Sampling in tomography

Transcription

1 in tomography Echantillonnage en tomographie L. Desbat TIMC-IMAG, UMR 5525, UJF-Grenoble 1 15 June 2012 L. Desbat in tomography

2 Outline CT, Radon transform and its inversion 1 CT, Radon transform and its inversion CT Radon Transform Radon and Fourier 2 on lattices the Radon Transform 3 Local approaches ROI approaches L. Desbat in tomography

3 CT, Radon transform and its inversion CT Radon transform Radon and Fourier CT scanner: principle ID = IS e RD S µ(l)dl Cormack, Hounsfield, 79 Nobel Price, [Cormack(1963), Hounsfield(1973)]. L. Desbat in tomography

4 From Radiology to Tomography CT Radon transform Radon and Fourier Definition Let µ L 1 ( R 2) then the Radon transform R of µ is defined by: R φ µ (s) def = Rµ (φ, s) def = where s R, θ(φ), ζ(φ) S 1, the unit circle in R 2, R ( µ sθ + lζ ) dl (1) ζ e2 θ s φ e1 L. Desbat in tomography

5 Center Slice Theorem and FBP CT Radon transform Radon and Fourier Theorem Let µ L 1 (R 2 ) then R φ µ(σ) = ˆµ(σ θ) Let µ L 1 (R 2 ) sufficiently smooth then π µ( x) = R φ µ(σ) σ e 2iπσ x θ dσdφ 0 R L. Desbat in tomography

6 on lattices the Radon Transform Petersen-Middleton - Shannon generalization Let g C 0 (Rn ), the F.T. of g ĝ( ξ) def = 1 n 2π R g( x)e i x ξ d x Let K R n, the non-overlapping Shannon condition associated to K for the sampling lattice L W = W Z n generated by the non singular n n matrix W is that the sets K + 2πW t l, l Z n are disjoint sets in R n. The Petersen-Middleton theorem yields the Fourier interpolation formula [Petersen and Middleton(1962), Faridani(1994)] (S W g)( x) = 1 2π n det W y L W g( y)ˇχ K ( x y), where χ K is the indicator function of the set K. L. Desbat in tomography

7 on lattices the Radon Transform Petersen-Middleton - Shannon generalization The interpolation error is given by [ 2 1 W I = h 0 1 S W g g 2 2π n ] ; 2πW t I = π h [ ξ K ĝ(ξ) dξ. ] W S = h [ ] x2 ξ2 x2 ξ2 x ξ1 x ξ Figure: 2D interlaced and parallel sampling. L. Desbat in tomography

8 FT of the RT: essential support on lattices the Radon Transform Let g L 1 ([0, 2π[ R), ĝ(k, σ) = (2π) 3/2 g(φ, s)e i(kφ+σs) dφds, k Z, σ R. Let µ C0 k (Ω), p(φ, s)def = Rµ(φ, s) then (k,σ)/ K R ˆp(k, σ) dσ < η(ϑ, b) µ L ϑ 2π ɛ 0(µ, b) where ɛ 0 (µ, b) = ξ >b K R = ( ) ˆµ ξ dξ and η(ϑ, b) < e c(ϑ)b { ( ( ) )} σ 1 (k, σ) Z R; σ < b; k < max ϑ, ϑ 1 b [Rattey and Lindgren(1981), Natterer(1986)] L. Desbat in tomography

9 Rµ essential support on lattices the Radon Transform σ b b ϑ k Figure: Essential support of the FT of the RT of a b-band limited function. L. Desbat in tomography

10 Standard W S = [ π P Q ] [ ; 2πW t 2P 0 S = 0 πq on lattices the Radon Transform ] s σ b φ b ϑ k Figure: 2D parallel sampling. conditions: P > b ϑ ; Q > 2 b π. Optimal relation: P slightly larger than π 2 Q L. Desbat in tomography

11 (Efficient) Interlaced W I = [ 2π P π P 0 2 Q ] on lattices the Radon Transform [ ; 2πW t P 0 I = π 2 Q πq ] s σ b φ b ϑ k Figure: 2D parallel sampling. conditions: Q > 2 b π ; P = π Q ϑ Optimal relation: P slightly larger than πq L. Desbat in tomography

12 on lattices the Radon Transform Generalization in 2D: Fan Beam Geometry m = ξ 2 b(r + ρ) m(= ξ2) R detector (angular width (α)) br b(r ρ) w b(r + ρ) 2br 1 v bρ k = ξ 1 br b(r ρ) αm t bρ w 2brρ bρ r+ρ v k(= ξ1) αm α v(t) bρ r ρ r+ρ Figure: 2D Fan Beam [Natterer(1996), Natterer(1999), Gratton.(2005), Izen et al.(2005)izen, Rohler, and Sastry, Faridani(2006)] L. Desbat in tomography

13 on lattices the Radon Transform Generalization in 3D: Parallel Beam Geometry Figure: 3D Parallel Beam [Desbat(1997)] L. Desbat in tomography

14 CT, Radon transform and its inversion on lattices the Radon Transform Generalization in 3D: Fan Beam Geometry m h idt m ) α) ( w lar gu or ect (an R det 2br b(r+ρ) 1 br b(r ρ ) t bρ w bρ k ν αm b( ρ r) α αm ~v (t) Time axis line { sθ+t e3 +IR ζ e3 L α,β, t Source t sθ+t e3 α n2 ζ ζ 3.5b/2 2.5b/2 2.5b/2 1.5b/2 θ s n2 3.5b/2 β X2 b n1 b n1 φ α X1 helical path Figure: 3D Fan Beam X-ray [Desbat and Gratton(2007)], [Desbat et al.(2004)desbat, Roux, Koennig, and Grangeat] L. Desbat in tomography 3b 3

15 Generalizations and applications on lattices the Radon Transform SPECT collimation geometries (nuclear imaging) Vector field tomography [Desbat(1995)] Doppler Imaging in astrophysics (generalized rotation invariant RT) [Desbat and Mennessier(1997)] Efficient scheme perturbations for industrial applications [Desbat(1993)]... L. Desbat in tomography

16 Non Local Filter CT, Radon transform and its inversion Local approaches ROI approaches We define p H the Hilbert transform of the parallel projection p p H (φ, s) = + p(φ, t)h(s t)dt where h(u) = 1 πu and ĥ(σ) = isgn(σ) (distribution). The ramp filtering p R of p is p R (φ, s) = 1 2π s p H (φ, s) The filter p φ (s) p F φ (σ) p filter φ (σ) σ p F 1 φfiltered (s) is non local!!! σ = 1 2π (2iπσ) ( isgn(σ)) is the Hilbert filtering composed by the derivation. L. Desbat in tomography

17 Λ-tomography ; wavelets Local approaches ROI approaches Reconstruct Λµ locally from p where ( ) ( ) Λµ ξ = ˆµ ξ ξ [Faridani et al.(1992)faridani, Ritman, and Smith, Bilgot et al.(2011)bilgot, Desbat, and Perrier] Wavelets [Delaney and Bresler(1995), Berenstein and Walnut(1996), Bonnet et al.(2000)bonnet, Peyrin, Turjman, and Prost, Bilgot(2007)] L. Desbat in tomography

18 Local approaches ROI approaches Assumption µ is compressive (e.g., can be represented by few coefficients in a wavelet base) then µ can be identified (with high probability) from few projection samples [Candes and Wakin(2008), Wang(2011)] L. Desbat in tomography

19 Local approaches ROI approaches The parallel Fan Beam Hilbert Projection Equality Theorem p H (φ, v t θ) = g H (v t, φ) (2) (idea compute p H (φ, s) from g H (v t, φ) with v t θ = s) [Noo et al.(2002)noo, Defrise, Clackdoyle, and Kudo, Clackdoyle and Defrise(2010)] v(λ) L. Desbat in tomography

20 Very Short Scan CT, Radon transform and its inversion Local approaches ROI approaches Figure: Very Short Scan... L. Desbat in tomography

21 Virtual Fean Beam Local approaches ROI approaches Figure: Data truncation: Hip... L. Desbat in tomography

22 Dynamic and ROI Tomography Local approaches ROI approaches ( ) µ Γt ( x) def = µ Γt ( x) det J Γt ( x) ROI FBP Left: dyn. phantom ; right: 1rst Col. NoCorr, 2nd Col. Dyn.Corr [Desbat et al.(2007)desbat, Roux, and Grangeat, Desbat et al.(2012)desbat, Mennessier, and Clackdoyle] L. Desbat in tomography

23 Thank you! Questions? Local approaches ROI approaches Thank you for your attention! Questions? L. Desbat in tomography

24 Bibliography CT, Radon transform and its inversion Local approaches ROI approaches C. Berenstein and D. Walnut. Wavelets and local tomography. In Aldroubi and Unser, editors, Wavelets in Medicine and Biology, pages CRC Press, A. Bilgot. Méthodes locales d identification de surfaces de discontinuité à partir de projections tronquées pour l imagerie interventionnelle. PhD thesis, Université J. Fourier, Grenoble I, A. Bilgot, L. Desbat, and V. Perrier. Fbp and the interior problem in 2d tomography. In IEEE, NSSMIC conference proceedings, pages MIC21 S57, S. Bonnet, F. Peyrin, F. Turjman, and R. Prost. Tomographic reconstruction using nonseparable wavelets. IEEE Transactions on image processing, 9(8): , E. Candes and M. Wakin. An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2):21 30, R. Clackdoyle and M. Defrise. Tomographic reconstruction in the 21st century. IEEE Signal Processing Magazine, 27(4):60 80, A.M. Cormack. Representation of a Function by Its Line Integrals, with Some Radiological Applications. J. Appl. Phys., 34: , L. Desbat in tomography

25 Bibliography CT, Radon transform and its inversion Local approaches ROI approaches A.H. Delaney and Y. Bresler. Multiresolution tomographic reconstruction using wavelets. IEEE Transactions on Image Processing, 4(6): , L. Desbat. Efficient sampling on coarse grids in tomography. Inverse Problems, 9: , L. Desbat. Efficient parallel sampling in vector field tomography. Inverse Problems, 11: , L. Desbat. Echantillonnage parallèle efficace en tomographie 3D. C.R. Acad. Sci. Paris, Série I, t. 324, pages , L. Desbat and L. Gratton. with the reflected lattice in helical fan beam ct. In Fully 3D tomography abtract book, pages , L. Desbat and C. Mennessier. Efficient sampling in Doppler imaging. In Theory and Applications, SampTA 97, pages 16 21, L. Desbat, S. Roux, A. Koennig, and P. Grangeat. conditions of 3D Parallel and Fan-Beam X-ray CT with application to helical tomography. Phys. Med. Biol., 49(11): , L. Desbat in tomography

26 Bibliography CT, Radon transform and its inversion Local approaches ROI approaches L. Desbat, S. Roux, and P. Grangeat. Compensation of some time dependent deformations in tomography. IEEE transaction on Medical Imaging, 26(2): , L. Desbat, C. Mennessier, and R. Clackdoyle. Dynamic tomography, mass preservation and ROI reconstruction. In International Conference on Image Formation in X-Ray Computed Tomography, Salt Lake City, USA, June Oral presentation. A. Faridani. A generalized sampling theorem for locally compact abelian groups. Math. Comp., 63(207): , A. Faridani. Fan-Beam Tomography and Theory. In J.J. Benedetto and A. I. Zayed, editors, Proceedings of Symposia in Applied Mathematics, volume 63, pages American Mathematical Society, A. Faridani, E.L. Ritman, and K.T. Smith. Local Tomography I. SIAM J. Appl. Math., 52(2): , L. Gratton. Nonuniform Problems in Computed Tomography. PhD thesis, Oregon State University, G.N. Hounsfield. Br. J. Radiol. Computerized transverse axial scanning tomography: Part I, description of the system, 46: , L. Desbat in tomography

27 Bibliography CT, Radon transform and its inversion Local approaches ROI approaches S.H. Izen, D.P. Rohler, and K.L.A. Sastry. Exploiting symmetry in fan beam ct: Overcoming third generation undersampling. SIAM J. Appl. Math., 65(3): , F. Natterer. The Mathematics of Computerized Tomography. Wiley, F. Natterer. Resolution and reconstruction for a helical CT-scanner, Mathematics Dept., University Münster, Germany ; 2/. F. Natterer. of functions with symmetries, Mathematics Dept., University Münster, Germany ; 1/. F. Noo, M. Defrise, R. Clackdoyle, and H. Kudo. Image reconstruction from fan-beam projections on less than a short-scan. Phys. Med. Biol., 47: , July D.P. Petersen and D. Middleton. and reconstruction of wavenumber-limited functions in N-dimensional euclidean space. Inf. Control, 5: , P.A. Rattey and A.G. Lindgren. the 2-D Radon transform. IEEE Trans. ASSP, 29: , L. Desbat in tomography

28 Local approaches ROI approaches H. Wang. Méthodes de reconstruction d images à partir d un faible nombre de projections en tomographie par rayons X PhD thesis, Université J. Fourier, Grenoble I, L. Desbat in tomography