Inverse problems in acoustic tomography

Acoustic tomography aims at recovering the unknown parameters that describe a field of interest by studying the physical characteristics of sound propagating through the considered field. The tomographic approach is appealing in that it is non-invasive and allows to obtain a significantly larger amount of data compared to the classical one-sensor one-measurement setup. It has, however, two major drawbacks which may limit its applicability in a practical setting: the methods by which the tomographic data are acquired and then converted to the field values are computationally intensive and often ill-conditioned. This thesis specifically addresses these two shortcomings by proposing novel acoustic tomography algorithms for signal acquisition and field reconstruction. The first part of our exposition deals with some theoretical aspects of the tomographic sampling problems and associated reconstruction schemes for scalar and vector tomography. We show that the classical time-of-flight measurements are not sufficient for full vector field reconstruction. As a solution, an additional set of measurements is proposed. The main advantage of the proposed set is that it can be directly computed from acoustic measurements. It thus avoids the need for extra measuring devices. We then describe three novel reconstruction methods that are conceptually quite different. The first one is based on quadratic optimization and does not require any a priori information. The second method builds upon the notion of sparsity in order to increase the reconstruction accuracy when little data is available. The third approach views tomographic reconstruction as a parametric estimation problem and solves it using recent sampling results on non-bandlimited signals. The proposed methods are compared and their respective advantages are outlined. The second part of our work is dedicated to the application of the proposed algorithms to three practical problems: breast cancer detection, thermal therapy monitoring, and temperature monitoring in the atmosphere. We address the problem of breast cancer detection by computing a map of sound speed in breast tissue. A noteworthy contribution of this thesis is the development of a signal processing technique that significantly reduces the artifacts that arise in very inhomogeneous and absorbent tissue. Temperature monitoring during thermal therapies is then considered. We show how some of our algorithms allow for an increased spatial resolution and propose ways to reduce the computational complexity. Finally, we demonstrate the feasibility of tomographic temperature monitoring in the atmosphere using a custom-built laboratory-scale experiment. In particular, we discuss various practical aspects of time-of-flight measurement using cheap, off-the-shelf sensing devices.

Reconstruction Methods for Cryo-Electron Microscopy: From Model-based to Data-driven

Laurène Donati

The topic of this thesis is the development of new reconstruction methods for cryo-electron microscopy (cryo-EM). Cryo-EM has revolutionized the field of structural biology over the last decade and now permits the regular discovery of biostructures. Yet, the technical challenges associated to cryo-EM are still numerous, and the measurements remain notoriously difficult to process. This calls for fast and robust algorithms that can reliably handle the challenging reconstruction task at hand. In this thesis, we investigated two reconstruction paradigms: model-based and data-driven. Model-based methods formulate the reconstruction task as an inverse problem and rely on a faithful model of the acquisition physics. By contrast, the central philosophy of data-driven approaches is to let the reconstruction algorithm be guided by the measured data through some learning procedure. Both paradigms share a tight link in all our works: their reliance on a rigorous mathematical formulation of the cryo-EM imaging model. The first cryo-EM method we considered is scanning transmission electron tomography (STET), a modality whose primary concern is to reduce the electron dosage required for accurate imaging. To handle this, we developed a tailored acquisition-reconstruction STET framework that relies on the principles of compressed sensing. This scheme permits high-quality reconstruction from a reduced number of measurements, hence greatly preserving the sample. We then designed several reconstruction algorithms for single-particle analysis (SPA), a popular cryo-EM method that enables the determination of structures at near-atomic resolution. A key challenge for the deployment of robust, iterative reconstruction methods in SPA is that they usually come with a prohibitive computational cost if not carefully engineered. To circumvent this problem, we developed a regularized reconstruction scheme whose cost-dominant operation is recast as a discrete convolution, which makes the use of our robust scheme feasible in SPA. Building on this development, we devised a joint optimization framework that efficiently alternates between the reconstruction and the estimation of the unknown orientations. We then explored a learning-based method to estimate the unknown orientations in SPA directly from the acquired dataset of projections. Capitalizing on our ability to model the cryo-EM procedure, we generated large synthetic SPA datasets to train a function---parametrized as a neural network---to predict the relative orientation between two projections based on their similarity. The framework relies on the postulate that it is possible to recover, from these estimated orientation distances, the orientations themselves through an appropriate minimization scheme, as supported by preliminary tests. Finally, we developed a completely new paradigm for SPA reconstruction that leverages the remarkable capability of deep neural networks to capture data distribution. The proposed algorithm uses a generative adversarial network to learn the 3D structure that has simulated projections that most closely match the real data in a distributional sense. By doing so, it can resolve a 3D structure in a single algorithmic run using only the dataset of projections and CTF estimations as inputs. Hence, it bypasses many processing steps that are necessary in the usual cryo-EM reconstruction pipeline, which opens new perspectives for reconstruction in SPA.

EPFL2020

Sampling the Multiple Facets of Light

Gilles Baechler

The theme of this thesis revolves around three important manifestations of light, namely its corpuscular, wave and electromagnetic nature. Our goal is to exploit these principles to analyze, design and build imaging modalities by developing new signal processing and algorithmic tools, based in particular on sampling and sparsity concepts. First, we introduce a new sampling scheme called variable pulse width, which is based on the finite rate of innovation (FRI) sampling paradigm. This new framework enables to sample and perfectly reconstruct weighted sums of Lorentzians; perfect reconstruction from sampled signals is guaranteed by a set of theorems. Second, we turn to the context of light and study its reflection, which is based on the corpuscular model of light. More precisely, we propose to use our FRI-based model to represent bidirectional reflectance distribution functions. We develop dedicated light domes to acquire reflectance functions and use the measurements obtained to demonstrate the usefulness and versatility of our model. In particular, we concentrate on the representation of specularities, which are sharp and bright components generated by the direct reflection of light on surfaces. Third, we explore the wave nature of light through Lippmann photography, a century-old photography technique that acquires the entire spectrum of visible light. This fascinating process captures interferences patterns created by the exposed scene inside the depth of a photosensitive plate. By illuminating the developed plate with a neutral light source, the reflected spectrum corresponds to that of the exposed scene. We propose a mathematical model which precisely explains the technique and demonstrate that the spectrum reproduction suffers from a number of distortions due to the finite depth of the plate and the choice of reflector. In addition to describing these artifacts, we describe an algorithm to invert them, essentially recovering the original spectrum of the exposed scene. Next, the wave nature of light is further generalized to the electromagnetic theory, which we invoke to leverage the concept of polarization of light. We also return to the topic of the representation of reflectance functions and focus this time on the separation of the specular component from the other reflections. We exploit the fact that the polarization of light is preserved in specular reflections and investigate camera designs with polarizing micro-filters with different orientations placed just in front of the camera sensor; the different polarizations of the filters create a mosaic image, from which we propose to extract the specular component. We apply our demosaicing method to several scenes and additionally demonstrate that our approach improves photometric stereo. Finally, we delve into the problem of retrieving the phase information of a sparse signal from the magnitude of its Fourier transform. We propose an algorithm that resolves the phase retrieval problem for sparse signals in three stages. Unlike traditional approaches that recover a discrete approximation of the underlying signal, our algorithm estimates the signal on a continuous domain, which makes it the first of its kind. The concluding chapter outlines several avenues for future research, like new optical devices such as displays and digital cameras, inspired by the topic of Lippmann photography.

EPFL2018

Sampling methods for parametric non-bandlimited signals

Irena Maravic

Sampling theory has experienced a strong research revival over the past decade, which led to a generalization of Shannon's original theory and development of more advanced formulations with immediate relevance to signal processing and communications. For example, it was recently shown that it is possible to develop exact sampling schemes for a large class of non-bandlimited signals, namely, certain signals with finite rate of innovation. A common feature of such signals is that they have a parametric representation with a finite number of degrees of freedom and can be perfectly reconstructed from a finite number of samples. The goal of this thesis is to advance the sampling theory for signals of finite rate of innovation and consider its possible extensions and applications. In the first part of the thesis, we revisit the sampling problem for certain classes of such signals, including non-uniform splines and piecewise polynomials, and develop improved schemes that allow for stable and precise reconstruction in the presence of noise. Specifically, we develop a subspace approach to signal reconstruction, which converts a nonlinear estimation problem into the simpler problem of estimating the parameters of a linear model. This provides an elegant and robust framework for solving a large class of sampling problems, while offering more flexibility than the traditional scheme for bandlimited signals. In the second part of the thesis, we focus on applications of our results to certain classes of nonlinear estimation problems encountered in wideband communication systems, most notably ultra-wideband (UWB) systems, where the bandwidth used for transmission is much larger than the bandwidth or rate of information being sent. We develop several frequency domain methods for channel estimation and synchronization in UWB systems, which yield high-resolution estimates of all relevant channel parameters by sampling a received signal below the traditional Nyquist rate. We also propose algorithms that are suitable for identification of more realistic UWB channel models, where a received signal is made up of pulses with different pulse shapes. Finally, we extend our results to multidimensional signals, and develop exact sampling schemes for certain classes of parametric non-bandlimited 2-D signals, such as sets of 2-D Diracs, polygons or signals with polynomial boundaries.

EPFL2004