Tomography of Highly Scattering Media with the Method of Rotated Reference Frames. Vadim Markel Manabu Machida George Panasyuk John Schotland

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1 Tomography of Highly Scattering Media with the Method of Rotated Reference Frames Vadim Markel Manabu Machida George Panasyuk John Schotland Radiology/Bioengineering UPenn, Philadelphia

2 MOTIVATION (The Inverse Problem Perspective)

3 ρ s S zs α(r) zd z Given a data function φ( ρ, ρ ) s d which is measured for multiple pairs ( ρ, ρ ), find the absorption s d D ρ d coefficient α( r) inside the slab 3

4 Linearized Integral Equation φ( ρ, ρ ) = Γ( ρ, ρ ; r) ( r)d 3 s d s d δα r φ ( ρ, ρ ) s d = I ( ρs, zs; ρd, zd) I0 ( ρs, zs; ρd, zd) I ( ρ, z ; ρ, z ) (measurable data-function) 0 α ( r) = α + δα( r) 0 s s d d Γ ( ρ, ρ ) = G ( ρ, z ; r) G ( r; ρ, z ) s d 0 s s 0 d d (first Born approximation)

5 How To Invert? Numerical SVD approach: φ = Γ δα i ij j j j φ i φ =Γδα Γ δα φ = Γ Γδα * * + * 1 * ( ) = ΓΓ Γ reg φ

6 Analytical SVD approach: Making use of the translational invariance φ ( q, q ) = φ ( ρ, ρ ) e d ρ d ρ i ( q s ρ s+ q d ρ d ) s d s d s d qs = q / + p, qd = q / p; Data function: ψ ( q, p) = φ ( q / + p, q / p) L ψ ( q, p) = g ( q / + p; z) g ( q / p; z) δα ( q; z)dz 0 iρ q δα ( q; ) = δα ( ρ, ) d ρ z dz d q G ( ρ, z ; ρ, z) = g s z e ( π ) 0 s s s ( π ) d ( q; z) e d q G0 ( ρ, z; ρd, zd) = g ( ; ) d q z e iq ( ρ ρ ) s iq ( ρ ρ) d 6

7 6cm S.D.Konecky, G.Y.Panasyuk, K.Lee, V.Markel, A.G.Yodh and J.C.Schotland Imaging complex structures with diffuse light Optics Express 16(7), (008)

8 In the case of RTE, we need the plane-wave decomposition of the Greens function, of the form d q G (,; rsr ˆ, sˆ ) = g(;,; qz sˆ z, sˆ ) e ( π ) 0 iq ( ρ ρ ) and the integral kernel Γ ( q, p; z) = g( q/ + p; z, zˆ ; z, sˆ ) g( q/ p; z,; sˆ z, zˆ )ds We then get the integral equations of the form z d ψ( qp, ) = Γ( qp, ; z) δα ( q; z)dz z s s d

9 Motivation Continued (The Forward Model Perspective)

10 Plane Wave Modes The Weyl Expanson The Helmholtz Equation ( + k ) u= 0 0 k = const e ikr k k = k 0 e r ik r 0 i = π Q= k q 0 Q e d q 1 i( q ρ+ Qz) The Diffusion Equation ( D + α) u= 0 D, α = const e k kr k = k = α 0 / D e r kr 0 1 = π Q= k + q 0 Q e d q 1 iq ρ Qz RTE ( sˆ + μ ) I( r, sˆ) = t Arnold Kim and us (the past ~10 years) Arnold Kim and us (the past ~10 years) = μ ( ˆˆ, ) (, ˆ s A ss I rs ) μ, μ = const t s

11 Spectral Method for the RTE? sˆ μ ˆ ˆ ˆ ˆ ˆ ˆ t I r s μs A s s I r s s ε r s RTE: ( + )(,) = (, )(, )d + (,) Where is the "spectral variable"? How can we write this equation in the form ( z+ W) I = ε? μ and μ do not qualify... t s We can try to expand I( rs, ˆ) into a 3D Fourier integral with respect to r and into the basis of ordinary spherical harmonics Y ( θϕ, ) with respect to sˆ... lm...and see if the equation can be cast into the desired form.

12 The Conventional Method of Spherical Harmonics This results in the following system of equations with respect to the vector of the expansion coefficients I( k) ( k - the Fourier variable) : ia k I( k) + ia k I( k) + ia k I( k) + D I( k) = ε ( k) ( x) ( y) ( z) x y z A, A, ( x) ( y) A ( z) are different matrices. ( x) * Alm, l m = sinθcos ϕylm( θϕ, ) Yl m ( θϕ, )sin θdθdϕ, etc.... This rather awe-inspiring set of equations has perhaps only academic interest. K.M.Case, P.F.Zweifel, Linear Transport Theory

13 Rotated Reference Frames The usual sperical harmonics are defined in the laboratory reference frame. Then θ and ϕ are the polar angles of the unit vector sˆ in that frame. THE MAIN IDEA: For each value of the Fourier variable k, use spherical harmonics defined in a reference frame whose z-axis is aligned with the direction of k. We call such frames "rotated". Spherical harmonics defined in the rotaded frame are denoted by Y ( sk ˆ; ˆ ).

14 Rotation of the Laboratory Frame (x,y,z). z y' z' l ˆ ˆ l Y (; ) (,,0) () ˆ lm sk = Dm m ϕk θk Ylm s m = l θ k y Wigner D-functions Euler angles x ϕ θ Spherical functions in the laboratory frame x'

15 ( sˆ + μ ) I = μ AI + ε t (,) ˆ (,) ˆ d ikr 3 I rs = I kse k ikr ε(,) rsˆ = ε( ks,) ˆ e d ( ik sˆ + μ ) I = μ AI + ε t I ( ks, ˆ) = F ( k) Y ( sk ˆ; ˆ ) lm, ε ( ks, ˆ) = E ( k) Y ( sk ˆ; ˆ ) lm, lm lm A(, ss) AY (; skˆ) Y (; skˆ) * ˆˆ = ˆ ˆ l lm lm lm, s s lm lm 3 k

16 lm lm l m l lm lm ik R F ( k) + σ F ( k) = E ( k) lm σ μ μ = + (1 A ) l a s l R = ˆ Y ˆ Y ˆ s= lm sˆ k * (; sk ˆ ) (; sk ˆ )d lm lm l m [ b( m) b ( m) ] = + δ δ δ mm l l = l 1 l+ 1 l = l+ 1 b( m) l = l m 4l 1

17 RTE in the Angular Basis of Rotated Spherical Functions ikr I( k) + D I( k) = E( k) Scalar spectral parameter Block-tridiagonal real symmetric matrix Source term Diagonal matrix Slm, l m = δ ll δmm [ μa + μs(1 Al )] Parameters of the phase function. (For the HG l model, A = g ) l

18 Let D W = SS = S RS 1 1 ikr I( k) + D I( k) = E( k) ikw S I k S E 1 ( + 1) ( ) = ( ) I k S ikw S E ( ) = (1 + ) ( ) k k W ψ = λ ψ μ μ μ I ( k) = μ 1 1 S ψ ψ μ μ S E 1+ ikλ μ ( k)

19 The Spectral Solution ˆ ˆ (,) d σ 1+ ikλ 1 1 lm ψμ ψμ S E( k) Ylm( s; k) ikr 3 I rsˆ = e k lm ε(,) rsˆ = δ( r r) δ( sˆ sˆ ) ikr 0 * ˆ E ˆ lm( k) = e Y ( 3 lm s0; k) ( π ) l μ μ The integral is not easy but doable. Details in J.Phys.A 39, 115 (006)

20 ˆ m * ˆ ˆ ˆ ˆ = lm χll l m 0 m= l, l = m G (,; rsr, s) Y (; sr) ( R) Y ( s; Rˆ) m m ( 1) χll ( R) = ( 1) π σ σ l l l M n 3 M= l n, λ > 0 λn l φ ( M) φ ( M) l ( M ) l l l + j,0 l l + j,0 R ClMl,,, MClml,,, m kl l + j j= 0 λn( M ) l = min( l, l ) μ = ( Mn, ) lm ψ ( ) Mn = δmm l φn M n n r ŝ ŝ 0 R r 0

21 The Plane-Wave Decomposition ˆ ˆ (,) d σ 1+ ikλ 1 1 lm ψμ ψμ S E( k) Ylm( s; k) ikr 3 I rsˆ = e k dq ( 0 ) * ˆ ˆ iq ρ ρ lm ˆ ˆ ˆ lm lm 0 l m 0 lm, l m G (,; rsr, s) = e Y (;) szg (;, qz z ) Y ( s;) zˆ π ( ) dk g q z z ˆ ( ) ˆ 0 = e D q+ zk K q + k D q+ zk π z ikz ( z z0 ) ( ;, ) ( ) ( ) Kw ( ) = lm μ S 1 l ψ μ ψ 1+ iwλ S μ z z z 1 μ μ iϕ J iθ J μ ; D ( kˆ ) = e cos θ = k / q + k k z k z e k y z

22 i( m m ) ϕq l l lm e g ( q; z, z ) = sgn( z z ) d [ iτ ( qλ )] lm Q ( q) = q + 1/ λ μ 0 0 σσ l l μ cos[ τ( )] 1, sin[ τ( )] l+ l + m+ m l [ ] mm1 m = l m = l μλ, > 0 1 Qμ ( q) z z0 e l 1ψ μ ψ μ mm τ λ μ λμqμ( q) lm l m d [ i ( q )] i x = + x i x = ix α D ( αβγ,, ) = e d ( β) e l im l im γ mm mm μ μ

23 Plane waves: Evanescent Waves kr I = e F sˆ k k = k = knˆ k k( ), 1/ λμ, nˆ - real unit vector F ( ˆ) = lm Y ( ˆ; ˆ k s ψ μ lm s k) lm Evanescent waves: k = q± zˆ + q zˆ = k k = i q 1/ λμ, 0, 1/ λμ iq k =± q + 1/ λ z z n

24 kˆ r exp( imϕ k ) l Iˆ = exp Y ( ˆ; ˆ lm ) dmm ( θ ) l φn ( M) Mn sz k k λmn lm σ l (plane wave modes) exp( imϕ ) I = exp i Q ( q) z Y ( sz ˆ; ˆ) ( 1) d ( θ ) l φ ( M) [ q ρ ] ( ± ) q l+ m l qmn Mn lm m, M k n lm σ l (evanescent modes) G V 0 dq ( ) ( ) (,; ˆ 0, ˆ ˆ ˆ 0) I ± rsr s = qmn(,) rs V Mn I q qmn( r0, s0) ( π ) qmn = Q Mn 1 ( q) λ Mn Mn

25 Half-space problem Z>0 Scattering medium Propagationd escribed by the RTE z<0 Nonscattering medium d q I = A ( q) I ( ) Mn π Mn ( + ) qmn

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29 RESULTS: SIMULATIONS

30 A set of 5 point absorbers in an L=6l* slab The field of view is 16l* RTE Diffusion approximation

31 A bar target in the center of the same slab RTE Diffusion approximation

32 RESULTS: EXPERIMENT

33 Two thin vertical wires in a 1cm thick slab filled with intralipid solution (looks like milk) RTE Diffusion approximation

34 CONCLUSIONS The method of rotated reference frames can be used in optical tomography of mesoscopic samples The images are of superior quality compared to those obtained by using the diffusion approximation The quality of images can be comparable to that in X-ray tomography because the RTE retains some information about ballistic rays, single-scattered rays, etc.

35 1. V.A.Markel, Modified spherical harmonics method for solving the radiative transport equation, Letter to the Editor, Waves in Random Media 14(1), L13-L19 (004).. G.Y.Panasyuk, J.C.Schotland, and V.A.Markel, Radiative transport equation in rotated reference frames, Journal of Physics A, 39(1), (006). 3. J.C.Schotland and V.A.Markel, Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation, Inverse Problems and Imaging 1(1), (007). 4. M.Machida, J.C.Schotland and V.A.Markel, MRRF with boundary conditios, under consideration in J.Phys.A 5. M.Machida, J.C.Schotland and V.A.Markel, MRRF with boundary conditios, under consideration in J.Phys.A 6. M.Machida, J.C.Schotland and V.A.Markel, Inverse problem with MRRF, to be submitted to Inv.Prob. in the near future..