A data-dependent regularization method based on the graph Laplacian

TL;DR

The $\texttt{graphLa+}\Psi$ approach significantly enhances the quality of the approximated solutions for each method $\Psi$, and is outperforming, offering a robust and stable application of deep neural networks in solving inverse problems.

Abstract

We investigate a variational method for ill-posed problems, named $\texttt{graphLa+}Ψ$, which embeds a graph Laplacian operator in the regularization term. The novelty of this method lies in constructing the graph Laplacian based on a preliminary approximation of the solution, which is obtained using any existing reconstruction method $Ψ$ from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data and noise. We demonstrate that $\texttt{graphLa+}Ψ$ is a regularization method and rigorously establish both its convergence and stability properties. We present selected numerical experiments in 2D computerized tomography, wherein we integrate the $\texttt{graphLa+}Ψ$ method with various reconstruction techniques $Ψ$, including Filter Back Projection ($\texttt{graphLa+FBP}$), standard Tikhonov ($\texttt{graphLa+Tik}$), Total Variation ($\texttt{graphLa+TV}$), and a trained deep neural network ($\texttt{graphLa+Net}$). The $\texttt{graphLa+}Ψ$ approach significantly enhances the quality of the approximated solutions for each method $Ψ$. Notably, $\texttt{graphLa+Net}$ is outperforming, offering a robust and stable application of deep neural networks in solving inverse problems.

Figures (12)

• Figure 1: A schematic representation of the $\texttt{graphLa+}\Psi$ method. The reconstructors $\Psi_\Theta$ do not necessarily need to be a regularization method, and this is represented by the piecewise linear path of $\Psi^\delta_\Theta$ as $\delta$ goes to $0$. However, when combined with the graph Laplacian in the Tikhonov method \ref{['model_eq2']}, it generates a convergent and stable regularization operator, that is, $\texttt{graphLa+}\Psi$, which is represented by the smooth red path. See \ref{['sec:model_setting', 'sec:regularization']} for more details on the notation.
• Figure 1: Simple outline of how to build a graph from an image ${\boldsymbol{x}}$. To be read from left to right. Left: a $7\times7$ pixels image made by orange-like and blue-like square pixels. The color intensity of each pixel is given by the pixel-wise evaluation of a function ${\boldsymbol{x}}$. Center-left: each pixel corresponds to one node, represented by a black circle. Since the pixels are disposed on a grid, each node can be associated to an ordered pair in $\mathbb{Z}^2$. Center-right: the geometric edge-weight function $w_{\textnormal{d}}$ in \ref{['def:induced_w']} is given by $\mathds{1}_{(0,1]}(\|p-q\|_1)$, that is, two nodes $p, q$ are connected if and only if $\|p-q\|_1 =1$, and in that case the magnitude of the connection is one. Right: the magnitude of an edge between two nodes is then weighted by $h_{\textnormal{i}}(\|{\boldsymbol{x}}(p)-{\boldsymbol{x}}(q)\|)\in (0,1]$, where $h_{\textnormal{i}}(t) = \exp\{-t^2/\sigma^2\}$ is the Gaussian function. The role of $h_{\textnormal{i}}$ is to measure the difference of intensity between two adjacent pixels, and it is close to zero when two pixels have very different color intensities. This is represented by the different thicknesses of the edges connecting two adjacent pixels, where a thick edge means a very similar color intensity and a thin edge means a very different color intensity.
• Figure 1: A diagram of the ResU-net architecture used in the experiments.
• Figure 1: (a): Example of a ${\boldsymbol{x}}_{\textnormal{gt}}$ image from COULE dataset. (b): The resulting sinogram
• Figure 1: Visual comparison: ground truth (a), NETT (b), FISTA (c) and ISTA (d).

Theorems & Definitions (44)

• Definition 2.2
• Definition 2.3: graph Laplacian
• Example 3.3
• Example 3.4
• Definition 3.5
• Remark 3.6
• Lemma 3.8
• Proof 1
• Proposition 3.11
• Proof 2