Ph.D Thesis


Ph.D StudentCohen Regev
SubjectVariational Inverse Problems with Application to
Medical Ultrasound: From Sparsity through Deep
Learning to Regularization by Denoising
DepartmentDepartment of Electrical and Computers Engineering
Supervisor PROF. Michael Elad
Full Thesis textFull thesis text - English Version


Abstract

Variational inverse problems arise in diverse of research areas such as astrophysics, signal processing, medical imaging and optics. Typically, inverse tasks lead to ill-posed problems whose solutions are unstable under data perturbations. Over the recent decade, numerous regularization methods have been developed, incorporating prior information into the mathematical formulation to ensure unique and stable solutions.

This research concerns with the development of inverse problems for medical imaging. In particular, we investigate the following regularization approaches: utilizing signal structure, exploiting sensor redundancy, integrating model-based methods with data-driven techniques, and employing powerful denoisers as regularization. We focus on the contribution of the aforementioned strategies to ultrasound imaging, commonly-used for visualizing inner body structures. Due to its non-ionizing nature and real-time capabilities, medical ultrasound impacts important clinical segments, however, compared to other prominent imaging modalities, it suffers from poor resolution, low signal-to-noise ratio, etc. These limitations with the demand for real-time processing reduce the performance of ultrasound devices and jeopardize their reliability.

We aim at demonstrating the effectiveness of our regularization techniques to ultrasound imaging. First, we rely on sparse representations to formulate inverse problems concerning different ultrasound imaging modes. We suggest iterative inverse solutions for these problems, leading to improved image quality, clutter suppression and data reduction. To obtain real-time capabilities, we exploit the structure of our iterative schemes to construct model-based network architectures. Thus, we combine the power of data with signal models to design task-tailored deep architectures. This leads to compact networks, trained with limited data, which outperform general-purpose networks in a given task.

Next, we consider sensor-based systems, widely-used in various fields such as radar, commutations, etc. These systems often rely on uniform arrays with equally-spaced elements. However, this uniformity introduces redundancy into the measurements, reducing the system efficiency. Hence, we study the geometric properties of ultrasound systems to show they allow the use of sparse arrays with non-uniform element spacing. We employ sparse arrays and develop simple recovery methods for different ultrasound modes, including 3D imaging, which achieve improved image quality while allowing significant data reduction. The latter suggests that the design of sparse arrays is of great interest. However, creating large sparse arrays with multiple properties leads to NP-hard problems that are intractable in large scale. Thus, we introduce a fractal array design in which a small generator array, optimized to meet given requirements, is recursively expanded to create large arrays proven to satisfy the same specifications.

The final part of this study is motivated by the Plug-and-Play Prior (PnP) and Regularization by Denoising (RED) frameworks, which utilize denoisers as regularization. While both RED and PnP have shown state-of-the-art results in various recovery tasks, their theoretical justification remains incomplete. In this work, we enrich the understanding of RED and its connection to PnP. We do so by reformulating RED as inverse problems utilizing a projection onto the fixed-point set of demicontractive denoisers. Then, we derive simple iterative solutions through which we unify RED and PnP and provide guarantees for their global convergence to optimal solutions.