Plane wave solutions for the radiative transport equation

Transcription

1 Plane wave solutions for the radiative transport equation Arnold D. Kim School of Natural Sciences University of California, Merced

2 Overview Introduction to plane wave solutions Green s function Connection to the diffusion approximation Boundary value problems for the radiative transport equation Applications to inverse problems Conclusions and future work Plane wave solutions for the radiative transport equation p.1

3 The radiative transport equation Light propagation in tissues is governed by the radiative transport equation Ω I + ρσa I ρσs L I = Q, with the scattering operator L defined as Z f (Ω Ω )I(Ω, r)dω. L I = I + S2 Plane wave solutions for the radiative transport equation p.2

4 Simplification We consider here the radiative transport equation: µ z I + ρσa I ρσs L I = 0, with µ = cos θ and L I = I + Z 1 h(µ, µ )I(µ, z)dµ. 1 Everything we discuss here can be extended to the full dimensional problem (in principle). Plane wave solutions for the radiative transport equation p.3

5 Plane wave solutions Plane wave solutions are special solutions of the homogeneous problem with constant coefficients of the following form: I(µ, z) = V (µ)eλz. Substitution of this ansatz into the radiative transport equation yields the generalized eigenvalue problem: λµv + ρσa V ρσs L V = 0. Plane wave solutions for the radiative transport equation p.4

6 Brief history The notion of plane wave solutions has been discussed in at least the following references. S. Chandrasekhar, Radiative Transfer (Dover, 1960). K. Case and P. Zweifel, Linear Transport Theory (Addison-Wesley, 1967). A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1978). Plane wave solutions for the radiative transport equation p.5

7 Symmetry Theorem. If the pair [λ, V (µ)] is a solution of the generalized eigenvalue problem above, then so is [ λ, V ( µ)]. If we replace λ by λ and µ by µ in the generalized eigenvalue problem, we obtain Proof. λµv ( µ) + ρσa V ( µ) + ρσs V ( µ) Z 1 ρσs h( µ, µ )V ( µ )dµ = 0. 1 Because h( µ, µ ) = h(µ, µ ), we find that [ λ, V ( µ)] satisfies the original generalized eigenvalue problem. Plane wave solutions for the radiative transport equation p.6

8 Interpreting this symmetry For each plane wave solution of the form V (µ)eλz there exists a complimentary plane wave solution of the form V ( µ)e λz. The relationship between these two plane wave solutions is simply a coordinate transformation: If z z then we must also change µ µ. Plane wave solutions for the radiative transport equation p.7

9 Orthogonality Suppose [λ, V (µ)] and [λ, V (µ)] are two different solutions of the generalized eigenvalue problem so that λµv + ρσa V ρσs L V = 0, λ µv + ρσa V ρσs L V = 0. From these two equations, we derive the orthogonality property Z 1 V (µ)v (µ)µdµ = 0. (λ λ ) 1 Plane wave solutions for the radiative transport equation p.8

10 Ording and indexing For each eigenvalue λ, we know that λ is also a eigenvalue. Let us order and index these eigenvalues as < λ j < < λ 1 < λ1 < < λj <. We denote the eigenfunction corresponding to eigenvalue λj by Vj (µ). By the symmetry property, we have λ j = λj, V j (µ) = Vj ( µ). Plane wave solutions for the radiative transport equation p.9

11 Normalization Since Z 1 V j (µ)v j (µ)µdµ = 1 Z 1 Vj ( µ)vj ( µ)µdµ 1 = Z 1 Vj (µ)vj (µ)µdµ, 1 we choose to normalize the eigenfunctions so that Z 1 Vj (µ)vj (µ)µdµ = sgn(j). 1 Plane wave solutions for the radiative transport equation p.10

12 Green s function Green s function G satisfies µ z G + ρσa G ρσs L G = δ(µ µ )δ(z z ) in [ 1, 1] (, ). We prescribe that G 0 as z z, and that lim+ µg z +ǫ µg z ǫ = δ(µ µ ). ǫ 0 Plane wave solutions for the radiative transport equation p.11

13 Computing Green s function For z > z we seek G as the following expansion in plane wave solutions: G= X aj V j (µ)e λj (z z ), z > z. j=1 For z < z we seek G as G= X bj Vj (µ)e λj (z z ), z < z. j=1 Plane wave solutions for the radiative transport equation p.12

14 Computing Green s function Substituting our plane wave expansions into lim+ µg z +ǫ µg z ǫ = δ(µ µ ), ǫ 0 we find that X [aj V j (µ)µ bj Vj (µ)µ] = δ(µ µ ). j=1 Plane wave solutions for the radiative transport equation p.13

15 Computing Green s function By multiplying the equation above by Vk (µ) and integrating over µ, we find that X j=1 aj Z 1 Vk (µ)v j (µ)µdµ bj 1 = Z Z 1 Vk (µ)vj (µ)µdµ 1 1 Vk (µ)δ(µ µ )dµ = Vk (µ ). 1 Due to the orthogonality property, we find that ak = Vk (µ ) when k < 0, bk = Vk (µ ) when k > 0. Plane wave solutions for the radiative transport equation p.14

16 Green s function Using plane wave solutions, we can write G(µ, z; µ, z ) explicitly as X λ, j (z z ) V (µ)v (µ )e, z < z j j j=1 G(µ, z; µ z ) = X λ (z z ). j V (µ)v (µ )e, z > z j j j=1 Plane wave solutions for the radiative transport equation p.15

17 Asymptotics Recall that the eigenvalues are ordered as < λ j < < λ 1 < λ1 < < λj <. That means as z z, we find that ( λ1 (z z ) V1 (µ)v1 (µ )e, G(µ, z; µ z ) λ1 (z z ) V 1 (µ)v 1 (µ )e, z < z, z > z. Plane wave solutions for the radiative transport equation p.16

18 Diffusion approximation When ρσa ρσs, we find that p λ1 3ρσa ρσs (1 g), V1 (µ) c0 + c1 µ, which is exactly what the diffusion approximation gives. The diffusion approximation corresponds to the slowest decaying plane wave solution. Plane wave solutions for the radiative transport equation p.17

19 Diffusion approximation V1(µ) λ µ Plane wave solution Diffusion approximation µ's/µa Plane wave solutions for the radiative transport equation p.18

20 Diffusion approximation µ's/µa = 100 µ's/µa = λj λj 2 diffusion mode 10 0 diffusion mode j j Plane wave solutions for the radiative transport equation p.19

21 Half-space Consider the boundary value problem in [ 1, 1] (0, ): µ z H + ρσa H ρσs L H = δ(µ µ )δ(z z ), H z=0 = 0 on (0, 1]. Green s function G satisfies the equation, but not the boundary condition, so we seek H as H = G Y, where Y satisfies the homogeneous problem and G z=0 = Y z=0 on (0, 1]. Plane wave solutions for the radiative transport equation p.20

22 Half-space Since Y is a solution to the homogeneous problem in the half-space, we represent it as Y (µ, z; µ, z ) = X yj (µ, z )V j (µ)e λj z. j=1 Substituting into the boundary condition, we obtain X j=1 Vj (µ)vj (µ )e λj z = X yj (µ, z )V j (µ), 0 < µ 1. j=1 Plane wave solutions for the radiative transport equation p.21

23 Half-space Through some elementary calculations, we can show that X λk z yj (µ, z ) = djk Vk (µ )e k=1 where djk satisfies the linear system X V j (µ)djk = Vk (µ) 0 < µ 1, k = 1, 2,. j=1 Plane wave solutions for the radiative transport equation p.22

24 Extensions We can use this approach to study other geometries, namely Plane-parallel slab Layered half-space composed of a slab situated on top of a half-space. Plane wave solutions for the radiative transport equation p.23

25 Recap So far, we have made use of plane wave solutions to compute Green s function and solve the half-space problem. To do these calculations, we need the eigenvalues λj and the eigenfunctions Vj (µ), but what are they? In general, we cannot calculate these eigenvalues and eigenfunctions analytically, so we calculate them numerically. Plane wave solutions for the radiative transport equation p.24

26 Numerics We use an M -point Gauss-Legendre quadrature rule of the form Z 1 M X f (µ)dµ f (µm )wm, 1 m=1 with µm and wm denoting the quadrature abscissas and weights, respectively. This quadrature rule is exact for integrating polynomials on [ 1, 1] of degree 2M 1. Plane wave solutions for the radiative transport equation p.25

27 Numerics Using this M -point Gauss-Legendre quadrature rule in the scattering operator, we obtain LM Vm = Vm + M X h(µm, µn )Vn wn, n=1 where Vm V (µm ). Then, we solve the M M generalized eigenvalue problem: λµm Vm + ρσa Vm ρσs LM Vm = 0, m = 1,, M. Plane wave solutions for the radiative transport equation p.26

28 Numerics We can extend this approach to study the full problem by computing a sequence of these matrix eigenvalue problems in the spatial frequency domain: p λµv + i 1 µ2 (kx cos ϕ + ky sin ϕ)v + ρσa V ρσs L V = 0. This method is so-called embarrassingly parallel since each of the eigenvalue problems for spatial frequency each pair (kx, ky ) is decoupled completely from the others. Plane wave solutions for the radiative transport equation p.27

29 Inverse source problem Consider the following problem in [ 1, 1] (0, ): µ z I + ρσa I ρσs L I = S0 δ(z z0 ), I z=0 = 0 on (0, 1]. Suppose, we measure the out-going intensity at z = 0: M (µ) = I(µ, 0) on 1 µ < 0, can we determine S0 and z0? Plane wave solutions for the radiative transport equation p.28

30 Inverse source problem The solution to forward model is Z 1 M (µ) = S0 H(µ, 0; µ, z0 )dµ on 1 µ < 0. 1 Substituting our plane wave expansions, we obtain M (µ) = S0 X j=1 with V j = Z " Vj (µ)v j e λj z0 X k=1 djk V k e λk z0! V j (µ) # 1 Vj (µ)dµ. 1 Plane wave solutions for the radiative transport equation p.29

31 Inverse source problem Using our explicit expression for M (µ), we can seek the unknown values: S0 and z0 by solving a nonlinear least-squares problem numerically. The key here is that we see the explicit dependence that the measured data has on the parameters. Plane wave solutions for the radiative transport equation p.30

32 Other inverse problems We have used this basic approach to study a variety of linear or linearized inverse problems. Registering a source in a half-space of tissue. Estimating optical properties in layered tissues. Reconstructing a thin absorber in a half-space of tissue. Reconstructing an irregular interface in layered tissues. Reconstructing an absorber in epithelial tissues. Plane wave solutions for the radiative transport equation p.31

33 Conclusions We have given an overview of plane wave solutions. We have shown how to construct explicit solutions to boundary value problems using expansions of plane wave solutions. These explicit solutions provide insight on how measured data depends on the optical properties of tissues. Plane wave solutions for the radiative transport equation p.32

34 Future work Plane wave solutions for the vector radiative transport equation. Using plane wave solutions to study data measured using spatial frequency domain methods. Implementation of the plane wave solution codes to various platforms for practical use in teaching and research, e.g. Virtual Photonics Simulator, Python, etc. Plane wave solutions for the radiative transport equation p.33