Analytical Reconstructions of Regions of Interest in Medical Imaging

Transcription

1 Analytical s of Regions of Interest in Medical Imaging Kévin Polisano Supervised by : Laurent Desbat 09/18/2012 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

2 1 Introduction Context Topic Principles of tomography and motivation 2 State of the art Incomplete data reconstruction Personal record Future improvements Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

3 Contents Introduction Context Topic Principles of tomography and motivation 1 Introduction Context Topic Principles of tomography and motivation Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

4 Context Introduction Context Topic Principles of tomography and motivation Manmachine interface GMCAO Robotics Clinical applications Soft tissus modelisation Tomography imaging Statistics modelisation Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

5 Topic Introduction Context Topic Principles of tomography and motivation What is the topic of this intership? To discover the field of Computed Tomography Imaging To improve the current way of reconstruction of images To reduce the X-ray exposure by decreasing the trajectory of the scanner around the patient To implement a first version in Matlab which could be reused then by Surgivisio Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

6 Topic Introduction Context Topic Principles of tomography and motivation What is the topic of this intership? To discover the field of Computed Tomography Imaging To improve the current way of reconstruction of images To reduce the X-ray exposure by decreasing the trajectory of the scanner around the patient To implement a first version in Matlab which could be reused then by Surgivisio Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

7 Topic Introduction Context Topic Principles of tomography and motivation What is the topic of this intership? To discover the field of Computed Tomography Imaging To improve the current way of reconstruction of images To reduce the X-ray exposure by decreasing the trajectory of the scanner around the patient To implement a first version in Matlab which could be reused then by Surgivisio Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

8 Topic Introduction Context Topic Principles of tomography and motivation What is the topic of this intership? To discover the field of Computed Tomography Imaging To improve the current way of reconstruction of images To reduce the X-ray exposure by decreasing the trajectory of the scanner around the patient To implement a first version in Matlab which could be reused then by Surgivisio Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

9 Principles of tomography Introduction Context Topic Principles of tomography and motivation Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

10 Contents Introduction State of the art Incomplete data reconstruction 1 Introduction 2 State of the art Incomplete data reconstruction Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

11 Mathematical formulation Projection State of the art Incomplete data reconstruction Projection p(φ,s) = f(rα+sβ)drforφ (0,π),s (, ) Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

12 Mathematical formulation Filters applied to projections State of the art Incomplete data reconstruction Ramp filter Hilbert filter p R (φ,.) = p(φ,.) r R(σ) = σ p H (φ,.) = p(φ,.) h H(σ) = j.sign(σ) Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

13 Mathematical formulation Filtred backprojection State of the art Incomplete data reconstruction Filtred backprojection f(x) = π p 0 R(φ,s) s=x β dφ f(x) = 1 p H (φ,s)dφ 2π s π 0 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

14 Parallel geometry problems State of the art Incomplete data reconstruction Some problems with parallel geometry The X-ray source doesn t cast parallel beams The parallel formulas require all projections p(φ, s) This geometry is not adapted to data truncation Work flow To adopt a new geometry called fanbeam geometry truncated parallel projections into fanbeam To apply Hilbert equality to evaluate p(φ, s) Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

15 Fanbeam geometry Projection and Hilbert equality Introduction State of the art Incomplete data reconstruction Projection g(λ,φ) = 0 f(v(λ)+lα)dl α = (cosφ,sinφ) β = ( sinφ,cosφ) Hilbert equality p H (φ,s) = g H (v λ,φ),s = v λ β Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

16 Contents Introduction 1 Introduction Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

17 Global vision of the architecture input acquisition.m fanbeam.m reconstruction.m h rebinning d ds G GH PH PHP f(x) = 1 M p 2M H(n(φk),x n(φk)) k=1 output Nvertex Nbeams M 2Q +1 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

18 Focus on the CT imaging process acquisition.m input Data available size Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

19 Focus on the CT imaging process acquisition.m 500 Data required image s position Rtraj = 150 Nvertex = 512 Nbeams = 1024 γ m = arcsin( FOV Rtraj ) 20 FOV = 50 integrale discretization=256 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

20 Focus on the CT imaging process acquisition.m input G Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

21 Focus on the CT imaging process acquisition.m input Calculation of g(λ i,α j ) G Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

22 Focus on the CT imaging process acquisition.m Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

23 Focus on the CT imaging process acquisition.m input G Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

24 Focus on the CT imaging process acquisition.m input Hilbert Filter applied to G G h GH Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

25 Filtered projections acquisition.m Introduction Numerical problems Calculation of g H = g h unstable around zero Solution = regularization 5 4 Hilbert function h h regularized Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

26 Focus on the CT imaging process fanbeam.m input G h GH Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

27 Focus on the CT imaging process fanbeam.m input acquisition.m G h GH Nvertex Nbeams Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

28 : fanbeam geometry to parallel geometry fanbeam.m input acquisition.m G h GH rebinning PH Nvertex Nbeams Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

29 : fanbeam geometry to parallel geometry Discretized plan s l n(φ k ) a(λ t ) Discretization φ k = k π M,k = 0..M 1 s l = l RROI Q,l = Q..Q Q = 512 M = 1607 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

30 : fanbeam geometry to parallel geometry The rebinning method step by step 500 Steps Intersection a(λ t ) of (n(φ k ),s l ) with path 2 Determine the angle α t between two lines 3 Give a bound of λ i λ t λ i+1 and α i α t α i+1 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

31 : fanbeam geometry to parallel geometry The rebinning method step by step 500 Steps Intersection a(λ t ) of (n(φ k ),s l ) with path 2 Determine the angle α t between two lines 3 Give a bound of λ i λ t λ i+1 and α i α t α i+1 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

32 : fanbeam geometry to parallel geometry The rebinning method step by step 500 Steps Intersection a(λ t ) of (n(φ k ),s l ) with path 2 Determine the angle α t between two lines 3 Give a bound of λ i λ t λ i+1 and α i α t α i+1 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

33 : fanbeam geometry to parallel geometry Bilinear interpolation of g H (λ t,α t ) 500 α i α t α i λ i λ t λ i GH Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

34 : fanbeam geometry to parallel geometry fanbeam.m input Hilbert equality gives: p H (n(φ k ),s l ) g H (λ t,α t ) acquisition.m G h GH rebinning PH Nvertex Nbeams Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

35 : fanbeam geometry to parallel geometry fanbeam.m input Derivationp H (n(φ k),s l ) p H (n(φ k ),s l+1 ) p H (n(φ k ),s l 1 ) s l+1 s l 1 acquisition.m G h GH rebinning PH d ds PHP Nvertex Nbeams Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

36 : fanbeam geometry to parallel geometry fanbeam.m input acquisition.m fanbeam.m G h GH rebinning PH d ds PHP Nvertex Nbeams M 2Q +1 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

37 reconstruction.m Introduction input Allp H (n(φ k),x n(φ k ))areinterpolatedandsumup acquisition.m fanbeam.m reconstruction.m h rebinning d ds G GH PH PHP f(x) = 1 M 2M k=1 p H (n(φk),x n(φk)) Nvertex Nbeams M 2Q +1 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

38 Interpolation Introduction x n(φ k ) a(λ t ) Linear interpolation y = x n(phi k ) p H (n(φ k),y) (y s l ) s l+1 s l p H (n(φ k),s l+1 )+ (s l+1 y) s l+1 s l p H (n(φ k),s l ) x n(φ k ) p H (n(φ k),s l+1 ) p H (n(φ k),x n(φ k )) p H (n(φ k),s l ) Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

39 Condition of accurate reconstruction x 2 x 1 n(φ 1 ) a(λ t ) n(φ 2 ) Fanbeam data condition The point x can be reconstructed from complete fanbeam projections provided a fanbeam vertex can be found on each line passing through x Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

40 reconstruction.m Introduction input acquisition.m fanbeam.m reconstruction.m h rebinning d ds G GH PH PHP f(x) = 1 M 2M k=1 p H (n(φk),x n(φk)) output Nvertex Nbeams M 2Q +1 size Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

41 Contents Introduction 1 Introduction Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

42 CT imaging results acquisition.m : sinogram and filtred sinogram Fanbeam measures On the top the sinogram G = (g(λ i,α j )) i,j On the bottom the filtred sinogram after applying the Hilbert filter h, GH = (g H (λ i,n(α j ))) i,j Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

43 results fanbeam.m : PH and PHP visualization Parallel geometry On the left PH = (p H (n(φ k ),s l )) k,l On the right PHP = (p H (n(φ k),s l )) k,l Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

44 results reconstruction.m : reconstructed image display Original image Reconstructed image Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

45 results reconstruction.m : reconstructed image display Original image Reconstructed image Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

46 results Error calculation mean/pixel standart deviation quadratic max > , , ,01 0,62 38/4900 Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

47 Contents Introduction Personal record Future improvements 1 Introduction Personal record Future improvements Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

48 Personal record Introduction Personal record Future improvements Scientific point of view Discover of a new field of science applied to health Learn a new mathematical theory and its recent advances Fight against numerical problems Acceleration of the execution of programs Human point of view Pluridisplinary within research teams Partnership with companies Business dimensions Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

49 Future improvements Introduction Personal record Future improvements Improvements Report will be use to teaching aid C++ implementation, real time Adaptation of this algorithm in 3D reconstruction Extent to fanbeam truncated data virtual fanbeam Integration in Surgivisio s softwares Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45

50 Thank you for your attention Personal record Future improvements Kévin Polisano Analytical s of Regions of Interest in Medical Imaging 09/18/ / 45