# Stochastic Proximal Gradient Algorithms for Multi-Source Quantitative Photoacoustic Tomography

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## Abstract

**:**

## 1. Introduction

#### 1.1. Multi-Source QPAT

#### 1.2. Stochastic Proximal Gradient Algorithms

#### 1.3. Outline

## 2. The Forward Problem in QPAT

#### 2.1. Mathematical Notation

#### 2.2. The Radiative Transfer Equation

**Theorem**

**1**(Well-posedness of the RTE).

**Proof.**

**Definition**

**1**(Solution operator for the RTE).

#### 2.3. Heating Operator

**Definition**

**2**(Heating operator).

#### 2.4. The Wave Equation

**Lemma**

**1.**

**Definition**

**3.**

#### 2.5. Analysis of the Forward Problem in Multi-Source QPAT

**Definition**

**4.**

**Theorem**

**2**(Continuity and Differentiability).

- (1)
- The operators ${\mathbf{T}}_{i}$, ${\mathbf{F}}_{i}$ and ${\mathbf{H}}_{i}$ are sequentially continuous and Lipschitz-continuous.
- (2)
- For every $\mu \in \mathbb{D}\left(\mathbf{T}\right)$, the one-sided directional derivatives ${\mathbf{T}}_{i}^{\prime}\left(\mu \right)\left(h\right)$, ${\mathbf{F}}_{i}^{\prime}\left(\mu \right)\left(h\right)$ of ${\mathbf{T}}_{i}$, ${\mathbf{F}}_{i}$ at μ in any feasible direction h exist, and are given by$$\begin{array}{cc}{\mathbf{T}}_{i}^{\prime}\left(\mu \right)\left(h\right)\hfill & =\mathbf{T}\left(0,-({h}_{a}+{h}_{s}-{h}_{s}\mathbf{K})\mathbf{T}\left(\mu \right),\mu \right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathbf{F}}_{i}^{\prime}\left(\mu \right)\left(h\right)& ={\mathbf{U}}_{{\mathrm{\Lambda}}_{i},{T}_{i}}\left({h}_{a}\mathbf{A}{\mathbf{T}}_{i}\left(\mu \right)+{\mu}_{a}\mathbf{A}\left({\mathbf{T}}_{i}^{\prime}\left(\mu \right)\left(h\right)\right)\right)\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$

**Proof.**

**Theorem**

**3**(Adjoint of ${\mathbf{F}}_{i}^{\prime}\left(\mu \right)$).

**Proof.**

**Theorem**

**4**(Lipschitz continuity of ∇F

_{I})

**Proof.**

## 3. The Stochastic Proximal Gradient Method for QPAT

#### 3.1. Formulation of the Inverse Problem

#### 3.2. Tikhonov Regularization in QPAT

**Theorem**

**5**(Well-posedness and convergence).

- (1)
- For any $v\in \mathbb{Y}$ and any $\lambda >0$, the Tikhonov functional ${T}_{\lambda ,v}$ has at least one minimizer.
- (2)
- Let $v\in \mathrm{ran}\left(\mathbf{F}\right)$, ${\left({\delta}_{m}\right)}_{m\in \mathbb{N}}\in {(0,\infty )}^{\mathbb{N}}$, ${\left({v}^{m}\right)}_{m\in \mathbb{N}}\in {\mathbb{Y}}^{\mathbb{N}}$ with $\u2225v-{v}^{m}\u2225\le {\delta}_{m}$. Suppose further that ${\left({\lambda}_{m}\right)}_{m\in \mathbb{N}}\in {(0,\infty )}^{\mathbb{N}}$ satisfies ${\lambda}_{m}\to 0$ and ${\delta}_{m}^{2}/{\lambda}_{m}\to 0$ as $m\to \infty $. Then:
- ■
- Every sequence ${\left({\mu}^{m}\right)}_{m\in \mathbb{N}}$ with ${\mu}^{m}\in \mathrm{arg}\phantom{\rule{0.166667em}{0ex}}\mathrm{min}{T}_{{v}^{m},{\lambda}_{m}}$ has a weakly converging subsequence.
- ■
- The limit of every weakly convergent subsequence of ${\left({\mu}^{m}\right)}_{m\in \mathbb{N}}$ is an $\u2225\mathbf{L}\left(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\right)\u2225$-minimizing solution of $\mathbf{F}\mu =v$.
- ■
- If the $\u2225\mathbf{L}\left(\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\right)\u2225$-minimizing solution of $\mathbf{F}\mu =v$ is unique and denoted by ${\mu}^{+}$, then $\left({\mu}^{m}\right)\rightharpoonup {\mu}^{+}$.

**Proof.**

#### 3.3. The Proximal Stochastic Gradient Algorithm for QPAT

**Proximal gradient algorithm:**The proximal gradient algorithm is a splitting method that iteratively computes explicit gradient steps for F and implicit proximal steps for G. In our context, we take F as the data fidelity term and

**Dykstra’s projection algorithm:**The constraint quadratic optimization problem (24) can efficiently be solved by a proximal variant of Dykstra’s projection algorithm [35,36,53]. For that purpose, we write ${s}_{k}{G}_{\lambda}={\chi}_{\mathbb{D}}+g$ with $g\left(x\right):=\frac{{s}_{k}\lambda}{2}{\u2225\mathbf{L}x\u2225}^{2}$. Setting ${x}_{0}=\mu $, ${p}_{0}=0$ and ${q}_{0}=0$, Dykstra’s projection algorithm for (24) reads, for $m\in \mathbb{N}$,

**Proximal stochastic gradient algorithm:**The methods described so far require in any iterative step the computation of the full gradient

#### 3.4. Iterative Regularization Methods

## 4. QPAT as Multilinear Inverse Problem

#### 4.1. Reformulation as Multilinear Inverse Problem

#### 4.2. Application of Tikhonov Regularization

#### 4.3. Solution of the MULL Formulation of QPAT Using Stochastic Gradient Methods

**MULL-projected stochastic gradient algorithm:**Here, we consider the form (38). For any iteration index $k\in \mathbb{N}$ choose $i\left(k\right)\in \{1,\dots ,N\}$ and $\ell \left(k\right)\in \{1,\dots ,4\}$ and define the sequence of iterates ${\left({\mathbf{z}}^{k}\right)}_{k\in \mathbb{N}}$ by

**MULL-proximal stochastic gradient algorithm:**Here, we consider the form (39). For any iteration index $k\in \mathbb{N}$ choose $i\left(k\right)\in \{1,\dots ,N\}$ and $\ell \left(k\right)\in \{1,\dots ,3\}$ and define sequence of iterates ${\left({\mathbf{z}}^{k}\right)}_{k\in \mathbb{N}}$ by

## 5. Numerical Simulations

#### 5.1. Numerical Solution of the RTE

#### 5.2. Test Scenario for Multiple Illumination

#### 5.3. Numerical Results

**Standard formulation of QPAT (19):**We assume that the scattering coefficient is known and we restrict ourself to reconstructing the absorption coefficient. Then, the proximal gradient and proximal stochastic gradient algorithm, respectively, read

**Novel MULL formulation of QPAT (35):**The multilinear approach overcomes the problem of solving the RTE by minimizing (38) or (39). In both cases, one selects an arbitrary functional and performs a steepest descent step, resulting in an iterative scheme for the variables ${\mu}_{a}$, ${\mu}_{s}$, $\mathrm{\Phi}$ and H. Recall that none of the partial gradients (40)–(48) requires solving the RTE (which is the most time-consuming part for the standard formulation of QPAT). In each iterative step, we take a random illumination number $i\in \{1,\dots ,4\}$ and a random functional number $\ell \in \{1,2,3\}$. The gradient step then consists of the update rule

**Remark**

**1.**

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Basic principles of PAT.

**Left**: the investigated object is illuminated with a short optical pulse;

**Middle**: due to the thermoelastic effect, the absorbed light distribution induces an acoustic pressure wave depending on internal tissue properties;

**Right**: the acoustic pressure wave is measured outside the object and used to reconstruct an image of the interior.

**Figure 2.**(

**a**) The phantom is defined on the square $\mathrm{\Omega}={[-1\mathrm{cm},1\mathrm{cm}]}^{2}$ and the acoustic pressure is measured on a semi-circle on the side of the illumination; (

**b**) the simulated pressure correspond to the phantom and the illumination in (a) and are represented as gray scale density.

**Figure 3.**Absorption coefficient distribution of the tissue sample used for the numerical examples. Background absorption of the tissue is taken as ${\mu}_{a}=0.3\text{}{\mathrm{cm}}^{-1}$, the blue obstacles have ${\mu}_{a}=1\text{}{\mathrm{cm}}^{-1}$ and the red stripes ${\mu}_{a}=2\text{}{\mathrm{cm}}^{-1}$. The area between the red stripes has absorption coefficient ${\mu}_{a}=0.5\text{}{\mathrm{cm}}^{-1}$. The scattering coefficient is constant in the whole sample and chosen to be ${\mu}_{s}=3\text{}{\mathrm{cm}}^{-1}$. Illuminations are applied consecutively from top, right, bottom and left. The corresponding boundary sources are given by ${q}_{o}(x,\theta )=\delta (\theta -{\theta}_{i}){\chi}_{i}\left(x\right)1\text{}\mathrm{m}\mathrm{J}\text{}{\mathrm{cm}}^{-1}$. The x- and y-axis cover $[-1\mathrm{cm},1\mathrm{cm}]$.

**Figure 4.**Reconstruction results based on standard formulation (19).

**Top**: proximal gradient method;

**Bottom**: proximal stochastic gradient method. The left images show the relative reconstruction errors of the reconstructed absorption coefficient as a function of the number of iterations, whereas the right pictures show the result after the final iteration. (The phantom is as described in Figure 3.)

**Figure 5.**Flowcharts of stochastic gradient algorithms for QPAT proposed in this paper.

**Left**: algorithm based on the standard formulation (19).

**Right**: algorithm based on the novel MULL formulation (35). The update (54) using the standard formulation requires solving the forward RTE and the adjoint RTE, which is not required by (55) with the MULL formulation. Simulations are performed with $N=4$.

**Figure 6.**Reconstruction results based on the novel MULL formulation (35).

**Top**: MULL-proximal stochastic gradient method based on the decomposition (38).

**Bottom**: MULL-projected stochastic gradient method based on the decomposition (39). The left images show the relative reconstruction errors of the reconstructed absorption coefficient as a function of the number of iterations, whereas the right pictures show the results after the last iterations. (The phantom is as described in Figure 3.)

**Figure 7.**Reconstruction results from noisy data.

**Top**: Proximal gradient method based on (19).

**Bottom**: MULL-proximal stochastic gradient method. The left images show the relative reconstruction errors of the reconstructed absorption coefficient as a function of the number of iterations, whereas the right pictures show the results after the last iterations. (The phantom is as described in Figure 3.)

**Figure 8.**Reconstruction results from noisy data for two illuminations.

**Left**: proximal gradient method based on (19) using 10 iterations.

**Right**: MULL-proximal stochastic gradient method using only 500 iterations. The phantom is as described in Figure 3 and, for the reconstruction methods, we use two consecutive illuminations (from the top and from the left). The reconstruction time has been about 14 h for the method based on the standard formulation (19) and 3 h for the proposed MULL-proximal stochastic gradient method.

**Table 1.**Reconstruction times for all methods. Recall that one iteration of the proximal stochastic gradient method is approximately four times cheaper than one iteration of the full proximal gradient method (both based on (19)). Further recall that one step in the methods based on the MULL formulation (35) is much less time consuming than for the methods based on (19); see Remark 1.

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**MDPI and ACS Style**

Rabanser, S.; Neumann, L.; Haltmeier, M. Stochastic Proximal Gradient Algorithms for Multi-Source Quantitative Photoacoustic Tomography. *Entropy* **2018**, *20*, 121.
https://doi.org/10.3390/e20020121

**AMA Style**

Rabanser S, Neumann L, Haltmeier M. Stochastic Proximal Gradient Algorithms for Multi-Source Quantitative Photoacoustic Tomography. *Entropy*. 2018; 20(2):121.
https://doi.org/10.3390/e20020121

**Chicago/Turabian Style**

Rabanser, Simon, Lukas Neumann, and Markus Haltmeier. 2018. "Stochastic Proximal Gradient Algorithms for Multi-Source Quantitative Photoacoustic Tomography" *Entropy* 20, no. 2: 121.
https://doi.org/10.3390/e20020121