The broken-ray transform and its generalizations

Transcription

1 The broken-ray transform and its generalizations Gaik Ambartsoumian University of Texas at Arlington 100 Years of the Radon Transform Linz, March 27-31, 2017

2 Acknowledgements The talk is based on results of collaborative work with Rim Gouia-Zarrad, American University of Sharjah, UAE Mohammad Latifi-Jebelli, University of Texas at Arlington Sunghwan Moon, UNIST, Korea Souvik Roy, Universität Würzburg, Germany Partially supported by NSF DMS Simons Foundation

3 Outline Some Motivating Imaging Modalities Prior Work and Terminology BRT in a Disc BRT in a Slab Geometry CRT in a Slab Geometry Partial Order and Positive Cones Cone Differentiation BRT in a Slab Geometry (revisited) Polyhedral CRT in a Slab Geometry Radon-Nikodym Theorem and Range Description

4 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.

5 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.

6 Compton Scattering 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)

7 Broken-Ray and Conical Transforms L. Florescu, V. Markel, J. Schotland, F. Zhao P. Grangeat, M. Morvidone, M. Nguyen, R. Régnier, T. Truong, H. Zaidi A. Katsevich, R. Krylov F. Terzioglu V. Palamodov R. Gouia-Zarrad M. Courdurier, F. Monard, A. Osses and F. Romero D. Finch, B. Sherson

8 Broken-Ray and Conical Transforms M. Cree, P. Bones and R. Basko, G. Zeng, G. Gullberg M. Allmaras, D. Darrow, Y. Hristova, G. Kanschat, P. Kuchment M. Haltmeier C. Jung, S. Moon V. Maxim D. Schiefeneder M. Lassas, M. Salo, G. Uhlmann M. Hubenthal, J. Ilmavirta

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, (1) 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 Full Data 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), (2) 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 (3) where H is the Hilbert transform defined by Hh (t) = i sgn (r) ĥ(r) eirt dr. (4) 2π R and ĥ(r) is the Fourier transform of h(t), i.e. ĥ(r) = 1 h(t) e irt dt. (5) 2π R

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

14 VRT in a Disc: Partial Data (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 Fourier Expansions 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 f n (ρ) = 1 2π 2π 0 n= f (φ, ρ) e inφ dφ, g n (t) = 1 2π 2π 0 g n (t) e inβ, g(β, t) e inβ dβ.

16 Inversion Formula Mf n (s) = Mg n (s 1), R(s) > 1 (6) 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. (7) T 2πi t Ti 1/(s 1) + Mh n (s 1)

17 Definition of h n 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 Numerical Reconstruction (G.A., S. Roy 2015)

19 Numerical Reconstruction (G.A., S.(b) rroy (a) r = M/8 = M/ ) Fig. 7: Reconstructions with 5% multiplicative Gaussian noise. (a) Phantom (b) Reconstructed (c) Reconstruction with 5% multiplicative Gaussian noise. Fig. 8: Reconstructions of a set of phantoms. In problems of invert that integrate a function standard results of microl of the object s (true) sin and which parts will be expect to recover stably be tangentially touched in the Radon data (e.g. However, if the generaliz image function along nonto do better than that. Fo produces images of excel basically two (angular) d the fact that the integratio corner at every point of reconstructions do not be The phantom edges do bl broken rays touch them away from the origin in mathematical study of th lack of it is not an easy ta by the authors and their c The artifacts of second of the (incomplete) Radon CT the integrals of the im lines with limited angular of full range [0, 2π]). Th streak artifacts along line equal to the endpoints of CT example above those

20 VRT in Slab Geometry (G.A., R. Gouia-Zarrad 2013) Theorem f C (R 2 ) in D = {(x, y) R 2 0 x x max, 0 y y max }. For (x v, y v ) R 2 and fixed β (0, π 2 ) consider the VRT g(x v, y v ). Then f (x, y) = cos β 2 ( ymax y g(x, y) + tan2 (β) y 2 ) g(x, t) dt. x 2

21 Numerical Implementation (2D)

22 Conical Surfaces of Various Flavors in 3D

23 CRT in Slab Geometry (G.A., R. Gouia-Zarrad 2013) Consider a function f C (R 3 ) supported in D = {(x, y, z) R 3 0 x x max, 0 y y max, 0 z z max }. For (x v, y v, z v ) R 3 we define the 3D conical Radon transform by g(x v, y v, z v ) = f (x, y, z) ds. C(x v,y v,z v )

24 3D Slab Geometry (G.A., R. Gouia-Zarrad 2013) Theorem An exact solution of the inversion problem for CRT is given by z [ ] d 2 2 x f λ,µ (z) = C(β) J 0 (u(z x)) z max dx 2 + u2 ĝ λ,µ (z v ) dz v dx z max where ĝ λ,µ (z v ) and f λ,µ (z) are the 2D Fourier transforms of the functions g(x v, y v, z v ) and f (x, y, z) with respect to the first two variables, C(β) = cos 2 β/(2π sin β) and u = tan β λ 2 + µ 2.

25 Partial Order in R n and Positive Cones A Partially Ordered Vector Space V is a vector space over R together with a partial order such that 1 if x y then x + z y + z for all z V 2 if x 0 then cx 0 for all c R + From the definition we have x y 0 y x and hence the order is completely determined by V + = {x V ; x 0} positive cone of V. Furthermore, for P V there is a partial order on V such that P = V + if and only if P ( P) = {0} P + P P c 0 cp P

26 Partial Order in R n and Positive Cones A Partially Ordered Vector Space V is a vector space over R together with a partial order such that 1 if x y then x + z y + z for all z V 2 if x 0 then cx 0 for all c R + From the definition we have x y 0 y x and hence the order is completely determined by V + = {x V ; x 0} positive cone of V. Furthermore, for P V there is a partial order on V such that P = V + if and only if P ( P) = {0} P + P P c 0 cp P

27 Partial Order in R n and Positive Cones A Partially Ordered Vector Space V is a vector space over R together with a partial order such that 1 if x y then x + z y + z for all z V 2 if x 0 then cx 0 for all c R + From the definition we have x y 0 y x and hence the order is completely determined by V + = {x V ; x 0} positive cone of V. Furthermore, for P V there is a partial order on V such that P = V + if and only if P ( P) = {0} P + P P c 0 cp P

28 Partial Order in R n and Negative Cones We consider partial orders in R n corresponding to negative cones (R n ) B generated by a set of fixed basis vectors B = {v 1,..., v n }, i.e. (R n ) B = { n i=1 c iv i ; c i 0}. In R 2 we will use linearly independent vectors u, v as a generating set for the negative cone. In this case the boundary of the negative cone is a broken line. For f L 1 (R n ) we define F on R n as F (x) = f (y)dµ y x where µ is the Lebesgue measure on R n and y x represents the negative cone at x with respect to partial order on R n.

29 Partial Order in R n and Negative Cones We consider partial orders in R n corresponding to negative cones (R n ) B generated by a set of fixed basis vectors B = {v 1,..., v n }, i.e. (R n ) B = { n i=1 c iv i ; c i 0}. In R 2 we will use linearly independent vectors u, v as a generating set for the negative cone. In this case the boundary of the negative cone is a broken line. For f L 1 (R n ) we define F on R n as F (x) = f (y)dµ y x where µ is the Lebesgue measure on R n and y x represents the negative cone at x with respect to partial order on R n.

30 Partial Order in R n and Negative Cones We consider partial orders in R n corresponding to negative cones (R n ) B generated by a set of fixed basis vectors B = {v 1,..., v n }, i.e. (R n ) B = { n i=1 c iv i ; c i 0}. In R 2 we will use linearly independent vectors u, v as a generating set for the negative cone. In this case the boundary of the negative cone is a broken line. For f L 1 (R n ) we define F on R n as F (x) = f (y)dµ y x where µ is the Lebesgue measure on R n and y x represents the negative cone at x with respect to partial order on R n.

31 Partial Order in R n and Negative Cones F (x) = y x f (y)dµ

32 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) If is the natural order on R, for an integrable function f and F (x) = f (y)dt y x we have F = f almost everywhere. Note that in this case F is absolutely continuous. Can we have a similar result in higher dimensions? We start from two dimensions. Let f be an integrable function on R 2 (with f < ) with respect to Lebesgue measure and define F (x) = y x f (y)dµ using the partial order made by u, v.

33 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) If is the natural order on R, for an integrable function f and F (x) = f (y)dt y x we have F = f almost everywhere. Note that in this case F is absolutely continuous. Can we have a similar result in higher dimensions? We start from two dimensions. Let f be an integrable function on R 2 (with f < ) with respect to Lebesgue measure and define F (x) = y x f (y)dµ using the partial order made by u, v.

34 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) If is the natural order on R, for an integrable function f and F (x) = f (y)dt y x we have F = f almost everywhere. Note that in this case F is absolutely continuous. Can we have a similar result in higher dimensions? We start from two dimensions. Let f be an integrable function on R 2 (with f < ) with respect to Lebesgue measure and define F (x) = y x f (y)dµ using the partial order made by u, v.

35 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) Define V t,s (x) as the average of f over the parallelogram centered at x, sides of length t, s and directions u, v. Note that the area of the parallelogram made with vectors tu, sv is equal to det (tu, sv) = ts det(u, v). Then V t,s (x) = 1 ts det(u, v) [F (x + t 2 u + s 2 v) F (x t 2 u + s 2 v) F (x + t 2 u s 2 v) + F (x t 2 u s 2 v)]

36 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) Define V t,s (x) as the average of f over the parallelogram centered at x, sides of length t, s and directions u, v. Note that the area of the parallelogram made with vectors tu, sv is equal to det (tu, sv) = ts det(u, v). Then V t,s (x) = 1 ts det(u, v) [F (x + t 2 u + s 2 v) F (x t 2 u + s 2 v) F (x + t 2 u s 2 v) + F (x t 2 u s 2 v)]

37 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) Define V t,s (x) as the average of f over the parallelogram centered at x, sides of length t, s and directions u, v. Note that the area of the parallelogram made with vectors tu, sv is equal to det (tu, sv) = ts det(u, v). Then V t,s (x) = 1 ts det(u, v) [F (x + t 2 u + s 2 v) F (x t 2 u + s 2 v) F (x + t 2 u s 2 v) + F (x t 2 u s 2 v)]

38 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) Likewise, for n dimensions by a geometric argument and induction over n we get the following averaging formula for f V t1,...,t n (x) = 1 t 1... t n det(v 1,..., v n ) α { 1 2, 1 2 }n sgn(α 1... α n )F (x + α 1 t 1 v α n t n v n ) In special case, to get a symmetric neighborhood of x we can let t 1 = = t n = t to get the average of f over P t, the parallelograms with sides of length t centered at x V t,...,t (x) = 1 t n det(v 1,..., v n ) P t f dµ = 1 µ(p t ) P t f dµ.

39 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) Likewise, for n dimensions by a geometric argument and induction over n we get the following averaging formula for f V t1,...,t n (x) = 1 t 1... t n det(v 1,..., v n ) α { 1 2, 1 2 }n sgn(α 1... α n )F (x + α 1 t 1 v α n t n v n ) In special case, to get a symmetric neighborhood of x we can let t 1 = = t n = t to get the average of f over P t, the parallelograms with sides of length t centered at x V t,...,t (x) = 1 t n det(v 1,..., v n ) P t f dµ = 1 µ(p t ) P t f dµ.

40 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) Now by averaging over this infinitesimal symmetric neighborhood of x and applying the Lebesgue Differentiation Theorem we have Theorem Let be an order in R n made from the positive cone of v 1,..., v n and for f L 1 (R n ) define F (x) = f (y)dµ. Then for almost every x we have y x f (x) = lim t 0 V t,...,t (x).

41 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) Theorem Let the hypothesis of the previous theorem be satisfied and f be continuous. Then 1 f (x) =... F (x), det(v 1,..., v n ) v 1 v n where v j denotes the directional derivative along vector v j. How does this help us with the inversion of the broken-ray or conical Radon transforms?

42 Cone Differentiation Th. (G.A., M. Latifi-Jebelli 2016) Theorem Let the hypothesis of the previous theorem be satisfied and f be continuous. Then 1 f (x) =... F (x), det(v 1,..., v n ) v 1 v n where v j denotes the directional derivative along vector v j. How does this help us with the inversion of the broken-ray or conical Radon transforms?

43 Inversion of BRT in 2D (G.A., M. Latifi-Jebelli 2016) Assume that L(x, y) is the unique broken ray with vertex at (x, y) and axis of symmetry α, where α = (α x, α y ) is a unit vector parallel to u+v 2. Also, let β be the angle between u and α. Theorem Let R be the broken ray transform on L 1 (R 2 ) defined by: (Rf )(x, y) = fdl. Then F (x, y) = 0 L(x,y) (Rf )(x + tα x, y + tα y ) sin β dt is the integral of f over the negative cone at (x, y). Hence f (x, y) = 1 det(u, v) u v 0 (Rf )(x + tα x, y + tα y ) sin β dt

44 Inversion of BRT in 2D (G.A., M. Latifi-Jebelli 2016) Assume that L(x, y) is the unique broken ray with vertex at (x, y) and axis of symmetry α, where α = (α x, α y ) is a unit vector parallel to u+v 2. Also, let β be the angle between u and α. Theorem Let R be the broken ray transform on L 1 (R 2 ) defined by: (Rf )(x, y) = fdl. Then F (x, y) = 0 L(x,y) (Rf )(x + tα x, y + tα y ) sin β dt is the integral of f over the negative cone at (x, y). Hence f (x, y) = 1 det(u, v) u v 0 (Rf )(x + tα x, y + tα y ) sin β dt

45 Inversion of BRT in 2D (G.A., M. Latifi-Jebelli 2016)

46 Polyhedral CRT in R n (G.A., M. Latifi-Jebelli 2016) For any x R n, we define (Rf )(x) to be integral over the boundary of polyhedral cone C generated by unit basis vectors u 1,..., u n starting from x, i.e. (Rf )(x) = f ds, where ds is n 1 dimensional Lebesgue measure on C. Assume that u i u j is constant for any i and j. Define w = u 1+ +u n u 1 + +u n. Let X i = span u 1,..., u i 1, u i+1,..., u n be the hyperplane containing a face of polyhedral cone and define y i to be a unit vector in X i. C

47 Polyhedral CRT in R n (G.A., M. Latifi-Jebelli 2016) For any x R n, we define (Rf )(x) to be integral over the boundary of polyhedral cone C generated by unit basis vectors u 1,..., u n starting from x, i.e. (Rf )(x) = f ds, where ds is n 1 dimensional Lebesgue measure on C. Assume that u i u j is constant for any i and j. Define w = u 1+ +u n u 1 + +u n. Let X i = span u 1,..., u i 1, u i+1,..., u n be the hyperplane containing a face of polyhedral cone and define y i to be a unit vector in X i. C

48 Polyhedral CRT in R n (G.A., M. Latifi-Jebelli 2016) For any x R n, we define (Rf )(x) to be integral over the boundary of polyhedral cone C generated by unit basis vectors u 1,..., u n starting from x, i.e. (Rf )(x) = f ds, where ds is n 1 dimensional Lebesgue measure on C. Assume that u i u j is constant for any i and j. Define w = u 1+ +u n u 1 + +u n. Let X i = span u 1,..., u i 1, u i+1,..., u n be the hyperplane containing a face of polyhedral cone and define y i to be a unit vector in X i. C

49 Polyhedral CRT in R n (G.A., M. Latifi-Jebelli 2016) For any x R n, we define (Rf )(x) to be integral over the boundary of polyhedral cone C generated by unit basis vectors u 1,..., u n starting from x, i.e. (Rf )(x) = f ds, where ds is n 1 dimensional Lebesgue measure on C. Assume that u i u j is constant for any i and j. Define w = u 1+ +u n u 1 + +u n. Let X i = span u 1,..., u i 1, u i+1,..., u n be the hyperplane containing a face of polyhedral cone and define y i to be a unit vector in X i. C

50 Polyhedral CRT in R n (G.A., M. Latifi-Jebelli 2016) Theorem Let R, w, y j be defined as above, then F (x) = 0 (Rf )(x + wt) w, y 1 dt is the integral of f over the polyhedral cone generated by u 1,... u n starting from x. Hence f (x) = 1... det(v 1,..., v n ) v 1 v n 0 (Rf )(x + wt) w, y 1 dt.

51 Range Description (G.A., M. Latifi-Jebelli 2016) Existence of f such that F (x) = y x f (y) dµ. What is the necessary and sufficient condition for a function F to be a cone integral of another function f 0 with respect to a given order structure in R n? In case of n = 1 the answer was provided by absolute continuity. We apply the Radon Nikodym Theorem to get the desired description of F. For a given F, we construct a corresponding measure ν that implies existence of f. We use the above conditions to obtain a range description for CRT.

52 Range Description (G.A., M. Latifi-Jebelli 2016) Existence of f such that F (x) = y x f (y) dµ. What is the necessary and sufficient condition for a function F to be a cone integral of another function f 0 with respect to a given order structure in R n? In case of n = 1 the answer was provided by absolute continuity. We apply the Radon Nikodym Theorem to get the desired description of F. For a given F, we construct a corresponding measure ν that implies existence of f. We use the above conditions to obtain a range description for CRT.

53 Range Description (G.A., M. Latifi-Jebelli 2016) Existence of f such that F (x) = y x f (y) dµ. What is the necessary and sufficient condition for a function F to be a cone integral of another function f 0 with respect to a given order structure in R n? In case of n = 1 the answer was provided by absolute continuity. We apply the Radon Nikodym Theorem to get the desired description of F. For a given F, we construct a corresponding measure ν that implies existence of f. We use the above conditions to obtain a range description for CRT.

54 Range Description (G.A., M. Latifi-Jebelli 2016) Existence of f such that F (x) = y x f (y) dµ. What is the necessary and sufficient condition for a function F to be a cone integral of another function f 0 with respect to a given order structure in R n? In case of n = 1 the answer was provided by absolute continuity. We apply the Radon Nikodym Theorem to get the desired description of F. For a given F, we construct a corresponding measure ν that implies existence of f. We use the above conditions to obtain a range description for CRT.

55 Range Description (G.A., M. Latifi-Jebelli 2016) Existence of f such that F (x) = y x f (y) dµ. What is the necessary and sufficient condition for a function F to be a cone integral of another function f 0 with respect to a given order structure in R n? In case of n = 1 the answer was provided by absolute continuity. We apply the Radon Nikodym Theorem to get the desired description of F. For a given F, we construct a corresponding measure ν that implies existence of f. We use the above conditions to obtain a range description for CRT.

56 Thanks for Your Attention!