Theoretically-exact CT-reconstruction from experimental data

Transcription

1 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

2 Can we reconstruct an object from X-ray projections?

3 Can we reconstruct an object from X-ray projections? In 2D: reconstruction of a function from collection of 1D line integrals. [J. Radon, 1918]

4 Can we reconstruct an object from X-ray projections? In 2D: reconstruction of a function from collection of 1D line integrals. [J. Radon, 1918]

5 Can we reconstruct an object from X-ray projections? In 2D: reconstruction of a function from collection of 1D line integrals. [J. Radon, 1918]

6 Can we reconstruct an object from X-ray projections? In 2D: reconstruction of a function from collection of 1D line integrals. [J. Radon, 1918]

7 Can we reconstruct an object from X-ray projections? In 2D: reconstruction of a function from collection of 1D line integrals. [J. Radon, 1918] 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.

8 Can we reconstruct a 3D object from 2D X-ray projections? 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)... 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.

9 Can we reconstruct a 3D object from 2D X-ray projections? 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)

10 Katsevich inversion formula exact reconstruction for helix trajectory (Katsevich 2002) shift-invariant filter + backprojection constrained by Tam-Danielsson window no artefacts at high cone-angle several generalisations available f(x) = 1 2π 2 I PI 1 x y(t) 2π 0 q D f (y(q), Θ(t, x, γ)) q=t dγ sin γ dt

11 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

12 Why helical micro-ct? L 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 R R Cone angle L Cone angle

13 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

14 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 Artefacts from sampling, not geometry!

15 Hardware Varian Paxscan flat panel 2048 x 1536 pixels ~ 400mm x 300mm Rail-mounted sample stage Phoenix nano-focused source rotation/translation axis Detector v u Source R L

16 Reservoir carbonate 556mm camera length 60mm sample distance 30 micron voxel size

17 But the results are not great... Berea sandstone ~2.8 micron voxel size

18 Awful results: why? discrete implementation Detector noise in the data rotation/translation axis v H hardware alignment P u 9 parameters for geometric alignment Source R L W consistent sensitivity analysis

19 Optimal units 1.0ou change in alignment parameter changes backprojected rays through the volume of the order of 1 detector pixel. rotation/translation axis Detector v H u P For example: Source R L W 1.0ou L = W/N L +(W/2) cos α f leads to for horizontal offset 1.0ou = W/N 1+(W/2L) cos α f = W/N 1 + sin α f

20 Optimal units Detector v 1.0ou change in alignment parameter changes backprojected rays through the volume of the order of 1 detector pixel. P rotation/translation axis u H Source W L=330mm, W=400mm, H=300mm, M=2048, N=1536 R L Require precision to within 0.5 detector pixel Hardware not sufficiently accurate

21 Sample distance misalignment

22 Detector offset misalignment

23 Optimisation-based reconstruction Image sharpness: sh(f) := f 2

24 Optimisation-based reconstruction Image sharpness: sh(f) := f 2 Alignment parameter search: p := argmax p P where f p (x) = 1 2π 2 I PI 1 x y(t) [sh(f p )] 2π 0 q Dp f (y(q), Θ(t, x, γ)) q=t dγ sin γ dt

25 Optimisation-based reconstruction Image sharpness: sh(f) := f 2 sharpness R [mm] D W [mm] 20

26 Fast Filtered Backprojection Feldkamp with a twist 150 Horizontal ramp filter Backproject entire projection without weighting factors Correct geometry backprojects edges to correct location Maximal inconsistency when geometry is incorrect v [mm] u [mm]

27 Horizontal detector offset FFBP Katsevich

28 Optimisation-based reconstruction Image sharpness: sh(f) := f 2 Normalised sharpness R [mm] D W [mm] 20 Normalised sharpness R [mm] D W [mm] Normalised sharpness Sample distance [mm] Normalised sharpness Horizontal detector offset [mm]

29 Optimisation-based reconstruction Image sharpness: 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

30 Auto-focus alignment post-processing of projection data reconstruct without exact geometry

31 Auto-focus alignment post-processing of projection data reconstruct without exact geometry

32 Sample distance 556mm camera length 8mm sample distance 2.8 micron voxel size

33 Sample distance 556mm camera length 8mm sample distance 2.8 micron voxel size

34 Sample distance 556mm camera length 8mm sample distance 2.8 micron voxel size

35 Sample distance 556mm camera length 8mm sample distance 2.8 micron voxel size

36 Comparison old vs. new system old: New: 11h acc.

37 Comparison old vs. new system old: New: 11h acc.

38 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 330mm camera length >10x speed-up Circular scan with same camera length

39 Summary better SNR at high cone angle good tomogram with 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