Posterior-Variance–Based Error Quantification for Inverse Problems in Imaging

Traditionally, analytical methods have been used to solve imaging problems such as image restoration, inpainting, and superresolution (SR). In recent years, the fields of machine and deep learning have gained a lot of momentum in solving such imaging problems, often surpassing the performance provided by analytical approaches. Unlike analytical methods for which the problem is explicitly defined and domain-knowledge carefully engineered into the solution, deep neural networks (DNNs) do not benefit from such prior knowledge and instead make use of large data sets to learn the unknown solution to the inverse problem. In this article, we review deep-learning techniques for solving such inverse problems in imaging. More specifically, we review the popular neural network architectures used for imaging tasks, offering some insight as to how these deep-learning tools can solve the inverse problem. Furthermore, we address some fundamental questions, such as how deeplearning and analytical methods can be combined to provide better solutions to the inverse problem in addition to providing a discussion on the current limitations and future directions of the use of deep learning for solving inverse problem in imaging.

This review provides an introduction to—and overview of—the current state of the art in neural‐network based regularization methods for inverse problems in imaging. It aims to introduce readers with a solid knowledge in applied mathematics and a basic understanding of neural networks to different concepts of applying neural networks for regularizing inverse problems in imaging. Distinguishing features of this review are, among others, an easily accessible introduction to learned generators and learned priors, in particular diffusion models, for inverse problems, and a section focusing explicitly on existing results in function space analysis of neural‐network‐based approaches in this context.

Effective representation methods and proper signal priors are crucial in most signal processing applications. In this thesis we focus on different structured models and we design appropriate schemes that allow the discovery of low dimensional latent structures that characterise and identify the signals. Motivated by the highly non-linear structure of most datasets, we firstly investigate the geometry of manifolds. Manifolds are low dimensional, non-linear structures that are naturally employed to describe sets of strongly related signals such as the images of a 3-D object captured from different viewpoints. However, the use of manifolds in applications is not straightforward due to their usually non-analytic and non-linear form. We propose here a way to `disassemble' a manifold into simpler components by approximating it with affine subspaces. Our objective is to discover a set of low dimensional affine subspaces that can represent manifold data accurately while preserving the manifold's structure. To this end, we employ a greedy technique that iteratively merges manifold samples into groups based on the difference of local tangents. We use our algorithm to approximate synthetic and real manifolds and to demonstrate that it is competitive to state-of-the-art techniques. Then, we consider structured sparse representations of signals and we propose a new sparsity model, where signals are essentially composed of a small number of structured {\it molecules }. We define the molecules to be linear combinations of a small number of atoms in a redundant dictionary. Our multi-level model takes into account the energy distribution of the significant signal components in addition to their support. It permits to define typical visual patterns and recognise them in prototypical or deformed form. We define a new structural difference measure between molecules and their deformed versions, which is based on their sparse codes and we create an algorithm for decomposing signals into molecules that can account for different deviations in the internal molecule structure. Our experiments verify the benefits of the new image model in various restoration tasks and they confirm that the development of proper models that extend the mere notion of sparsity can be very useful for various inverse problems in imaging. Finally, we investigate the problem of learning molecule representations directly in the sparse code domain. We constrain sparse codes to be linear combinations of a few, possibly deformed, molecules and we design an algorithm that can learn the structure from the codes without transforming them back into the signal domain. To this end, we take advantage of our structural difference which is based on the sparse codes and we devise a scheme for representing the codes with molecules and learn the molecules at the same time. To illustrate the effectiveness of our proposed algorithm we apply it to various synthetic and real datasets and we compare the results with traditional sparse coding and dictionary learning techniques. From the experiments, we verify the superior performance of our scheme in interpreting and recognising correctly the underlying structure. In short, in this thesis we are interested in low-dimensional, structured models. Among the various choices, we focus on manifolds and sparse representations and we propose schemes that enhance their structural properties and highlight their effectiveness in signal representations.

Ultrasound computed tomography (USCT) receives increasing attention because of its capability to reconstruct quantitative information about the material property distribution as images with superior resolution. However, one roadblock for the wide adoption of relevant techniques is the high demand for computational resources and the long processing time for solving a large inverse problem in imaging. To alleviate the associated challenges, a two-stage inversion scheme is proposed: 1) the ultrasound scanning signals are first processed using a full waveform inversion (FWI) technique with a single iteration to rapidly create a model (image) with embedded wave speed distribution; 2) the corresponding image will be further improved by feeding into a pre-trained deep neural network. The deep learning models presented in this paper are built upon two architectures to instantaneously solve the associated inverse problems and to produce a high-resolution image in real-time. The first is based on 1D convolutional neural network (1D-CNN) layers with an autoencoder structure. The second implements additional layers and skip connections inspired by a U-Net architecture. The resultant superior reconstructions from both CNNs demonstrate that the proposed framework produces a high-resolution image from a rapidly-generated, low-resolution image in real-time, with dramatically improved results.

A two-dimensional inverse problem in imaging the thermal conductivity of a non-homogeneous medium

The concept of region of sensitivity is central to the field of diffuse optics and is closely related to the Jacobian matrix used to solve the inverse problem in imaging. It is well-known that, in diffuse reflectance, the region of sensitivity associated with a given source-detector pair is shaped as a banana, and features maximal sensitivity to the portions of the sample that are closest to the source and the detector. We have recently introduced a dual-slope method based on a special arrangement of two sources and two detectors, which results in deeper and more localized regions of sensitivity, resembling the shapes of different kinds of nuts. Here, we report the regions of sensitivity associated with a variety of source-detector arrangements for dual-slope measurements of intensity and phase with frequency-domain spectroscopy (modulation frequency: 140 MHz) in a medium with absorption and reduced scattering coefficients of 0.1 cm-1 and 12 cm-1, respectively. The main result is that the depth of maximum sensitivity, considering only cases that use source-detector separations of 25 and 35 mm, progressively increases as we consider single-distance intensity (2.0 mm), dual-slope intensity (4.6 mm), single-distance phase (7.5 mm), and dual-slope phase (10.9 mm). These results indicate the importance of dual-slope measurements, and even more so of phase measurements, when it is desirable to selectively probe deeper portions of a sample with diffuse optics. This is certainly the case in non-invasive optical studies of brain, muscle, and breast tissue, which are located underneath superficial tissue at variable depths.

Wavelet decompositions of integral operators have proven their efficiency in reducing computing times for many problems, ranging from the simulation of waves or fluids to the resolution of inverse problems in imaging. Unfortunately, computing the decomposition is itself a hard problem which is oftentimes out of reach for large-scale problems. The objective of this work is to design fast decomposition algorithms based on another representation called product-convolution expansion. This decomposition can be evaluated efficiently, assuming that a few impulse responses of the operator are available, but it is usually less efficient than the wavelet decomposition when incorporated in iterative methods. The proposed decomposition algorithms, run in quasi-linear time and we provide some numerical experiments to assess its performance for an imaging problem involving space-varying blurs.

This book provides researchers and engineers in the imaging field with the skills they need to effectively deal with nonlinear inverse problems associated with different imaging modalities, including impedance imaging, optical tomography, elastography, and electrical source imaging. Focusing on numerically implementable methods, the book bridges the gap between theory and applications, helping readers tackle problems in applied mathematics and engineering. Complete, self-contained coverage includes basic concepts, models, computational methods, numerical simulations, examples, and case studies. Provides a step-by-step progressive treatment of topics for ease of understanding. Discusses the underlying physical phenomena as well as implementation details of image reconstruction algorithms as prerequisites for finding solutions to non linear inverse problems with practical significance and value. Includes end of chapter problems, case studies and examples with solutions throughout the book. Companion website will provide further examples and solutions, experimental data sets, open problems, teaching material such as PowerPoint slides and software including MATLAB m files. Essential reading for Graduate students and researchers in imaging science working across the areas of applied mathematics, biomedical engineering, and electrical engineering and specifically those involved in nonlinear imaging techniques, impedance imaging, optical tomography, elastography, and electrical source imaging

Continuous-time image reconstruction using differential equations for computed tomography

Image reconstruction in computed tomography can be treated as an inverse problem, namely, obtaining pixel values of a tomographic image from measured projections. However, a seriously degraded image with artifacts is produced when a certain part of the projections is inaccurate or missing. A novel method for simultaneously obtaining a reconstructed image and an estimated projection by solving an initial-value problem of differential equations is proposed. A system of differential equations is constructed on the basis of optimizing a cost function of unknown variables for an image and a projection. Three systems described by nonlinear differential equations are constructed, and the stability of a set of equilibria corresponding to an optimized solution for each system is proved by using the Lyapunov stability theorem. To validate the theoretical result given by the proposed method, metal artifact reduction was numerically performed.

An alternative differential framework for the existing integral-based electromagnetic inverse problems is proposed. This differential form, supplemented by the corresponding boundary conditions, overcomes the existing limitations of the integral-based inverse problems, i.e. considering the imaging antennas as point sources, the essential homogeneity along one axis, and the necessity of using a background matching medium. These limitations result in reconstructing two-dimensional images with certain level of error, and the practical problems associated with selecting suitable dielectric properties of the background medium. Thus, the ability of the proposed technique in reconstructing accurate three-dimensional images, including the dimensions and real structure of the imaging antennas and avoiding the essential use of matching medium, are unique advancements in solving inverse electromagnetic problems. To validate this proposed novel method, a biomedical inverse electromagnetic problem in head imaging is successfully solved.

The potentialities offered by the Quantum Computing framework in the solution of microwave imaging problems are discussed and validated in a proof-of-concept fashion in this work. Towards this end, a tomographic inverse problem is firstly formulated within the first-order Born approximation under transverse-magnetic illumination. The resulting linear system of equations is solved by a customized version of the Harrow-Hassidim-Lloyd (HHL) quantum algorithm. Preliminary numerical results are shown to illustrate the possibility to dramatically decrease the retrieval time by leveraging on the exponential computational improvement made possible by quantum algorithms.

In imaging inverse problems, we would like to know how close the recovered image is to the true image in terms of full-reference image quality (FRIQ) metrics like PSNR, SSIM, LPIPS, etc. This is especially important in safety-critical applications like medical imaging, where knowing that, say, the SSIM was poor could potentially avoid a costly misdiagnosis. But since we don't know the true image, computing FRIQ is non-trivial. In this work, we combine conformal prediction with approximate posterior sampling to construct bounds on FRIQ that are guaranteed to hold up to a user-specified error probability. We demonstrate our approach on image denoising and accelerated magnetic resonance imaging (MRI) problems. Code is available at https://github.com/jwen307/quality_uq.

Addressing the Inverse Problem of Far-Field Imaging: A Noniterative Exact Solution for Phase in Imaging

This paper proposes a diffusion-model-based method for addressing inverse problems in optical sound-field imaging. Optical sound-field imaging, known for its high spatial resolution, measures sound by detecting small variations in the refractive index of air caused by sound but often suffers from unavoidable noise contamination. Therefore, we present a diffusion model-based approach for sound-field inverse problems, including denoising, noisy sound-field reconstruction and extrapolation. During inference, sound-field degradation is introduced into the inverse denoising process, with range-null space decomposition used as a solver to handle degradation, iteratively generating degraded sound-field information. Numerical experiments show that our method outperforms other deep-learning-based methods in denoising and reconstruction tasks, and obtains effective results in extrapolation task. The experimental results demonstrate the applicability of our model to the real world.