Learning a projection term on a manifold to constrain tomographic reconstruction

Inverse problems, such as signal recovery from noisy measurements, have traditionally been addressed through optimisation techniques that balance data fidelity with regularisation. Classical regularisation methods, including total variation (TV), promote sparsity in a predefined transform domain, such as the Fourier domain or learned dictionaries [1]. However, with the advances in deep learning and large-scale datasets, significantly more expressive regularisation techniques have emerged [2]. Deep neural network post-processors have gained popularity, yet they do not inherently ensure that the reconstructed signals adhere to the underlying physical constraints, which is unacceptable for non destructive applications [3]. More recent approaches, such as unrolled optimisation methods-including primal-dual hybrid gradient techniques with learned parameters-integrate data fidelity terms directly within the optimisation framework alongside regularisation [4]. Additionally, neural networks can serve as priors within variational inference frameworks, such as diffusion model based methods [5]. In this study, we focus on a family of priors using an adversarial regulariser. The core idea behind this latter approach is to construct a distance term capable of distinguishing between the distribution of ground-truth images and that of corrupted images by learning the manifold M on which ground-truth samples reside [6]. This class of models is generally simpler to implement than generative adversarial networks (GANs), as it does not require access to corrupted samples. Adversarial regularisers and GAN-based methods have already been employed in various ways for ill-posed problems, such as direct optimisation within a latent space [7] or through projection onto this space [8].