et al.

Theoretically-exact CT-reconstruction 

from experimental data 

T Varslot, A Kingston, G Myers, A Sheppard 

Dept. Applied Mathematics 

Research School of Physics and Engineering 

Australian National University

Can we reconstruct an object from X-ray 

projections?


• In 2D: reconstruction of a function 

from collection of 1D line 

integrals. [J. Radon, 1918] 

projections?





projections?





projections?





projections?





• In 3D: Radon and X-ray 

transforms are different 

• 3D Radon transform: 1D 

projection data resulting from 

2D integrals 

• 3D X-ray transform: 2D 

projection data resulting from 

1D line integrals. 

projections?

Can we reconstruct a 3D object from 2D 

• Tuy 1983, Kirillow 1961 

• condition for trajectory to facilitate exact 

reconstruction from cone-beam projections 

• Feldkamp et al., 1984 

• extension of 2D filtered backprojection with 

circle trajectory to 3D (cone-beam) 

• works well in many practical applications 

• artefacts for high cone-angle 

• Danielsson et al., 1997 

• Each object point lies on a unique PI-line 

(helix trajectory) 

X-ray projections? 

“ ... To accurately reconstruct an image for a ROI, all the 

planes passing through the ROI should intersect the 

source trajectory at least once in a nontangential and 

nonended way. ... “ 

Tuy, “An inverse formula for cone-beam reconstruction,” SIAM J Appl. Math., 1983.

Can we reconstruct a 3D object from 2D 

• Tuy 1983, Kirillow 1961 

• condition for trajectory to facilitate exact 

reconstruction from cone-beam projections 

• Feldkamp et al., 1984 

• extension of 2D filtered backprojection with 

circle trajectory to 3D (cone-beam) 

• works well in many practical applications 

• artefacts for high cone-angle 

• Danielsson et al., 1997 

• Each object point lies on a unique PI-line 

(helix trajectory) 

X-ray projections?

Why helical micro-CT? 

• Image long objects 

• multiple circular scans 

• one helical scan 

• Acquisition time 

• better use of X-ray flux 

• improved SNR, faster imaging 

• Imaging artefacts 

• circular micro-CT: 10 degrees 

• our helical micro-CT: 50 degrees










• our helical micro-CT: 50 degrees 

R 

R 

L 

Cone 

angle 

L 

Cone 

angle










• our helical micro-CT: 50 degrees










• our helical micro-CT: 50 degrees 

Artefacts from sampling, not geometry!

P 

• Varian Paxscan flat panel 

• 2048 x 1536 pixels 

• ~ 400mm x 300mm 

• Rail-mounted sample stage 

• Phoenix nano-focused source 

Source 

rotation/translation axis 

R 

L 

Hardware 

Detector 

v 

u

Reservoir 

carbonate 

• 556mm camera length 

• 60mm sample distance 

• 30 micron voxel size

But the results are not great ... 

Berea sandstone ~2.8 micron voxel size

• 

• 

• 

discrete implementation 

noise in the data 

hardware alignment 

• 9 parameters for geometric 

alignment 

• consistent sensitivity analysis 

Awful results: why? 

P 

Source 


R 

L 


W 

v 

u 

H

1.0ou change in alignment parameter 

changes backprojected rays through the 

volume of the order of 1 detector pixel. 

For example: 

1.0ou 

L = 

leads to for horizontal offset 

1.0ou = 

W/N 

L +(W/2) cos αf 

W/N 

1+(W/2L) cos αf 

Optimal units 

= W/N 

1 + sin αf 

P 

Source 


R 

L 


W 

v 

u 

H

1.0ou change in alignment parameter 

changes backprojected rays through the 

volume of the order of 1 detector pixel. 

L=330mm, W=400mm, H=300mm, M=2048, N=1536 

Optimal units 

P 

Source 


R 

L 


Require precision to within 0.5 

detector pixel 

Hardware not sufficiently accurate 

W 

v 

u 

H

Sample distance misalignment

Detector offset misalignment

Optimisation-based 

reconstruction 

Image sharpness: 

sh(f) :=||∇f||2




sharpness 

1.4 

1.2 

1 

0.8 

8 

sh(f) :=||∇f||2 

6 

R [mm] 

4 

−10 

0 

10 

D W [mm] 

20

Fast Filtered Backprojection 

• “Feldkamp with a twist” 

• Horizontal ramp filter 

• Backproject entire projection without weighting 

factors 

• Correct geometry backprojects 

edges to correct location 

• Maximal inconsistency when 

geometry is incorrect 

v [mm] 

−150 

−100 

−50 

0 

50 

100 

150 

−200 −150 −100 −50 0 

u [mm] 

50 100 150 200

Katsevich 

FFBP 

Horizontal detector offset




Normalised sharpness 


1.4 

1.2 

1 

0.8 

8 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

sh(f) :=||∇f||2 

6 

R [mm] 

4 

−10 

0 

10 

D W [mm] 

0 

4 5 6 

Sample distance [mm] 

7 8 

20 



1 

0.5 

8 0 

1 

0.8 

0.6 

0.4 

0.2 

6 

R [mm] 

4 

−10 

0 

10 

D W [mm] 

0 

−5 0 5 10 15 20 

Horizontal detector offset [mm] 

20




sh(f) :=||∇f||2 

• use FFBP to reconstruct image 

• structured search 

• “sequentially decoupled” parameters 

• slice-based reconstruction 

• computation O(N 4 ) becomes O(kN 3 ) 

• full projection set needed 

• local cache reduces MPI overhead 

• multi-resolution optimization: 

• computation O(kN 3 ) becomes O(k[N/q] 3 ) 

• memory requirement O(N 3 ) becomes O([N/q] 3 ) 

• trivially parallel at lowest level

• 

• 

Auto-focus alignment 

post-processing of projection data 

reconstruct without exact geometry

• 

• 

Auto-focus alignment 

post-processing of projection data 

reconstruct without exact geometry

Sample distance 



• 2.8 micron voxel size













Comparison 

old vs. new system 

old: 60mm@190mm. New: 400mm@382mm, 11h acc.

Comparison 

old vs. new system 

old: 60mm@190mm. New: 400mm@382mm, 11h acc.

Carbonate sample 

• 20mm by 5 mm sample 

• 4 revolutions of data 

• 1760 x 1760 x 6000 voxels 

• 3.5 micron voxel size 

• 400mm wide detector 


• >10x speed-up 

Circular scan with same camera length

Summary 

• better SNR at high cone angle 

• good tomogram with 2048 3 from 4 

hours acquisition 

• routine imaging of long objects 

• successfully reconstructed tomogram: 

2048 x 2048 x 8192 

• limited by 

• acquisition time 

• disk space 

• computer memory 

• 100mm vertical travel 

• currently ~1.5 micron resolution

et al.

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