# Discretization of Learned NETT Regularization for Solving Inverse Problems

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Reconstruction with Learned Regularizers

- (T1)
- Choose a family of desired reconstructions ${\left({x}_{i}\right)}_{i=1}^{n}$.
- (T2)
- For some $\mathbf{B}:\mathbb{Y}\to \mathbb{X}$, construct undesired reconstructions ${\left(\mathbf{B}\mathbf{A}{x}_{i}\right)}_{i=1}^{n}$.
- (T3)
- Choose a class ${\left({\Phi}_{\theta}\right)}_{\theta \in \Theta}$ of functions (networks) ${\Phi}_{\theta}:\mathbb{X}\to \mathbb{X}$.
- (T4)
- Determine ${\theta}^{\star}\in \Theta $ with ${\Phi}_{{\theta}^{\star}}\left({x}_{i}\right)\simeq {x}_{i}\wedge {\Phi}_{{\theta}^{\star}}\left(\mathbf{B}\mathbf{A}{x}_{i}\right)\simeq {x}_{i}$.
- (T5)
- Define $\mathcal{R}\left(x\right)=r(x,\Phi (x\left)\right)$ with $\Phi ={\Phi}_{{\theta}^{\star}}$ for some $r:\mathbb{Y}\times \mathbb{Y}\to [0,\infty ]$.

#### 1.2. Discrete NETT

#### 1.3. Outline

## 2. Convergence Analysis

#### 2.1. Well-Posedness

**Assumption**

**1**

- (W1)
- $\mathbb{X}$, $\mathbb{Y}$ are Banach spaces, $\mathbb{X}$ reflexive, $\mathbb{D}\subseteq \mathbb{X}$ weakly sequentially closed.
- (W2)
- The distance measure $\mathcal{D}:\mathbb{Y}\times \mathbb{Y}\to [0,\infty ]$ satisfies
- (a)
- $\exists \tau \ge 1:\forall {y}_{1},{y}_{2},{y}_{3}\in \mathbb{Y}:\mathcal{D}({y}_{1},{y}_{2})\le \tau \mathcal{D}({y}_{1},{y}_{3})+\tau \mathcal{D}({y}_{3},{y}_{2})$.
- (b)
- $\forall {y}_{1},{y}_{2}\in \mathbb{Y}:\mathcal{D}({y}_{1},{y}_{2})=0\iff {y}_{1}={y}_{2}$.
- (c)
- $\forall y,\tilde{y}\in \mathbb{Y}:\mathcal{D}(y,\tilde{y})<\infty \wedge \parallel \tilde{y}-{y}_{k}\parallel \to 0\Rightarrow \mathcal{D}(y,{y}_{k})\to \mathcal{D}(y,\tilde{y})$.
- (d)
- $\forall y\in \mathbb{Y}:\parallel {y}_{k}-y\parallel \to 0\Rightarrow \mathcal{D}({y}_{k},y)\to 0$.
- (e)
- $\mathcal{D}$ is weakly sequentially lower semi-continuous (wslsc).

- (W3)
- $\mathcal{R}:\mathbb{X}\to [0,\infty ]$ is proper and wslsc.
- (W4)
- $\mathbf{A}:\mathbb{D}\subseteq \mathbb{X}\to \mathbb{Y}$ is weakly sequentially continuous.
- (W5)
- $\forall y,\alpha ,C:\{x\in \mathbb{X}\mid {\mathcal{T}}_{y,\alpha}\le C\}$ is nonempty and bounded.
- (W6)
- ${\left({\mathbb{X}}_{n}\right)}_{n\in \mathbb{N}}$ is a sequence of subspaces of $\mathbb{X}$.
- (W7)
- ${\left({\mathbf{A}}_{n}\right)}_{n\in \mathbb{N}}$ is a family of weakly sequentially continuous ${\mathbf{A}}_{n}:\mathbb{D}\to \mathbb{Y}$.
- (W8)
- ${\left({\mathcal{R}}_{n}\right)}_{n\in \mathbb{N}}$ is a family of proper wslsc regularizers ${\mathcal{R}}_{n}:\mathbb{X}\to [0,\infty ]$.
- (W9)
- $\forall y,\alpha ,C,n:\{x\in {\mathbb{X}}_{n}\mid {\mathcal{T}}_{y,\alpha ,n}\le C\}$ is nonempty and bounded.

**Theorem**

**1**

- (a)
- $argmin{\mathcal{T}}_{y,\alpha ,n}\ne \varnothing $.
- (b)
- Let ${\left({y}_{k}\right)}_{k\in \mathbb{N}}\in {\mathbb{Y}}^{\mathbb{N}}$ with ${y}_{k}\to y$ and consider ${x}_{k}\in argmin{\mathcal{T}}_{{y}_{k},\alpha ,n}$.
- ${\left({x}_{k}\right)}_{k\in \mathbb{N}}$ has at least one weak accumulation point.
- Every weak accumulation point ${\left({x}_{k}\right)}_{k\in \mathbb{N}}$ is a minimizer of ${\mathcal{T}}_{y,\alpha ,n}$.

- (c)
- The statements in (a),(b) also hold for ${\mathcal{T}}_{y,\alpha}$ in place of ${\mathcal{T}}_{y,\alpha ,n}$,

**Proof.**

**Lemma**

**1**

**Proof.**

#### 2.2. Convergence

**Assumption**

**2**

- (C1)
- $\exists \left({z}_{n}\right)\in {\prod}_{n\in \mathbb{N}}(\mathbb{D}\cap {\mathbb{X}}_{n})$ with ${\lambda}_{n}:=\left|{\mathcal{R}}_{n}\left({z}_{n}\right)-\mathcal{R}\left({x}^{+}\right)\right|\to 0$.
- (C2)
- ${\rho}_{n}:={sup}_{x\in {\mathbb{D}}_{n,M}}|{\mathcal{R}}_{n}\left(x\right)-\mathcal{R}\left(x\right)|\to 0$.
- (C3)
- ${\gamma}_{n}:=\mathcal{D}({\mathbf{A}}_{n}{z}_{n},\mathbf{A}{x}^{+})\to 0$.
- (C4)
- ${a}_{n}:={sup}_{x\in {\mathbb{D}}_{n,M}}|\mathcal{D}({\mathbf{A}}_{n}x,\mathbf{A}{x}^{+})-\mathcal{D}(\mathbf{A}x,\mathbf{A}{x}^{+})|\to 0$.

**Theorem**

**2**

- (a)
- ${\left({x}_{k}\right)}_{k\in \mathbb{N}}$ has a weakly convergent subsequence ${\left({x}_{\sigma \left(k\right)}\right)}_{k\in \mathbb{N}}$
- (b)
- The weak limit of ${\left({x}_{\sigma \left(k\right)}\right)}_{k\in \mathbb{N}}$ is an $\mathcal{R}$-minimizing solution of $\mathbf{A}x=y$.
- (c)
- ${\mathcal{R}}_{\sigma \left(k\right)}\left({x}_{\sigma \left(k\right)}\right)\to \mathcal{R}\left({x}^{\star}\right)$, where ${x}^{\star}$ is the weak limit of ${\left({x}_{\sigma \left(k\right)}\right)}_{k\in \mathbb{N}}$.
- (d)
- If the $\mathcal{R}$-minimizing solution of $\mathbf{A}x=y$ is unique, then ${\left({x}_{k}\right)}_{k\in \mathbb{N}}\rightharpoonup {x}^{+}$.

**Proof.**

#### 2.3. Convergence Rates

**Assumption**

**3**

- (R1)
- Items (C1), (C2) hold.
- (R2)
- ${\gamma}_{n,\delta}:={sup}_{{y}^{\delta}}\left|\mathcal{D}({\mathbf{A}}_{n}{z}_{n},{y}^{\delta})-\mathcal{D}(\mathbf{A}{x}^{+},{y}^{\delta})\right|\to 0$.
- (R3)
- ${a}_{n,\delta}:={sup}_{{y}^{\delta}}{sup}_{x\in {\mathbb{D}}_{n,M}}|\mathcal{D}({\mathbf{A}}_{n}x,{y}^{\delta})-\mathcal{D}(\mathbf{A}x,{y}^{\delta})|\to 0$.
- (R4)
- $\mathcal{R}$ is Gâteaux differentiable at ${x}^{+}$
- (R5)
- There exist a concave, continuous, strictly increasing $\phi :[0,\infty )\to [0,\infty )$ with $\phi \left(0\right)=0$ and $\u03f5,\beta >0$ such that for all $x\in \mathbb{X}$$$|\mathcal{R}\left(x\right)-\mathcal{R}\left({x}^{+}\right)|\le \u03f5\Rightarrow \beta {\mathcal{B}}_{\mathcal{R}}(x,{x}^{+})\le \mathcal{R}\left(x\right)-\mathcal{R}\left({x}^{+}\right)+\phi \left(\mathcal{D}(\mathbf{A}x,\mathbf{A}{x}^{+})\right)\phantom{\rule{0.166667em}{0ex}}.$$

**Proposition**

**1**

**Proof.**

**Remark**

**1.**

**Theorem**

**3**

**Proof.**

**Lemma**

**2**

**Proof.**

**Corollary**

**1**

**Proof.**

## 3. Application to a Limited Data Problem in PAT

#### 3.1. Discrete Forward Operator

#### 3.2. Discrete NETT

Algorithm 1: NETT optimization. |

#### 3.3. Numerical Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Top from left to right: phantom, masked phantom, and initial reconstruction ${\mathbf{A}}^{+}\mathbf{A}x$. The difference between the phantoms on the left and the middle one shows the mask region $I\subseteq {D}_{1}$ where no data is generated. Bottom from left to right: data without noise, low noise $\sigma =0.01$, and high noise $\sigma =0.1$.

**Figure 2.**Top row: reconstructions using post-processing network ${\Phi}^{\left(1\right)}$. Middle row: NETT reconstructions using ${\mathcal{R}}^{\left(1\right)}$. Bottom row: NETT reconstructions using ${\mathcal{R}}^{\left(3\right)}$. From Left to Right: Reconstructions from data without noise, low noise ($\sigma =0.01$) and high noise ($\sigma =0.1)$.

**Figure 3.**Semilogarithmic plot of the mean squared errors of the NETT using ${\mathcal{R}}^{\left(1\right)}$ and ${\mathcal{R}}^{\left(3\right)}$ depending on the noise level. The crosses are the values for the phantoms in Figure 2.

**Figure 4.**Left column: phantom with a structure not contained in the training data (

**top**) and pseudo inverse reconstruction (

**bottom**). Middle column: Post-processing reconstructions with ${\Phi}^{\left(3\right)}$ using exact (

**top**) and noisy data (

**bottom**). Right column: NETT reconstructions with ${\mathcal{R}}^{\left(3\right)}$ using exact (

**top**) and noisy data (

**bottom**).

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Antholzer, S.; Haltmeier, M. Discretization of Learned NETT Regularization for Solving Inverse Problems. *J. Imaging* **2021**, *7*, 239.
https://doi.org/10.3390/jimaging7110239

**AMA Style**

Antholzer S, Haltmeier M. Discretization of Learned NETT Regularization for Solving Inverse Problems. *Journal of Imaging*. 2021; 7(11):239.
https://doi.org/10.3390/jimaging7110239

**Chicago/Turabian Style**

Antholzer, Stephan, and Markus Haltmeier. 2021. "Discretization of Learned NETT Regularization for Solving Inverse Problems" *Journal of Imaging* 7, no. 11: 239.
https://doi.org/10.3390/jimaging7110239