Compressive imaging (CI) consists of reconstructing images from incomplete observed data. The reconstruction process involves solving an ill-posed inverse problem which is highly dependent on the number of real measurements, with a greater number of measurements typically leading to more accurate reconstructions. Due to their ability to learn data distributions, diffusion models (DM) have emerged as promising techniques for various inverse problems. Mainly, DMs solve inverse problems by conditioning the generation process to the acquired measurements. In this work, we introduce a new approach to improve this conditioning by exploiting synthetic measurements, which come from a synthetic sensing matrix. Synthetic measurements are estimated from real data via a neural network. The combined real and synthetic measurements form an augmented set, which is input into the conditional DM to enhance reconstruction capacity. Computational experiments demonstrate that augmenting measurements with the conditional DM improves performance compared to using only real measurements.

In compressive imaging systems, the scene is acquired via linear coded noisy projections, known as measurements, requiring a recovery process to estimate the underlying signal. This recovery is inherently ill-posed, posing a challenge for accurate signal recovery. Existing methods that employ prior information about the signal often fail in practical scenarios. In this work, instead of developing a new prior over the signal, we exploit the structure of the low-dimensional measurements to synthesize an augmented measurement set that can be used in various recovery methods to improve its performance. We used a deep neural network to generate the synthetic measurements from the acquired data. We show the benefits of this approach in two schemes, deep learning-based recovery and the plug-and-play (PnP) algorithm. Particularly, our method is interpreted as a non-linear preconditioning technique for the PnP algorithm. We show improved performance for different sensing matrices.

In computational optical imaging and wireless communications, signals are acquired through linear coded and noisy projections, which are recovered through computational algorithms. Deep model-based approaches, i.e., neural networks incorporating the sensing operators, are the state-of-the-art for signal recovery. However, these methods require exact knowledge of the sensing operator, which is often unavailable in practice, leading to performance degradation. Consequently, we propose a new recovery paradigm based on knowledge distillation. A teacher model, trained with full or almost exact knowledge of a synthetic sensing operator, guides a student model with an inexact real sensing operator. The teacher is interpreted as a relaxation of the student since it solves a problem with fewer constraints, which can guide the student to achieve higher performance. We demonstrate the improvement of signal reconstruction in computational optical imaging for single-pixel imaging with miscalibrated coded apertures systems and multiple-input multiple-output symbols detection with inexact channel matrix.

Imaging inverse problems aim to recover high-dimensional signals from undersampled, noisy measurements, a fundamentally ill-posed task with infinite solutions in the null-space of the sensing operator. To resolve this ambiguity, prior information is typically incorporated through handcrafted regularizers or learned models that constrain the solution space. However, these priors typically ignore the task-specific structure of that null-space. In this work, we propose Non-Linear Projections of the Null-Space (NPN), a novel class of regularization that, instead of enforcing structural constraints in the image domain, promotes solutions that lie in a low-dimensional projection of the sensing matrix's null-space with a neural network. Our approach has two key advantages: (1) Interpretability: by focusing on the structure of the null-space, we design sensing-matrix-specific priors that capture information orthogonal to the signal components that are fundamentally blind to the sensing process. (2) Flexibility: NPN is adaptable to various inverse problems, compatible with existing reconstruction frameworks, and complementary to conventional image-domain priors. We provide theoretical guarantees on convergence and reconstruction accuracy when used within plug-and-play methods. Empirical results across diverse sensing matrices demonstrate that NPN priors consistently enhance reconstruction fidelity in various imaging inverse problems, such as compressive sensing, deblurring, super-resolution, computed tomography, and magnetic resonance imaging, with plug-and-play methods, unrolling networks, deep image prior, and diffusion models.