A gradient-based optimisation scheme for optical tomography

Simon R. Arridge; Martin Schweiger

doi:10.1364/OE.2.000213

Optics Express
Vol. 2,
Issue 6,
pp. 213-226
(1998)
•https://doi.org/10.1364/OE.2.000213

A gradient-based optimisation scheme for optical tomography

Simon R. Arridge and Martin Schweiger

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Simon R. Arridge and Martin Schweiger, "A gradient-based optimisation scheme for optical tomography," Opt. Express 2, 213-226 (1998)

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Abstract

Optical tomography schemes using non-linear optimisation are usually based on a Newton-like method involving the construction and inversion of a large Jacobian matrix. Although such matrices can be efficiently constructed using a reciprocity principle, their inversion is still computationally difficult. In this paper we demonstrate a simple means to obtain the gradient of the objective function directly, leading to straightforward application of gradient-based optimisation methods.

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Supplementary Material (8)

Media 1: MOV (415 KB)

Media 2: MOV (611 KB)

Media 3: MOV (279 KB)

Media 4: MOV (602 KB)

Media 5: MOV (744 KB)

Media 6: MOV (485 KB)

Media 7: MOV (880 KB)

Media 8: MOV (436 KB)

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Figure 1. Target image (col. 1), reconstruction after 100 iterations with conjugate gradient method (col. 2), with steepest descent method (col. 3) and with ART method (col. 4). In all cases, the top image is μ_a , and the bottom image is μ_s . The animations linked to the figure show the first 50 reconstruction iterations, and the final 50 in steps of 10. [Media 1] [Media 2] [Media 3] [Media 4] [Media 5] [Media 6]

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Figure 2. L₂ data norms as a function of iteration number, for different algorithms, as a function of iteration number (left) and of runtime (right).

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Figure 3. L₂ solution norms as a function of iteration number, for different algorithms. Left: absorption, right: scatter.

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Figure 4. Target image (left col.), reconstruction after 100 iterations with conjugate gradient method (right col.). Top row is μ_a , bottom row is μ_s . The animations linked to the figure show the first 50 reconstruction iterations, and the final 50 in steps of 10. [Media 7] [Media 8]

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Tables (2)

Table 1. Optical parameters of circular test object.

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Table 2. Optical parameters of neonatal head model.

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Equations (44)

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\vec{F} = \vec{𝛲} [μ, κ]

Ψ = \frac{1}{2} \sum_{j = 1}^{S} \sum_{i = 1}^{M_{j}} {(\frac{y_{j, i} - 𝛲_{j, i} [μ, κ]}{σ_{j, i}})}^{2}

Ψ = \frac{1}{2} {(\vec{y} - \vec{F})}^{T} R^{- 2} (\vec{y} - \vec{F}) = \frac{1}{2} {\vec{b}}^{T} \vec{b}

R = diag (R_{1}, R_{2}, \dots R_{S}) = diag (σ_{1,1}, σ_{1,2}, \dots σ_{j, i}, \dots σ_{S, M_{s}})

- \nabla ∙ κ (\vec{r}) \nabla \hat{Φ} (\vec{r}, ω) + μ (\vec{r}) \hat{Φ} (\vec{r}, ω) + \frac{ιω}{c} \hat{Φ} (\vec{r}, ω) = {\hat{q}}_{0} (\vec{r}, t),

- \nabla ∙ κ (\vec{r}) \nabla Φ (\vec{r}, t) + μ (\vec{r}) Φ (\vec{r}, t) + \frac{1}{c} \frac{\partial Φ (\vec{r}, t)}{\partial t} = q_{0} (\vec{r}, t),

Γ (ξ) = - cκ (ξ) \vec{\hat{n}} ∙ \nabla Φ (ξ),

Φ (ξ) + \frac{κ}{α} \vec{\hat{n}} ∙ \nabla Φ (ξ) = 0,

(K (κ) + C (μ) + αA + ιω B) \vec{Φ} (ω) = \vec{Q} (ω)

(K (κ) + C (μ) + αA) \vec{Φ} (t) + B \frac{\partial \vec{Φ} (t)}{\partial t} = \vec{Q} (t)

K_{ij} = \int_{Ω} κ (\vec{r}) \nabla u_{i} (\vec{r}) ∙ \nabla u_{j} (\vec{r}) d^{n} \vec{r}

C_{ij} = \int_{Ω} μ (\vec{r}) u_{i} (\vec{r}) u_{j} (\vec{r}) d^{n} \vec{r}

B_{ij} = {\frac{1}{c} \int}_{Ω} u_{i} (\vec{r}) u_{j} (\vec{r}) d^{n} \vec{r}

A_{ij} = \int_{\partial Ω} u_{i} (\vec{r}) u_{j} (\vec{r}) d (\partial Ω)

{\vec{F}}_{j} = 𝚳 [{\vec{Φ}}_{j}]

\overset{̅}{𝚳} = \frac{τ}{ε}

time - gated intensity : 𝚳_{\bar{E} (T)} [Φ (t)] = \frac{1}{ε [Φ (t)]} \int_{0}^{T} B [Φ (t)] dt,

n - th temporal moment : 𝚳_{〈 t^{n} 〉} [Φ (t)] = \frac{1}{ε [Φ (t)]} \int_{0}^{\infty} t^{n} B [Φ (t)] dt,

n - th central moment : 𝚳_{c_{n}} [Φ (t)] = \frac{1}{ε [Φ (t)]} \int_{0}^{\infty} {(t - 〈 t 〉)}^{n} B [Φ (t)] dt,

normalised Laplace transform : 𝚳_{\overset{̅}{L} (s)} [Φ (t)] = \frac{1}{ε [Φ (t)]} \int_{0}^{\infty} e^{- st} B [Φ (t)] dt .

\frac{\partial Ψ}{\partial x_{k}} = \sum_{j = 1}^{S} \sum_{i = 1}^{M_{j}} (\frac{y_{j, i} - P_{j, i [μ, κ]}}{σ_{j, i}^{2}}) (- \frac{\partial P_{j, i} [μ, κ]}{\partial x_{k}})

\vec{z} = - \sum_{j = 1}^{S} {P'}_{j}^{T} R_{j}^{- 1} {\vec{b}}_{j} = - {\vec{P}'}^{T} R^{- 1} \vec{b}

= - \sum_{j = 1}^{S} {J_{j}}^{T} {\vec{b}}_{j} = - J^{T} \vec{b}

∣ \begin{matrix} {\vec{b}}_{1} \\ {\vec{b}}_{2} \\ . \\ . \\ . \\ {\vec{b}}_{S} \end{matrix} ∣ = ∣ \begin{matrix} J_{1, (μ)} & J_{1, (κ)} \\ J_{2, (μ)} & J_{2, (κ)} \\ . & . \\ . & . \\ . & . \\ J_{S, (μ)} & J_{S, (κ)} \end{matrix} ∣ ∣ \begin{matrix} Δ μ \\ Δ κ \end{matrix} ∣

J_{j_{i}} = {P \vec{MD} F}_{(j, i)}^{T} = [{P \vec{MD} F}_{(j, i), (μ)}^{T} {P \vec{MD} F}_{(j, i), (κ)}^{T}]

\vec{z} = - J^{T} \vec{b}

= - \sum_{j = 1}^{S} J_{j}^{T} {\vec{b}}_{j}

= - \sum_{j = 1}^{S} \sum_{j = 1}^{M_{j}} {P \vec{MD} F}_{(j, i)}^{T} b_{j, i}

(K + C + αA - ι ω B) {\vec{Φ}}_{m}^{+} (ω) = {\vec{Q}}_{i}^{+} (ω)

{\vec{z}}_{(μ)} = - \sum_{j = 1}^{S} \sum_{i = 1}^{M_{j}} \frac{b_{j, i}}{σ_{j, i}} {\vec{Φ}}_{i}^{+} (ω) \times {\vec{Φ}}_{j} (ω)

{\vec{z}}_{(κ)} = - \sum_{j = 1}^{S} \sum_{i = 1}^{M_{j}} \frac{b_{j, i}}{σ_{j, i}} \nabla {\vec{Φ}}_{i}^{+} (ω) \times \nabla {\vec{Φ}}_{j} (ω)

{\vec{ν}}_{j}^{+} (ω) = \sum_{i = 1}^{M_{j}} \frac{b_{j, i}}{σ_{j, i}} {\vec{Q}}_{i}^{+} (ω)

(K + C + αA - ι ω B) {\vec{η}}_{j}^{+} (ω) = {\vec{ν}}_{j}^{+} (ω)

{\vec{z}}_{(μ)} = - \sum_{j = 1}^{S} {\vec{η}}_{j}^{+} (ω) \times {\vec{Φ}}_{j} (ω)

{\vec{z}}_{(κ)} = - \sum_{j = 1}^{S} \nabla {\vec{η}}_{j}^{+} (ω) \times \nabla {\vec{Φ}}_{j} (ω)

J_{j_{i}}^{\overset{̅}{𝚳}} = \frac{1}{F_{j, i}^{ε}} (J_{j_{i}}^{𝚳} - F_{j, i}^{\overset{̅}{𝚳}} J_{j_{i}}^{ε})

J^{{\overset{̅}{𝚳}}^{T}} {\vec{b}}^{\overset{̅}{𝚳}} = \sum_{j}^{S} \sum_{i}^{M_{j}} J_{j_{i}}^{\overset{̅}{𝚳}} b_{j, i}^{\overset{̅}{𝚳}}

= \sum_{j}^{S} \sum_{i}^{M_{j}} \frac{1}{F_{j, i}^{ε}} (J_{j_{i}}^{𝚳} - F_{j, i}^{\overset{̅}{𝚳}} J_{j_{i}}^{ε}) b_{j, i}^{\overset{̅}{𝚳}}

{\vec{z}}_{(μ)}^{\overset{̅}{𝚳}} = \sum_{j}^{S} \sum_{i}^{M_{j}} \frac{b_{j, i}^{\overset{̅}{𝚳}}}{σ_{j, i}^{\overset{̅}{𝚳}} F_{j, i}^{ε}} (τ [{\vec{Φ}}_{i}^{+} \times {\vec{Φ}}_{j}] - F_{j, i}^{\overset{̅}{𝚳}} {\vec{Φ}}_{i}^{+} \times {\vec{Φ}}_{j})

{\vec{z}}_{(κ)}^{\overset{̅}{𝚳}} = \sum_{j}^{S} \sum_{i}^{M_{j}} \frac{b_{j, i}^{\overset{̅}{𝚳}}}{σ_{j, i}^{\overset{̅}{𝚳}} F_{j, i}^{ε}} (τ [\nabla {\vec{Φ}}_{i}^{+} \times \nabla {\vec{Φ}}_{j}] - F_{j, i}^{\overset{̅}{𝚳}} {\nabla \vec{Φ}}_{i}^{+} \times \nabla {\vec{Φ}}_{j})

{\vec{ν}}_{j}^{{\overset{̅}{𝚳}}^{+}} (0) = \sum_{i = 1}^{M_{j}} \frac{b_{j, i}^{\overset{̅}{𝚳}} F_{j, i}^{\overset{̅}{𝚳}}}{σ_{j, i}^{\bar{𝚳}} F_{j, i}^{ε}} {\vec{Q}}_{i}^{+}

{\vec{ν}}_{j}^{{\overset{̅}{𝚳}}^{+}} (1) = \sum_{i = 1}^{M_{j}} \frac{b_{j, i}^{\overset{̅}{𝚳}}}{σ_{j, i}^{\bar{𝚳}} F_{j, i}^{ε}} {\vec{Q}}_{i}^{+}

{\vec{z}}_{(μ)}^{\overset{̅}{𝚳}} = - \sum_{j}^{S} τ [{\vec{η}}_{j}^{{\overset{̅}{𝚳}}^{+}} (1) \times {\vec{Φ}}_{j}] - {\vec{η}}_{j}^{{\bar{𝚳}}^{+}} (0) \times {\vec{Φ}}_{j}

{\vec{z}}_{(κ)}^{\overset{̅}{𝚳}} = - \sum_{j}^{S} τ [\nabla {\vec{η}}_{j}^{{\overset{̅}{𝚳}}^{+}} (1) \times \nabla {\vec{Φ}}_{j}] - \nabla {\vec{η}}_{j}^{{\bar{𝚳}}^{+}} (0) \times \nabla {\vec{Φ}}_{j}

		μ_a [mm^-1]	μ_s [mm^-1]	n
background		0.025	2	1.4
	top	0.05	2	1.4
objects	bottom right	0.025	4	1.4
	bottom left	0.05	4	1.4

		μ_a [mm^-1]	μ_s [mm^-1]	n
normal	skin	0.022	1	1.4
tissue	bone	0.025	2	1.4
	grey matter	0.02	0.55	1.4
	white matter	0.01	2	1.4
lesions	centre	0.1	0.55	1.4
	centre left	0.05	2	1.4
	top left	0.02	6	1.4
	top right	0.05	4	1.4
	bottom right	0.05	0.55	1.4
	bottom left	0.01	4	1.4

		μ_a [mm^-1]	μ_s [mm^-1]	n
background		0.025	2	1.4
	top	0.05	2	1.4
objects	bottom right	0.025	4	1.4
	bottom left	0.05	4	1.4

		μ_a [mm^-1]	μ_s [mm^-1]	n
normal	skin	0.022	1	1.4
tissue	bone	0.025	2	1.4
	grey matter	0.02	0.55	1.4
	white matter	0.01	2	1.4
lesions	centre	0.1	0.55	1.4
	centre left	0.05	2	1.4
	top left	0.02	6	1.4
	top right	0.05	4	1.4
	bottom right	0.05	0.55	1.4
	bottom left	0.01	4	1.4

Abstract

Supplementary Material (8)

Cited By

Figures (4)

Tables (2)

Equations (44)

Optics Express