[Paper Review] Deep neural networks for inverse problems with pseudodifferential operators: an application to limited-angle tomography

Tatiana A. Bubba, Mathilde Galinier+4 more|arXiv (Cornell University)|Jun 2, 2020

Numerical methods in inverse problems36 references22 citations

TL;DR

This paper introduces a novel convolutional neural network, $\Psi$ DONet, that learns pseudodifferential operators for solving linear inverse problems, specifically limited-angle computed tomography (LA-CT). By interpreting the iterative soft thresholding algorithm (ISTA) as a deep network architecture, the method leverages wavelet-based decomposition and convolutional operations derived from the geometric and analytical properties of the X-ray transform, achieving stable and accurate reconstructions from sparse angular data.

ABSTRACT

We propose a novel convolutional neural network (CNN), called $\\Psi$DONet, designed for learning pseudodifferential operators ($\\Psi$DOs) in the context of linear inverse problems. Our starting point is the Iterative Soft Thresholding Algorithm (ISTA), a well-known algorithm to solve sparsity-promoting minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling and convolution, which characterize our $\\Psi$DONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited angle X-ray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of $\\Psi$DONet on simulated data from limited-angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are $\\Psi$DOs or Fourier integral operators.

Motivation & Objective

• To develop a deep learning framework that systematically learns pseudodifferential operators (PDOs) arising in linear inverse problems.
• To address the ill-posedness of limited-angle computed tomography (LA-CT), where standard methods like filtered backprojection fail due to angular data deficiency.
• To establish a theoretical link between iterative algorithms like ISTA and deep neural network architectures, enabling interpretable and stable learning.
• To exploit the convolutional structure of the limited-angle X-ray transform and orthogonal wavelet systems to design a network with analytically determined layers.
• To demonstrate the feasibility and robustness of the proposed method on simulated LA-CT data using the ellipse phantom dataset.

Proposed method

• The method is based on unfolding the Iterative Soft Thresholding Algorithm (ISTA) into a deep residual network architecture, where each ISTA iteration corresponds to a network layer.
• The network, named $\Psi$ DONet, is designed to learn the action of a pseudodifferential operator (PDO) by modeling the forward operator as a convolutional operator with a kernel derived from the Calderón-Zygmund kernel and a cone indicator function.
• The architecture incorporates upscaling, downscaling, and convolutional operations that are analytically determined by the properties of the limited-angle X-ray transform and orthogonal wavelet systems.
• The network is trained using a sparsity-promoting variational formulation with $\ell^1$ regularization, minimizing a Tikhonov-type functional involving the data fidelity and sparsity terms.
• Theoretical analysis shows that the network parameters can be bounded under mild assumptions, and convergence is established via stability estimates involving the projection operators $\mathbb{P}_p$ and $\mathbb{P}_q$.
• The method is validated on simulated LA-CT data using the ellipse phantom, with two distinct implementations of $\Psi$ DONet tested for performance and robustness.

Experimental results

Research questions

• RQ1 Can the iterative soft thresholding algorithm (ISTA) be systematically unfolded into a convolutional neural network architecture that preserves theoretical stability and convergence?
• RQ2 To what extent can the convolutional layers of a deep network be analytically derived from the mathematical structure of the forward operator in limited-angle tomography?
• RQ3 How do wavelet-based decompositions and pseudodifferential operator theory enable the design of interpretable and stable deep learning models for inverse problems?
• RQ4 Can a deep network trained on limited-angle data achieve superior reconstruction quality compared to classical methods like filtered backprojection, especially in the presence of angular gaps?
• RQ5 What is the theoretical relationship between the learned network parameters and the underlying forward operator, particularly in terms of filter coefficient bounds and convergence?

Key findings

• The $\Psi$ DONet framework successfully maps the iterative ISTA algorithm into a deep residual network architecture, enabling the learning of pseudodifferential operators in inverse problems.
• The upscaling, downscaling, and convolutional operations in $\Psi$ DONet are analytically determined by the convolutional nature of the limited-angle X-ray transform and the properties of orthogonal wavelets.
• Two implementations of $\Psi$ DONet were tested on simulated ellipse phantom data, both yielding equally good and noteworthy preliminary results in reconstructing images from limited-angle projections.
• Theoretical analysis establishes bounds on the filter coefficients and convergence rates, showing that the network error decays exponentially with depth and is stable under perturbations of the forward operator.
• The method achieves stable reconstructions even when the data is noisy or angular coverage is severely restricted, outperforming classical methods like filtered backprojection.
• The framework provides a generalizable approach for other convolutional operators that are pseudodifferential or Fourier integral operators, paving the way for broader applications in medical and industrial imaging.