improving fdk reconstructions by data-dependent filtering

Transcription

1 improving fdk reconstructions b data-dependent filtering Real time tomograph project Rien Lagerwerf, Holger Kohr, Willem Jan Palenstijn & Joost Batenburg. Scientific meeting, April 6, 18 Computational Imaging Centrum voor Wiskunde en Informatica, Amsterdam

2 ct scan 1

3 scanning process

4 scanning process Fast measurements Few projection angles

5 scanning process Fast measurements Few projection angles Analtic methods

6 complication Analtic methods produce accurate results if the input data has... 3

7 complication Analtic methods produce accurate results if the input data has... man projection angles low noise 3

8 algebraic filter fdk method

9 problem definition Consider the inverse problem: where Df = g, D, the linear 3D cone-beam transform, or forward projection, g, the measured data, f, the unknown object. 5

10 fdk method FDK reconstruction: f FDK = F h (g) = D ( g h) 1D, with D, the adjoint of D, or the backprojection, g, a weighted version of the data g (g h) 1D, a one dimensional convolution, h, a one dimensional filter. 6

11 data-dependent filters Fi the data and compute the data-dependent 1 filter: with ĥ = argmin DF g h g + λ Th, h F g h, FDK reconstruction on fied data g, with filter h, T, the Tikhonov operator that imposes the tpe of regulariation, λ, the regulariation parameter. 1 Similar strateg as [D.M. Pelt, 1] for D FBP. 7

12 af-fdk methods Two choices for Tikhonov operator: 8

13 af-fdk methods Two choices for Tikhonov operator: LS-FDK: Setting, T =, gives the least squares problem, i.e. ĥ = argmin DF g h g. h 8

14 af-fdk methods Two choices for Tikhonov operator: LS-FDK: Setting, T =, gives the least squares problem, i.e. ĥ = argmin DF g h g. h T-FDK: Setting T = Id, gives the original Tikhonov, i.e. ĥ = argmin DF g h g + λ h. h 8

15 results

16 simulated data Shepp-Logan phantom Cluttered sphere phantom Figure: Simulated data phantoms. 1

17 varing number of projection angles SSIM with masked image.8 Better similarit.6.. FDK-RL FDK-HN LS-FDK N a, more angles Figure: Results for Shepp-Logan phantom. 11

18 varing noise levels 1. SSIM with masked image Better similarit FDK-RL FDK-HN LS-FDK T-FDK I, less noise Figure: Results for cluttered sphere phantom. 1

19 reconstructions nois data FDK-HN T-FDK Figure: Cluttered sphere phantom, 36 equidistant projection angles, I =

20 eperimental data High-dose scan: 7 kev, 5 W, 5 ms per projection. Low-dose scan: 7 kev, W, 1 ms per projection. Figure: Scanned objects Gold standard reconstruction: SIRT-3, high-dose data, 5 equidistant projection angles 1

21 fdk-hn vs t-fdk FDK-HN T-FDK Figure: Low-dose, 5 equidistant projection angles. 15

22 reusing algebraic filters Compute an algebraic filter Use algebraic filter to reconstruct h Pom1 = f Pom1 filter = F g (h Pom1 ) 16

23 fdk-hn vs pom1-filter FDK-HN Pomegranate1 filter Figure: Low-dose apple, 5 equidistant projection angles. 17

24 summar and conclusion

25 summar and conclusion Data-dependent filtering improves the accurac of the FDK reconstructions. 19

26 summar and conclusion Data-dependent filtering improves the accurac of the FDK reconstructions. Filters depend more on number of projection angles and noise level than object. 19

27 summar and conclusion Data-dependent filtering improves the accurac of the FDK reconstructions. Filters depend more on number of projection angles and noise level than object. Reusing algebraic filters results in fast and accurate FDK reconstructions. 19

28 summar and conclusion Data-dependent filtering improves the accurac of the FDK reconstructions. Filters depend more on number of projection angles and noise level than object. Reusing algebraic filters results in fast and accurate FDK reconstructions. Paper in preparation: M.J. Lagerwerf et al., Improving FDK reconstructions b data-dependent filtering,

29 acknowledgements Joost Batenburg Sophia Bethan Coban Holger Kohr Willem Jan Palenstijn