3D optical tomography in the presence of void regions

J Riley; H Dehghani; M Schweiger; S R Arridge; J Ripoll; M Nieto-Vesperinas

doi:10.1364/OE.7.000462

Optics Express
Vol. 7,
Issue 13,
pp. 462-467
(2000)
•https://doi.org/10.1364/OE.7.000462

3D optical tomography in the presence of void regions

J Riley, H Dehghani, M Schweiger, S R Arridge, J Ripoll, and M Nieto-Vesperinas

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J Riley, H Dehghani, M Schweiger, S R Arridge, J Ripoll, and M Nieto-Vesperinas, "3D optical tomography in the presence of void regions," Opt. Express 7, 462-467 (2000)

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Abstract

We present an investigation of the effect of a 3D non-scattering gap region on image reconstruction in diffuse optical tomography. The void gap is modelled by the Radiosity-Diffusion method and the inverse problem is solved using the adjoint field method. The case of a sphere with concentric spherical gap is used as an example.

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Media 2: MOV (3760 KB)

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Fig. 1. Left : cutaway of spherical mesh. Right : location of sources and detectors on the sphere surface

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Fig. 2. distribution of photon density, photons/mm³ (top row) and mean time of photon flight, picoseconds (bottom row) over sphere surface. Left to right, solid sphere (no gap), gap widths 3mm, 4mm, 5mm.

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Fig. 3. Sensitivity functions for the 3mm gap case. Left intensity (photons/mm²), right mean time(picoseconds mm). The functions plotted are cross-sections through the equatorial plane of the sphere. Also available as a QuickTime movie, pmdf.mov. (3.8MB)

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Fig. 4. Target images (top row) and reconstructions (bottom row) for the 3mm gap case. The images are transverse, sagittal and coronal slices through the true centre of the blob, orientated according to the diagram in the top right panel. Bottom right shows a profile along the equatorial diameter through the blob centre. A movie showing a rotating orthographic view is attached rotating3d.mov (3.8MB).

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Equations (14)

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- \nabla \cdot κ (r) \nabla Φ (r; ω) + (µ_{a} (r) + \frac{i ω}{c}) Φ (r; ω) = 0 r ∊ Ω^{d} ∖ (\partial Ξ^{d} \cup \partial Ω^{d})

Φ (m; ω) + 2 A κ (m) \frac{\partial Φ (m; ω)}{\partial ν} = η (m; ω) m ∊ \partial Ω_{1}^{+}

Φ (m; ω) + 2 A κ \frac{\partial Φ (m; ω)}{\partial ν} = \frac{1}{π} \int_{\partial Ξ} \cos θ \cos θ^{'} \frac{Φ (m^{'}; ω)}{2 A} h (m, m^{'}) \times

\frac{\exp [- (μ_{a} + i \frac{ω}{c}) ∣ m - m^{'} ∣]}{{∣ m - m^{'} ∣}^{2}} d m^{'} m, m^{'} ∊ \partial Ξ

\cos θ = \hat{ν} (m) \cdot \frac{m - m^{'}}{∣ m - m^{'} ∣}, \cos θ^{'} = \hat{ν} (m^{'}) \cdot \frac{m - m^{'}}{∣ m - m^{'} ∣}

y_{η} (m; ω) = 𝓟_{η} (\begin{matrix} μ_{a} \\ κ \end{matrix}) ≔ - κ (m) \frac{\partial Φ_{η} (m; ω)}{\partial ν}, m ∊ \partial Ω_{1}^{+}

C = \frac{1}{2} \sum_{j = 1}^{s} {〈 g_{j} - 𝓟_{j} (\binom{μ_{a}}{κ}), g_{j} - 𝓟_{j} (\begin{matrix} μ_{a} \\ κ \end{matrix}) 〉}_{L 2 (\partial Ω)}

(\begin{matrix} α \\ β \end{matrix}) = (\begin{matrix} - Re (\bar{Φ_{j}} Ψ_{j}) \\ - Re (\nabla \bar{Φ_{j}} \cdot Ψ_{j}) \end{matrix})

- \nabla \cdot κ (r) \nabla Ψ_{j} (r; ω) + (μ_{a} (r) - \frac{i ω}{c}) Ψ_{j} (r; ω) = 0 r ∊ Ω^{d} \ (\partial Ξ^{d} \cup \partial Ω^{d})

Ψ_{j} (m; ω) + 2 A κ (m) \frac{\partial Ψ_{j} (m; ω)}{\partial ν} = g_{j} (m; ω) - 𝓟_{j} (\begin{matrix} μ_{a} \\ κ \end{matrix}) m ∊ \partial Ω_{1}^{+}

Ψ_{j} (m; ω) + 2 A κ \frac{\partial Ψ_{j} (m; ω)}{\partial ν} = \frac{1}{π} \int_{\partial Ξ} \cos θ \cos θ' \frac{Ψ_{j} (m^{'}; ω)}{2 A} h (m, m^{'}) x

\frac{\exp [- (μ_{a} - i \frac{ω}{c}) ∣ m - m^{'} ∣]}{{∣ m - m^{'} ∣}^{2}} d m^{'} m, m^{'} ∊ \partial Ξ

f (m, m^{'}) = \frac{\cos θ (m) \cos θ (m^{'})}{π {∣ m - m^{'} ∣}^{2}}

f_{τ_{α}, τ_{α'}} (m, m^{'}) = \sum_{k = 1}^{N} \sum_{k^{'} = 1}^{N^{'}} u_{n (k)} (m) u_{n (k^{'})} (m^{'}) \frac{\cos θ_{n (k)} \cos θ_{n (k^{'})}}{π {∣ N_{n (k)} - N_{n (k^{'})} ∣}^{2}}

Abstract

Supplementary Material (2)

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Equations (14)

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