Phase problem and the Radon transform

Transcription

1 Phase problem and the Radon transform Andrei V. Bronnikov Bronnikov Algorithms The Netherlands

2 The Radon transform and applications Inverse problem of phase-contrast CT Fundamental theorem Image reconstruction algorithms Phantom studies Experimental data

3 The Radon transform (J. Radon, 1917) fˆ, s = Line f dl (, s) Projection s Object

4 The inverse 3D (Radon) transform (H.A. Lorentz, 1905) 2 1 f = fˆ( s,, ω) ddω 4π 2 2 s s ω

5 Applications Medical imaging Laboratory animals Materials research Non-destructive testing Reverse engineering Geophysics

6 Principle of x-ray tomography Detector plane g Object f X-ray source

7 Fourier slice theorem fˆ ( u, v ) = gˆ ( u, v) Fourier transform in the central plane is the Fourier transform of the projection Detector plane g Object f

8 Image reconstruction Bronnikov Algorithms Filling up the Fourier space Filtered backprojection: Backprojection 1D Filtering f π = 0 q g d

9 Density reconstruction; non-planar orbits (Bronnikov, ) Bronnikov Algorithms

10 Wavelet-based contrast enhancement Original image Processed image Bronnikov and Duifhuis, 1998

11 Contrast-enhanced imaging X-ray images of the rat cerebellum Absorption contrast Phase contrast A. Momose, T. Takeda, Y. Itai and K. Hirao, Nature Medicine, 2, (1996).

12 Phase-contrast imaging Coherent x-raysx Microfocus tube

13 Phase-contrast CT vs. absorption-contrast CT ( ESRF ) Bronnikov Algorithms

14 Insect knee E=18 KeV Snigirev et al, ESRF

15 Problems the lack of the linear theory comparable to that of conventional absorption-based based CT no quantitative solution (only edges of the objects are reconstructed) huge amount of high-resolution data

16 Goals to develop an accurate and practical mathematical tool for quantitative phase- contrast image reconstruction to suggest the fastest image reconstruction algorithm for phase-contrast CT

17 Inverse problem of phase-contrast tomography z find f( x1, x2, x3) from I ( x, y), 0 < π Phase retrieval (holographic) approach (Cloetens et al 1999): - find the phase from intensity - invert Radon transform to reconstruct the object Reconstruction formula (CT approach): - a straightforward linear relation between the intensity and the object function

18 Phase retrieval Nonlinear iterative methods (Gershberg, Saxton 1972; Fienup, 1982) Linear TIE method (Gureyev et al, 2004) Linear Wigner transform (Maier, Preobrazshenski 1990; Raymer et al 1994)

19 Wigner transform W z ( x, ξ) * x x = Vz ( x ) Vz ( x+ ) e dx 2 2 2π iξ x Radon transform Inverse Wigner * I( x, z) W0 ( x, ξ) V0 ( x) V0 (0) ϕ( x)

20 Radon-Wigner transform W 0( x, ξ) δ( x zξ x ) dx dξ = I( x, z) I( x, z) x' = arctan z W ( x, 0 ξ ) ξ

21 Implementation (Bronnikov et al 1991) n - number of samples of the discrete Wigner function Number of detection planes π n 2 Detector positions z Object Radon transform Inverse Wigner * I( x, z) W0 ( x, ξ) V0 ( x) V0 (0) ϕ( x)

22 Inverse problem of phase-contrast tomography z find f( x1, x2, x3) from I ( x, y), 0 < π Phase retrieval (holographic) approach: - find the phase from intensity - invert Radon transform to reconstruct the object - multiple detector positions along the beam Reconstruction formula (CT approach): - a straightforward linear relation between the intensity and the object function - single detector position

23 Mathematical model: the Fresnel propagator Bronnikov Algorithms

24 Theoretical background T ( x, y) Transmission function 1 µ ( x, y) i ( x, y) 2 e + ϕ = Transmission function U x, y) T ( x, y) ( U i = Wavefield z I ( x, y) = hz U, 0 < π 2 Fresnel propagation λd < D 2 x, 0 y µ µ Conditions I d ( x, y) 0 λd 2 = I 1 ϕ ( x, y) 2π Intensity equation

25 Radon transforms 2D transform (lines) 3D transform (planes) 2D transform (lines) 3D transform (planes) dl f f s Line s = ), (, ˆ ω ω σ ω ω d f f s Plane s = ),, (,, ˆ ω s s ω

26 Fundamental theorem (Bronnikov 2002) Second derivative of 3D Radon transform of the object is proportional to 2D Radon transform of the phase-contrast projection s 2 2 λd < D 1 fˆ( s,, ω) = g ˆ ( s, ω ) d 2 Object f Detector plane g d

27 Inversion formula (Bronnikov 1999) 3D backprojection of the 2D Radon 3D backprojection of the 2D Radon transform of the phase transform of the phase-contrast projection contrast projection Object ω ω π ω ω π d d s g d d d s f s f ), ( ˆ 4 1 ),, ˆ( = = f g d ω

28 FBP algorithm (Bronnikov 1999, 2002) π 1 f = q g 2 4π d 0 d Backprojection 2D Filtering

29 Analytical filter function (Bronnikov 1999, 2002) π π d g q d f = y x y q + = MTF of the 2D filter: MTF of the 2D filter: 2 2 η ξ ξ + Q =

30 Implementation FFT-based linear filtering 2D parallel-beam backpojection FBP structure (parallelization) Hardware acceleration The fastest algorithm possible

31 Phase object g Object d ( x, y) I / I 1 = i Detector π 1 f = q g 2 4π d 0 d d z Near-field intensity d I Reconstruction

32 Phantom studies: polystyrene sphere d = 2.5 cm λ = 0.1 nm, = 1 µm 3D computer phantom 3D reconstruction Phase-contrast projections

33 Quantification and the limits of the linear approximation d = 2.5 cm 1.5 < d < 20 cm

34 Stability to noise 0% 1% 2% 5%

35 Mixed phase and amplitude object Absorption function µ/2 Phase function ϕ T ( x, y) = e 1 2 µ ( x, y) + iϕ ( x, y)

36 Mixed phase and amplitude object Contact intensity Near-field intensity Bronnikov Algorithms g Object ( x, y) d = I / I 0 1 π 1 f = q g 2 4π d 0 d d z Reconstruction 0 I d I

37 Comparison with heuristic methods (Anastasio et al, PMB 2003) Backprojection BP Absorption FBP Absorption LFBP Bronnikov method

38 Polypropylene foam (Schena et al; Elettra,, 2004) Conventional FPB Bronnikov method

39 Polyethylene tube with polymer fibers (Groso and Stampanoni; ; SLS, 2005) 721 projections; 15 cm distance sample detector; Energy=13.5 kev Pixel size 1.75 microns Bronnikov Algorithms

40 Summary A fundamental theorem has been proved. This is the counterpart of the Fourier slice theorem used in absorption-based based CT A fast image reconstruction algorithm has been implemented in the form of filtered backprojection (FBP)