Reconstructing a Function from its V-line Radon Transform in a D

Transcription

1 Reconstructing a Function from its V-line Radon Transform in a Disc University of Texas at Arlington Inverse Problems and Application Linköping, Sweden April 5, 2013

2 Acknowledgements The talk is based on results of collaborative work with Sunghwan Moon, Texas A&M University Some current work is being done with Venky Krishnan, Tata Institute of Fundamental Research The work is partially supported by NSF DMS NHARP

3 Single Scattering Optical Tomography (SSOT) Compton Camera Imaging and Other Examples Single Scattering Optical Tomography (SSOT) Uses light, transmitted and scattered through an object, to determine the interior features of that object. If the object has moderate optical thickness it is reasonable to assume the majority of photons scatter once. Using collimated emitters/receivers one can measure the intensity of light scattered along various broken rays. Need to recover the spatially varying coefficients of light absorption and/or light scattering.

4 Single Scattering Optical Tomography (SSOT) Compton Camera Imaging and Other Examples Florescu, Schotland and Markel (2009, 2010, 2011) Mesoscopic Radiative Transport. [ŝ + µ a (r) + µ s (r)] I (r, ŝ) = µ s (r) A(ŝ, ŝ ) I (r, ŝ ) dŝ. (1) I (r, ŝ) is the light intensity at point r in the direction ŝ. µ a (r) and µ s (r) are the absorption and scattering coefficients. A(ŝ, ŝ ) is the probability that a photon travelling in the direction of ŝ is scattered in the direction of ŝ. I (r, ŝ) = I 0 (r, ŝ), ŝ ˆn(r) < 0, r V. (2)

5 Single Scattering Optical Tomography (SSOT) Compton Camera Imaging and Other Examples Florescu, Schotland and Markel (2009, 2010, 2011) Using Born approximation (1), (2) can be reduced to an integral geometry problem: [ ] µs (R 21 ) φ(r 2, ŝ 2, r 1, ŝ 1 ) = µ t [r(l)]dl ln, (3) µ s BR(r 2,ŝ 2,r 1,ŝ 1 ) where [ r21 sin θ 1 sin θ 2 Is (r 2, ŝ 2, r 1, ŝ 1 )dϕŝ2 φ(r 2, ŝ 2, r 1, ŝ 1 ) = ln I 0 µ s A(ŝ 2, ŝ 1 ) ]. (4)

6 Single Scattering Optical Tomography (SSOT) Compton Camera Imaging and Other Examples Florescu, Schotland and Markel (2009, 2010, 2011) So if the scattering coefficient is known, then the reconstruction of the absorption coefficient is reduced to inversion of a generalized Radon transform integrating along the broken rays.

7 Compton Scattering Single Scattering Optical Tomography (SSOT) Compton Camera Imaging and Other Examples cos β = 1 mc2 E (E E)E R. Basko, G. Zeng, and G. Gullberg (1997, 1998) M. Nguyen, T. Truong, et al (2000 s)

8 Broken Rays and Broken Geodesics Single Scattering Optical Tomography (SSOT) Compton Camera Imaging and Other Examples M. Hubenthal J. Ilmavirta M. Lassas M. Salo G. Uhlmann...

9 V-line Radon Transform (VRT) in 2D Definition The V-line Radon transform of function f (x, y) is the integral Rf (β, t) = f ds, (5) BR(β,t) of f along the broken ray BR(β, t) with respect to line measure ds. The problem of inversion is over-determined, so it is natural to consider a restriction of Rf to a two-dimensional set.

10 Geometry: Slab vs Disc Available directions Stability of reconstruction Hardware implementation (?)

11 (G.A. 2012) Theorem If f (x, y) is a smooth function supported in the disc D(0, R sin θ), then f is uniquely determined by Rf (φ, d), φ [0, 2π], d [0, 2R].

12 Inversion Formula Rf (ψ φ, t d ) = Rf (φ, d) + Rf (φ + π, 2R d) Rf (φ, 2R), (6) for all values φ [0, 2π] and d [0, 2R]. f (x, y) = 1 4π 2π 0 H ( Rf t ) (ψ, x cos ψ + y sin ψ) dψ (7) where H is the Hilbert transform defined by Hh (t) = i sgn (r) ĥ(r) eirt dr. (8) 2π R and ĥ(r) is the Fourier transform of h(t), i.e. ĥ(r) = 1 h(t) e irt dt. (9) 2π R

13 Inversion Formula Issues with the support Interior problem Other methods without loss of information Rotation invariance

14 (G.A., S. Moon 2013) Theorem If f (x, y) is a smooth function supported in the disc D(0, R), then f is uniquely determined by Rf (φ, d), φ [0, 2π], d [0, R].

15 Denote g(β, t) := Rf (β, t). f (φ, ρ) = n= f n (ρ) e inφ, g(β, t) = where the Fourier coefficients are given by n= g n (t) e inβ, f n (ρ) = 1 2π f (φ, ρ) e inφ dφ, g n (t) = 1 2π g(β, t) e inβ dβ. 2π 0 2π 0

16 Inversion Formula Mf n (s) = Mg n (s 1), R(s) > 1 (10) 1/(s 1) + Mh n (s 1) where MF denotes the Mellin transform of function F MF (s) = 0 p s 1 F (p) dp, and h n is some fixed function. Hence for any t > 1 we have 1 t+ti f n (ρ) = lim ρ s Mg n (s 1) ds. (11) T 2πi t Ti 1/(s 1) + Mh n (s 1)

17 If 1 < t < 1 sin θ then 1 + t cos[ψ(t)] + t 2 sin[ψ(t)] h n (t) = ( 1) n e inψ(t) 1 + t 2 + 2t cos(ψ(t)) sin θ 1 t 2 sin 2 θ 1 t cos[2θ ψ(t)] + t 2 sin[2θ ψ(t)] e in[2θ ψ(t)] 1 + t 2 2t cos[2θ ψ(t)] sin θ 1 t 2 sin 2 θ, 1 + t cos[ψ(t)] + t 2 sin[ψ(t)] h n (t) = ( 1) n e inψ(t) 1 + t 2 + 2t cos[ψ(t)] sin θ 1 t 2 sin 2 θ, 0 < t 1 and h n (t) 0, for all t > 1 sin θ. Here ψ(t) = arcsin(t sin θ) + θ.

18 Current Work and Open Problems Numerical implementation Stability and microlocal analysis Range description Over-determined setups

19 Thanks for Your Attention!

20 Ambartsoumian, G., Inversion of the V-line Radon transform in a disc and its applications in imaging, Computers and Mathematics with Application, (2013). Ambartsoumian, G., Moon, S., A series formula for inversion of the V-line Radon transform in a disc, Computers and Mathematics with Application, (2012). Florescu, L., Schotland, J.C., Markel, V.A., Single scattering optical tomography, Phys. Rev. E 79, (2009). Florescu, L., Markel, V.A., Schotland, J.C., Single-scattering optical tomography: Simultaneous reconstruction of scattering and absorption, Phys. Rev. E 81, (2010). Florescu, L., Markel, V.A., Schotland, J.C., Inversion formulas for the broken-ray Radon transform, Inverse Problems 27, (2011).