3D reconstruction of a phantom (cone angle 30 deg):

Bronnikov 1995, 2000

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ROI reconstruction

Two-step data acquisition

Position 1

Position 2

Source ROI Sample

Sample

Detector

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Non-planar source orbits Non-planar 3D reconstructions of a phantom:

- two

orthogonal circles - two circles and line - helix (most feasible mechanically) - saddle

Non-planar orbits satisfy the Kirillov-Tuy condition, but special reconstruction algorithms are required BronnikovAlgorithms

Katsevich algorithm for a non-planar source orbit Katsevich, 2002

f a( 2)

PI-line (“segment”, “chord”) between a( 1) and a( 2):

a

g

1. Differentiation of data 2. Hilbert transform along the filtration lines inside the Tam-Danielson window 3. Backprojection

a( 1)

Helix:

1 * A H 2

h R cos , R sin , 2

2

1=2

T

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BPF algorithms for ROI reconstruction f

1 H A* 2

g

1. Differentiation of data 2. Backprojection onto the PI chord (locality!) 3. Hilbert transform along the PI chord

Using that f has the finite support and

HHg

g

a( ) Zou, Pan, Sidky, 2005 derived:

f

1 H 2

1 * A ,a1 a 2

g

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Ultra-fast implementation • Graphic card (GPU) • CPU Reconstruction of a 512x512x512 image from 360 projections:

CPU (~2 GHz) :

Single

Dual core

Quad core

Twin quad-core

Time :

~80 sec

~40 sec

~20 sec

~10 sec

Reconstruction of a 1024x1024x1024 image from 800 projections:

CPU (~2 GHz) :

Dual core

Twin quad-core

Time :

~480 sec

~120sec

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Phase-contrast microtomography (Free propagation mode)

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Phase contrast Interference of the phase-shifted wave with the unrefracted waves

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I nline phase-contrast imaging

Snigirev et al, 1995

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Polychromatic x-ray phase contrast

Wilkins et al, 1996

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Phase-contrast tomography w ith Radon inversion: edges

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I nverse problem of phasecontrast microtomography Object function: f = n – 1

find

f ( x1 , x2 , x3 ) from

• CTF (Cloetens et al, 1999) • TIE (Paganin and Nugent, 1998)

z

I ( x, y ), 0

Phase retrieval, more than one detection plane

• Weak-absorption TIE (Bronnikov, 1999)

FBP, single detection plane

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Radon transform solution of TI E d

I ( x, y )

I

Bronnikov, 1999

g ( x, y )

f

1 4

2

d

0

1

d

2

2

( x, y )

I d / Ii 1

gˆ ( s, )d d

g Object f d

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Phase-contrast reconstruction in the form of the FBP algorithm Bronnikov, 1999, 2002, 2006

f

Q

Q

2

d

q

g d

q

y x2

y2

0

2D Filtering 2

2

2

2

Gureyev et al, 2004: choice of for linearly dependent absorption and refraction

Backprojection BronnikovAlgorithms

I mplementation at SLS “MBA: Modified Bronnikov Algorithm” Groso, Abela, Stampanoni, 2006

Phase tomography reconstruction (a) and the 3D rendering (b) of a 350 microns thin wood sample using modified filter given in the Eq. (8). The length of the scale bar is 50 µm.

Q

2

2

Validation of the MBA method: (a) Phase tomographic reconstruction of sample consisting of polyacrylate, starch and cross-linked rubber matrix obtained using DPC and (b) using MBA. The length of the scale bar is 100 µm. BronnikovAlgorithms

I mplementation at Ghent University

De Witte, Boone, Vlassenbroeck, Dierick, and Van Hoorebeke, 2009

Radon inversion

“Modified Bronnikov Algorithm” “Bronnikov-Aided Correction” BronnikovAlgorithms

Polychromatic source, mixed phase and amplitude object Air bubbles in epoxy, relatively strong absorption:

6 mm

Reconstruction by the “Bronnikov Filter” with correction

Data provided by Xradia BronnikovAlgorithms

Summary • New developments in theory: parallel-beam CT (synchrotron): the use of NFFT cone-beam CT (microfocus): exact reconstruction with nonplanar 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

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