Abstract

We consider the inverse problem of reconstructing the absorption and diffusion coefficients of an inhomogeneous highly scattering medium probed by diffuse light. The role of boundary conditions in the derivation of Fourier–Laplace inversion formulas is considered. Boundary conditions of a general mixed type are discussed, with purely absorbing and purely reflecting boundaries obtained as limiting cases. Four different geometries are considered with boundary conditions imposed on a single plane and on two parallel planes and on a cylindrical and on a spherical surface.

© 2002 Optical Society of America

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References

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  1. V. A. Markel, J. C. Schotland, “The inverse problem in optical diffusion tomography. I.  Fourier-Laplace inversion formulas,” J. Opt. Soc. Am. A 18, 1336–1347 (2001).
    [CrossRef]
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vol. 1.
  3. J. Ripoll, M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse–diffuse interfaces,” J. Opt. Soc. Am. A 16, 1947–1957 (1999).
    [CrossRef]
  4. R. Aronson, “Boundary conditions for diffuse light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  5. An explicit characterization of admissible scattering data for which integral equation (38) is solvable is difficult to state. An implicit characterization consists of the closure of the image under the integral operator defined by Eq. (38) of all functions with compact support in L2.
  6. V. A. Markel, J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E 64, R035601 (2001).
    [CrossRef]
  7. J. C. Schotland, V. A. Markel, “Inverse scattering with diffusing waves,” J. Opt. Soc. Am. A 18, 2767–2777 (2001).
    [CrossRef]

2001

1999

1995

Aronson, R.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vol. 1.

Markel, V. A.

Nieto-Vesperinas, M.

Ripoll, J.

Schotland, J. C.

J. Opt. Soc. Am. A

Phys. Rev. E

V. A. Markel, J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E 64, R035601 (2001).
[CrossRef]

Other

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, San Diego, Calif., 1978), Vol. 1.

An explicit characterization of admissible scattering data for which integral equation (38) is solvable is difficult to state. An implicit characterization consists of the closure of the image under the integral operator defined by Eq. (38) of all functions with compact support in L2.

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Figures (2)

Fig. 1
Fig. 1

Schematic illustration of the boundary conditions imposed on the surface S and the local coordinate system used in Section 2.

Fig. 2
Fig. 2

Kernel κA that appears in the transverse parts of integral equations (34) (for the half-space geometry) and (53) (for the slab geometry) as a function of z for (a) purely absorbing and (b) purely reflecting boundary conditions. The kernels are calculated for Q(q1)=Q(q2)=2η. In both cases, sources and detectors are located on the plane z=0.

Equations (122)

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u(r, t)t=·[D(r)u(r, t)]-α(r)u(r, t)+S(r, t),
(2-k2)u(r)=1D0[δα(r)-·δD(r)]u(r)-γ(A)D0S(r),
γ(A)=Aifω>01+Aifω=0,
k2=α0-iωD0.
u(r)=γ(A)G(r, r)S(r)d3r,
r2-k2-1D0δα(r)+1D0r·δD(r)rG(r, r)
=-1D0δ(r-r).
G(r1, r2)=G0(r1, r2)-G0(r1, r)[δα(r)-r·δD(r)r]G(r, r2)d3r-D0S[G(r, r2)G0(r1, r)-G0(r1, r)G(r, r2)]·nˆd2r,
(r2-k2)G0(r, r)=-1D0δ(r-r).
G(r1, r2)=G0(r1, r2)-G0(r1, r)[δα(r)-r·δD(r)r]G(r, r2)d3r.
V(r)δα(r)-r·δD(r)r.
G=G0-G0VG,
G(r1, r2)=G0(r1, r2)-G0(r1, r)V(r)G0(r, r2)d3r.
I(r, sˆ)=14π(cu-3Dsˆ·u),
(u+lnˆ·u)|rS=0.
IS(r)=IS(r, sˆ=nˆ)=c4π1+l*lu(r),
S(r;r1)=γ(A)S0δ(ρ)f(|z-z1|).
0f(x)dx=1,0xf(x)dx=l*.
G(r2, r)S(r; r1)d3r=γ(A)S01+l*lG(r2, r1).
G(r2, r)=G(r2, r1)-|z-z1|nˆ·rG(r2, r)|r=r1+.
G(r2, r)=1+|z-z1|lG(r2, r1).
IS(r2)=cγ(A)S04π1+l*l2G(r1, r2).
ϕ(r1, r2)=1+l*l2[G0(r1, r2)-G(r1, r2)].
ϕ(r1, r2)=1+l*l2G0(r1, r)V(r)G0(r, r2)d3r,
G0(r, r)=d2q(2π)2g(q; z, z)exp[iq·(ρ-ρ)].
2z2-Q2(q)g(q;z, z)=-δ(z-z)D0,
Q(q)(q2+k2)1/2.
g(q; 0, z)-lg(q; 0, z)=0,
g(q; z+, z)-g(q; z-, z)=0,
g(q; z+, z)-g(q; z-, z)=-1/D0,
g(q; , z)<,
g(q; z, z)=12Q(q)D0Q(q)l-1Q(q)l+1exp[-Q(q)|z+z|]+exp[-Q(q)|z-z|].
g(q; z, 0)=g(q; 0, z)lD0q˜(q; z)
g˜(q; z)=exp[-Q(q)|z|]Q(q)l+1.
ϕ(ρ1, ρ2)=l+l*D02d3rd2q1d2q2(2π)4g˜(q1; z)×exp[iq1·(ρ-ρ1)]V(r)g˜(q2; z)×exp[iq2·(ρ2-ρ)].
ϕ(q1, q2)=d2ρ1d2ρ2ϕ(ρ1, ρ2)exp[i(q1·ρ1+q2·ρ2)]
ϕ(q1, q2)=l+l*D02d3rg˜(q1; z)×exp(iq1·ρ)V(r)g˜(q2; z)exp(iq2·ρ).
ϕ(q1, q2)=d3r[κA(q1, q2; z)δα(r)+κD(q1, q2; z)δD(r)]exp[i(q1+q2)·ρ],
κA(q1, q2; z)
=l+l*D02exp{-[Q(q1)+Q(q2)]z}[Q(q1)l+1][Q(q2)l+1],
κD(q1, q2; z)
=l+l*D02
×[Q(q1)Q(q2)-q1·q2]exp{-[Q(q1)+Q(q2)]z}[Q(q1)l+1][Q(q2)l+1],
ψ(q1, q2)=D02[Q(q1)l+1][Q(q2)l+1](l+l*)2ϕ(q1, q2).
ψ(q1, q2)=d3r exp{i(q1+q2)·ρ-[Q(q1)+Q(q2)]z}{δα(r)+[Q(q1)Q(q2)-q1·q2]δD(r)}.
q=q1+q2,
η1=Q(q1),
η2=Q(q2),
d3r exp[iq·ρ-(η1+η2)z]δα(r)
+η1η2-12(q2-η12-η22+2k2)δD(r)=0.
d3r exp[iq·ρ-(η1+η2)z]δα(r)=0,
d3r exp[iq·ρ-(η1+η2)z]δD(r)=0.
ψ(q, η/2, η/2)=d3r exp(iq·ρ-ηz)×δα(r)+12(η2-q2-2k2)δD(r).
δα(r)=1k12-k22d2q(2π)2dη2πiexp(iq·ρ+ηz)×{k12ψk2(q, η/2, η/2)-k22ψk1(q, η/2, η/2)+(η2/2-q2/2)[ψk1(q, η/2, η/2)-ψk2(q, η/2, η/2)]},
δD(r)=1k12-k22d2q(2π)2dη2πiexp(iq·ρ+ηz)×[ψk2(q, η/2, η/2)-ψk1(q, η/2, η/2)],
g(q; L, z)+lg(q; L, z)=0.
g(q; z, z)=[1+(Ql)2]cosh[Q(L-|z-z|)]-[1-(Ql)2]cosh[Q(L-|z+z|)]+2Ql sinh[Q(L-|z-z|)]2D0Q[sinh(QL)+2Ql cosh(QL)+(Ql)2 sinh(QL)].
g(q; z, z)|z=0,L=g(q; z, z)|z=0,LlD0g˜(q; z, z).
g˜(q; z, z)
=sinh[Q(L-|z-z|)]+Ql cosh[Q(L-|z-z|)]sinh(QL)+2Ql cosh(QL)+(Ql)2 sinh(QL).
ϕ(q1, zs, q2, zd)=l+l*D02d3rg˜(q1; zs, z)×exp(iq1·ρ)V(r)g˜(q2; z, zd)×exp(iq2·ρ).
ϕ(q1, zs; q2, zd)=d3r[κA(q1, q2, zs, zd; z)δα(r)+κD(q1, q2, zs, zd; z)δD(r)]×exp[i(q1+q2)·ρ],
κA(q1, q2, zs, zd; z)
=l+l*D02g˜(q1; zs, z)g˜(q2; z, zd),
κD(q1, q2, zs, zd; z)
=l+l*D02g˜(q1; zs, z)zg˜(q2; z, zd)z-q1·q2g˜(q1; zs, z)g˜(q2; z, zd).
κA(q1, q2, zs, zd; z)=l*D02sinh[Q(q1)(L-|z-zs|)]sinh[Q(q2)(L-|zd-z|)]sinh[Q(q1)L]sinh[Q(q2)L],
κD(q1, q2, zs, zd; z)=l*D02Q(q1)Q(q2)Δ(zs,zd)cosh[Q(q1)(L-|z-zs|)]cosh[Q(q2)(L-|zd-z|)]sinh[Q(q1)L]sinh[Q(q2)L]-q1·q2 sinh[Q(q1)(L-|z-zs|)]sinh[Q(q2)(L-|zd-z|)]sinh[Q(q1)L]sinh[Q(q2)L].
κA(q1, q2, zs, zd; z)=cosh[Q(q1)(L-|z-zs|)]cosh[Q(q2)(L-|zd-z|)]D02Q(q1)Q(q2)sinh[Q(q1)L]sinh[Q(q2)L],
κD(q1, q2, zs, zd; z)=1D02Δ(zs, zd)sinh[Q(q1)(L-|z-zs|)]sinh[Q(q2)(L-|zd-z|)]sinh[Q(q1)L]sinh[Q(q2)L]-q1·q2 cosh[Q(q1)(L-|z-zs|)]cosh[Q(q2)(L-|zd-z|)]Q(q1)Q(q2)sinh[Q(q1)L]sinh[Q(q2)L].
κA(q, q, 0, 0; z)+a(q)κA(q, q, 0, L; z)
+b(q)κA(q, q, L, L; z)=c(q)exp(-2Qz),
κD(q, q, 0, 0; z)+a(q)κD(q, q, 0, L; z)
+b(q)κD(q, q, L, L; z)=[Q2(q)-q1·q2]c(q)exp(-2Qz),
a(q)=-2h,b(q)=h2,
c(q)=l+l*D0(Ql+1)2,
hQl-1Ql+1exp(-QL).
ψ(q1, q2)=1c(q)[ϕ(q1, 0; q2, 0)+a(q)ϕ(q1, 0; q2, L)+b(q)ϕ(q1, L; q2, L)]|q1=q2=q,
ψ(q1, q2)=d3r{δα(r)+[Q2(q)-q1·q2]δD(r)}×exp[i(q1+q2)·ρ-2Q(q)z].
a(q)=2 exp[-Q(q)L],
b(q)=exp[-2Q(q)L],
c(q)=(l*/D0)2.
a(q)=-2 exp[-Q(q)L],
b(q)=exp[-2Q(q)L],
c(q)=[1/D0Q(q)]2.
G0(r, r)=12πm=-dq2πexp[im(φ-φ)]×exp[iq(z-z)]g(m, q; ρ, ρ),
1ρρρρ-m2ρ2-Q2(q)g(m, q; ρ, ρ)=-δ(ρ-ρ)D0ρ.
g(m, q; 0, ρ)<,
g(m, q; ρ+, ρ)-g(m, q; ρ-, ρ)=0,
g(m, q; ρ+, ρ)-g(m, q; ρ-, ρ)=-1/D0ρ,
g(m, q; R, ρ)+lg(m, q; R, ρ)=0.
g(m, q; ρ, ρ)
=1D0Km(Qρ>)Im(Qρ<)
-Km(QR)+QlKm(QR)Im(QR)+QlIm(QR)Im(Qρ)Im(Qρ),
g(m, q; ρ, R)=g(m, q; R, ρ)lD0g˜(m, q; ρ)
g˜(m, q; ρ)=1RIm(Qρ)Im(QR)+QlIm(QR),
Km(x)Im(x)-Km(x)Im(x)=1/x.
ϕ(m1, q1, φ˜1; m2, q2, φ˜2)
=-dz1dz202πdφ1dφ2ϕ(φ1, z1; φ2, z2)exp{i[q1z1+q2z2+m1(φ1-φ˜1)+m2(φ2-φ˜2)]},
ϕ(m1, q1, φ˜1; m2, q2, φ˜2)
=l+l*D02d3r exp{i[m1(φ-φ˜1)+q1z]}g˜(m1, q1; ρ)Vexp{i[m2(φ-φ˜2)+q2z]}g˜(m2, q2; ρ).
ψ(q1, φ˜1; q2, φ˜2)=D0Rl+l*2m1,m2=-{Im1[Q(q1)R]+QlIm1[Q(q1)R]}{Im2[Q(q2)R]+QlIm2[Q(q2)R]}×ϕ(m1, q1, φ˜1; m2, q2, φ˜2)
m=-Im(w)exp(imθ)=exp(w cos θ)
ψ(q1, eˆ1; q2, eˆ2)=d3r exp[Q(q1)ρ·eˆ1+iq1z]V×exp[Q(q2)ρ·eˆ2+iq2z].
ψ(q1, eˆ1; q2, eˆ2)=d3r{δα(r)+[Q(q1)Q(q2)eˆ1·eˆ2-q1q2]δD(r)}exp{[Q(q1)eˆ1+Q(q2)eˆ2]·ρ+i(q1+q2)z}.
G0(r, r)=l=0 m=-llg(l; r, r)Ylm(rˆ)Ylm*(rˆ),
1r2rr2r-l(l+1)r2-k2g(l; r, r)=-δ(r-r)D0r2.
g(l; 0, r)<,
g(l; r+, r)-g(l; r-, r)=0,
g(l; r+,r)-g(l; r-, r)=-1/D0(r)2,
g(l; R, r)+lg(l; R, r)=0.
g(l; r, r)=2kπD0il(kr<)kl(kr>)-kl(kR)+klkl(kR)il(kR)+klil(kR)il(kr)il(kr),
g(l; r, R)=g(l; R, r)lD0g˜(l; r),
g˜(l; r)=1R2il(kr)il(kR)+klil(kR),
kl(x)il(x)-kl(x)il(x)=π2x2.
ϕ(rˆ1, rˆ2)=l+l*D02l1,l2m1,m2d3rg˜(l1; r)Yl1m1(rˆ1)×Yl1m1*(rˆ)Vg˜(l2; r)Yl2m2(rˆ)Yl2m2*(rˆ2).
ϕ(l1, m1; l2, m2)
=ϕ(rˆ1, rˆ2)Yl1m1*(rˆ1)Yl2m2(rˆ2)d2rˆ1d2rˆ2.
ϕ(l1, m1; l2, m2)=l+l*D02d3rg˜(l1; r)×Yl1m1*(rˆ)Vg˜(l2; r)Yl2m2(rˆ).
exp(a·b)=4πl=0m=-llil(ab)Ylm*(aˆ)Ylm(bˆ)
ψ(eˆ1, eˆ2)=(4π)2R4l1,l2 m1,m2[il1(kR)+klil1(kR)]×[il2(kR)+klil2(kR)]×Yl1m1(eˆ1)Yl2m2(eˆ2)ϕ(l1, m1; l2, m2)
ψ(eˆ1, eˆ2)=d3r exp(keˆ1·r)V exp(keˆ2·r)=d3r[δα(r)+k2eˆ1·eˆ2δD(r)]×exp[k(eˆ1+eˆ2)·r],

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