Optical image reconstruction based on the third-order diffusion equations

Huabei Jiang

doi:10.1364/OE.4.000241

Optics Express
Vol. 4,
Issue 8,
pp. 241-246
(1999)
•https://doi.org/10.1364/OE.4.000241

Optical image reconstruction based on the third-order diffusion equations

Huabei Jiang

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Huabei Jiang, "Optical image reconstruction based on the third-order diffusion equations," Opt. Express 4, 241-246 (1999)

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Abstract

This paper presents a third-order diffusion equations-based optical image reconstruction algorithm. The algorithm has been implemented using finite element discretizations coupled with a hybrid regularization that combines both Marquardt and Tikhonov schemes. Numerical examples are used to compare between the third- and first-order reconstructions. The results show that the third-order reconstruction codes are more stable than the first-order codes, and are capable of reconstructing void-like regions. From the examples given, it has also been shown that the first-order codes fail to both qualitatively and quantitatively reconstruct the void-like regions.

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Fig. 1. (a) Geometry of the test case under study; (b) reconstructed D image for the first test case; (c) reconstructed μ_a image for the first test case.

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Fig. 2. (a) Recovered D image for the second case using the third-order codes; (b) recovered μ_a image for the second case using the third-order codes; (c) recovered D image for the second case using the first-order codes; (b) recovered μ_a image for the second case using the firs-order codes;

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\nabla \cdot D (r) \nabla Φ^{(1)} (r) - μ_{a} (r) Φ^{(1)} (r) - \nabla \cdot D (r) \nabla Φ^{(2)} (r) + 6 \nabla \cdot D (r) {\nabla_{1} Φ}^{(3)} (r) + 6 \nabla \cdot D (r) {\nabla_{2} Φ}^{(4)} (r) = - S (r)

- \nabla \cdot D (r) \nabla Φ^{(1)} (r) + \frac{25}{7} \nabla \cdot D (r) {\nabla Φ}^{(2)} (r) - 5 {μ'}_{t} (r) Φ^{(2)} (r) - \frac{60}{7} \nabla \cdot D (r) {\nabla_{1} Φ}^{(3)} (r) - \frac{60}{7} \nabla \cdot D (r) {\nabla_{2} Φ}^{(4)} (r) = 0

\nabla \cdot D (r) \nabla_{1} Φ^{(1)} (r) - \frac{10}{7} \nabla \cdot D (r) {\nabla_{1} Φ}^{(2)} (r) + \frac{90}{7} \nabla \cdot D (r) {\nabla Φ}^{(3)} (r) - 10 {μ'}_{t} (r) Φ^{(3)} (r) = 0

\frac{1}{2} \nabla \cdot D (r) \nabla_{2} Φ^{(1)} (r) - \frac{5}{7} \nabla \cdot D (r) {\nabla_{2} Φ}^{(2)} (r) + \frac{45}{7} \nabla \cdot D (r) {\nabla Φ}^{(4)} (r) - 5 {μ'}_{t} (r) Φ^{(4)} (r) = 0

\sum_{j = 1}^{N} {Φ_{j}^{(1)} [〈 - \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla ϕ_{j} \cdot \nabla ϕ_{i} 〉 - 〈 \sum_{q = 1}^{Q} μ_{p} ϕ_{p} ϕ_{j} ϕ_{i} 〉] + Φ_{j}^{(2)} 〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla ϕ_{j} \cdot \nabla ϕ_{i} 〉 - {6 Φ}_{j}^{(3)} 〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla_{1} ϕ_{j} \cdot \nabla ϕ_{i} 〉 - 6 Φ_{j}^{(4)} 〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla_{2} ϕ_{j} \cdot \nabla ϕ_{i} 〉}

= - 〈 S ϕ_{i} 〉 - \oint D \hat{n} \cdot \nabla Φ^{(1)} ϕ_{i} ds + \oint D \hat{n} \cdot \nabla Φ^{(2)} ϕ_{i} ds - 6 \oint D \hat{n} \cdot \nabla_{1} Φ^{(3)} ϕ_{i} ds - 6 \oint D \hat{n} \cdot \nabla_{2} Φ^{(4)} ϕ_{i} ds

\sum_{j = 1}^{N} {Φ_{j}^{(1)} 〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla ϕ_{j} \cdot \nabla ϕ_{i} 〉 - \frac{25}{7} Φ_{j}^{(2)} [〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla ϕ_{j} \cdot \nabla ϕ_{i} 〉 + \frac{5}{3} 〈 \sum_{p = 1}^{Q} D_{p}^{- 1} ϕ_{q} ϕ_{j} ϕ_{i} 〉] + \frac{60}{7} Φ_{j}^{(3)} 〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla_{1} ϕ_{j} \cdot \nabla ϕ_{i} 〉 - \frac{60}{7} Φ_{j}^{(4)} 〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla_{2} ϕ_{j} \cdot \nabla ϕ_{i} 〉}

= \oint D \hat{n} \cdot \nabla Φ^{(1)} ϕ_{i} ds + \frac{25}{7} \oint D \hat{n} \cdot \nabla Φ^{(2)} ϕ_{i} ds - \frac{60}{7} \oint D \hat{n} \cdot \nabla_{1} Φ^{(3)} ϕ_{i} ds - \frac{60}{7} \oint D \hat{n} \cdot \nabla_{2} Φ^{(4)} ϕ_{i} ds

\sum_{j = 1}^{N} {Φ_{j}^{(1)} 〈 - \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla_{1} ϕ_{j} \cdot \nabla ϕ_{i} 〉 + \frac{10}{7} Φ_{j}^{(2)} 〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla_{1} ϕ_{j} \cdot \nabla ϕ_{i} 〉 - \frac{90}{7} Φ_{j}^{(3)} [〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla ϕ_{j} \cdot \nabla ϕ_{i} 〉 + \frac{10}{3} 〈 \sum_{p = 1}^{P} D_{p}^{- 1} ϕ_{p} ϕ_{j} ϕ_{i} 〉]}

= - \oint D \hat{n} \cdot \nabla_{1} Φ^{(1)} ϕ_{i} ds + \frac{10}{7} \oint D \hat{n} \cdot \nabla_{1} Φ^{(2)} ϕ_{i} ds - \frac{90}{7} \oint D \hat{n} \cdot \nabla Φ^{(3)} ϕ_{i} ds

\sum_{j = 1}^{N} {Φ_{j}^{(1)} 〈 - \frac{1}{2} \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla_{2} ϕ_{j} \cdot \nabla ϕ_{i} 〉 + \frac{5}{7} Φ_{j}^{(2)} 〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla_{2} ϕ_{j} \cdot \nabla ϕ_{i} 〉 - \frac{45}{7} Φ_{j}^{(4)} [〈 \sum_{p = 1}^{P} D_{p} ϕ_{p} \nabla ϕ_{j} \cdot \nabla ϕ_{i} 〉 + \frac{5}{3} 〈 \sum_{p = 1}^{P} D_{p}^{- 1} ϕ_{p} ϕ_{j} ϕ_{i} 〉]}

= - \frac{1}{2} \oint D \hat{n} \cdot \nabla_{2} Φ^{(1)} ϕ_{i} ds + \frac{5}{7} \oint D \hat{n} \cdot \nabla_{2} Φ^{(2)} ϕ_{i} ds - \frac{45}{7} \oint D \hat{n} \cdot \nabla Φ^{(4)} ϕ_{i} ds

(ℑ^{T} ℑ + λI) Δχ = ℑ^{T} (Φ^{o} - Φ^{c})

ℑ = [\begin{matrix} \frac{\partial Φ_{1}^{(1)}}{\partial D_{1}} & \frac{\partial Φ_{1}^{(1)}}{\partial D_{2}} & \dots & \frac{\partial Φ_{1}^{(1)}}{\partial D_{N}} & \frac{\partial Φ_{1}^{(1)}}{\partial μ_{a, 1}} & \frac{\partial Φ_{1}^{(1)}}{\partial μ_{a, 2}} & \dots & \frac{\partial Φ_{1}^{(1)}}{\partial μ_{a, N}} \\ \frac{\partial Φ_{2}^{(1)}}{\partial D_{1}} & \frac{\partial Φ_{2}^{(1)}}{\partial D_{2}} & \dots & \frac{\partial Φ_{2}^{(1)}}{\partial D_{N}} & \frac{\partial Φ_{2}^{(1)}}{\partial μ_{a, 1}} & \frac{\partial Φ_{2}^{(1)}}{\partial μ_{a, 2}} & \dots & \frac{\partial Φ_{2}^{(1)}}{\partial μ_{a, N}} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ \frac{\partial Φ_{M}^{(1)}}{\partial D_{1}} & \frac{\partial Φ_{M}^{(1)}}{\partial D_{2}} & \dots & \frac{\partial Φ_{M}^{(1)}}{\partial D_{N}} & \frac{\partial Φ_{M}^{(1)}}{\partial μ_{a, 1}} & \frac{\partial Φ_{M}^{(1)}}{\partial μ_{a, 2}} & \dots & \frac{\partial Φ_{M}^{(1)}}{\partial μ_{a, N}} \end{matrix}]

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