Variational and learning models for image and time series inverse problems

Inverse problems are at the core of many challenging applications. Variational and learning models provide estimated solutions of inverse problems as the outcome of specific reconstruction maps. In the variational approach, the result of the reconstruction map is the solution of a regularized minimization problem encoding information on the acquisition process and prior knowledge on the solution. In the learning approach, the reconstruction map is a parametric function whose parameters are identified by solving a minimization problem depending on a large set of data. In this thesis, we go beyond this apparent dichotomy between variational and learning models and we show they can be harmoniously merged in unified hybrid frameworks preserving their main advantages. We develop several highly efficient methods based on both these model-driven and data-driven strategies, for which we provide a detailed convergence analysis. The arising algorithms are applied to solve inverse problems involving images and time series. For each task, we show the proposed schemes improve the performances of many other existing methods in terms of both computational burden and quality of the solution. In the first part, we focus on gradient-based regularized variational models which are shown to be effective for segmentation purposes and thermal and medical image enhancement. We consider gradient sparsity-promoting regularized models for which we develop different strategies to estimate the regularization strength. Furthermore, we introduce a novel gradient-based Plug-and-Play convergent scheme considering a deep learning based denoiser trained on the gradient domain. In the second part, we address the tasks of natural image deblurring, image and video super resolution microscopy and positioning time series prediction, through deep learning based methods. We boost the performances of supervised, such as trained convolutional and recurrent networks, and unsupervised deep learning strategies, such as Deep Image Prior, by penalizing the losses with handcrafted regularization terms.

Regularization meets GreenAI: a new framework for image reconstruction in life sciences applications

Ill-conditioned inverse problems frequently arise in life sciences, particularly in the context of image deblurring and medical image reconstruction. These problems have been addressed through iterative variational algorithms, which regularize the reconstruction by adding prior knowledge about the problem's solution. Despite the theoretical reliability of these methods, their practical utility is constrained by the time required to converge. Recently, the advent of neural networks allowed the development of reconstruction algorithms that can compute highly accurate solutions with minimal time demands. Regrettably, it is well-known that neural networks are sensitive to unexpected noise, and the quality of their reconstructions quickly deteriorates when the input is slightly perturbed. Modern efforts to address this challenge have led to the creation of massive neural network architectures, but this approach is unsustainable from both ecological and economic standpoints. The recently introduced GreenAI paradigm argues that developing sustainable neural network models is essential for practical applications. In this thesis, we aim to bridge the gap between theory and practice by introducing a novel framework that combines the reliability of model-based iterative algorithms with the speed and accuracy of end-to-end neural networks. Additionally, we demonstrate that our framework yields results comparable to state-of-the-art methods while using relatively small, sustainable models. In the first part of this thesis, we discuss the proposed framework from a theoretical perspective. We provide an extension of classical regularization theory, applicable in scenarios where neural networks are employed to solve inverse problems, and we show there exists a trade-off between accuracy and stability. Furthermore, we demonstrate the effectiveness of our methods in common life science-related scenarios. In the second part of the thesis, we initiate an exploration extending the proposed method into the probabilistic domain. We analyze some properties of deep generative models, revealing their potential applicability in addressing ill-posed inverse problems.

Tensor-Train decomposition for image classification problems

In these last years a great effort has been put in the development of new techniques for automatic object classification, also due to the consequences in many applications such as medical imaging or driverless cars. To this end, several mathematical models have been developed from logistic regression to neural networks. A crucial aspect of these so called classification algorithms is the use of algebraic tools to represent and approximate the input data. In this thesis, we examine two different models for image classification based on a particular tensor decomposition named Tensor-Train (TT) decomposition. The use of tensor approaches preserves the multidimensional structure of the data and the neighboring relations among pixels. Furthermore the Tensor-Train, differently from other tensor decompositions, does not suffer from the curse of dimensionality making it an extremely powerful strategy when dealing with high-dimensional data. It also allows data compression when combined with truncation strategies that reduce memory requirements without spoiling classification performance. The first model we propose is based on a direct decomposition of the database by means of the TT decomposition to find basis vectors used to classify a new object. The second model is a tensor dictionary learning model, based on the TT decomposition where the terms of the decomposition are estimated using a proximal alternating linearized minimization algorithm with a spectral stepsize.

A skin surface characterization system based on capacitive image analysis

During the last few years, several methods have been proposed in order to study and to evaluate characteristic properties of the human skin by using non-invasive approaches. Mostly, these methods cover aspects related to either dermatology, to analyze skin physiology and to evaluate the effectiveness of medical treatments in skin diseases, or dermocosmetics and cosmetic science to evaluate, for example, the effectiveness of anti-aging treatments. To these purposes a routine approach must be followed. Although very accurate and high resolution measurements can be achieved by using conventional methods, such as optical or mechanical profilometry for example, their use is quite limited primarily to the high cost of the instrumentation required, which in turn is usually cumbersome, highlighting some of the limitations for a routine based analysis. This thesis aims to investigate the feasibility of a noninvasive skin characterization system based on the analysis of capacitive images of the skin surface. The system relies on a CMOS portable capacitive device which gives 50 micron/pixel resolution capacitance map of the skin micro-relief. In order to extract characteristic features of the skin topography, image analysis techniques, such as watershed segmentation and wavelet analysis, have been used to detect the main structures of interest: wrinkles and plateau of the typical micro-relief pattern. In order to validate the method, the features extracted from a dataset of skin capacitive images acquired during dermatological examinations of a healthy group of volunteers have been compared with the age of the subjects involved, showing good correlation with the skin ageing effect. Detailed analysis of the output of the capacitive sensor compared with optical profilometry of silicone replica of the same skin area has revealed potentiality and some limitations of this technology. Also, applications to follow-up studies, as needed to objectively evaluate the effectiveness of treatments in a routine manner, are discussed.

Quantifying the Economics of Medical Malpractice - A View from a Civil Law Perspective

Life is full of uncertainties. Legal rules should have a clear intention, motivation and purpose in order to diminish daily uncertainties. However, practice shows that their consequences are complex and hard to predict. For instance, tort law has the general objectives of deterring future negligent behavior and compensating the victims of someone else's negligence. Achieving these goals are particularly difficult in medical malpractice cases. To start with, when patients search for medical care they are typically sick in the first place. In case harm materializes during the treatment, it might be very hard to assess if it was due to substandard medical care or to the patient's poor health conditions. Moreover, the practice of medicine has a positive externality on the society, meaning that the design of legal rules is crucial: for instance, it should not result in physicians avoiding practicing their activity just because they are afraid of being sued even when they acted according to the standard level of care. The empirical literature on medical malpractice has been developing substantially in the past two decades, with the American case being the most studied one. Evidence from civil law tradition countries is more difficult to find. The aim of this thesis is to contribute to the empirical literature on medical malpractice, using two civil law countries as a case-study: Spain and Italy. The goal of this thesis is to investigate, in the first place, some of the consequences of having two separate sub-systems (administrative and civil) coexisting within the same legal system, which is common in civil law tradition countries with a public national health system (such as Spain, France and Portugal). When this holds, different procedures might apply depending on the type of hospital where the injury took place (essentially whether it is a public hospital or a private hospital). Therefore, a patient injured in a public hospital should file a claim in administrative courts while a patient suffering an identical medical accident should file a claim in civil courts. A natural question that the reader might pose is why should both administrative and civil courts decide medical malpractice cases? Moreover, can this specialization of courts influence how judges decide medical malpractice cases? In the past few years, there was a general concern with patient safety, which is currently on the agenda of several national governments. Some initiatives have been taken at the international level, with the aim of preventing harm to patients during treatment and care. A negligently injured patient might present a claim against the health care provider with the aim of being compensated for the economic loss and for pain and suffering. In several European countries, health care is mainly provided by a public national health system, which means that if a patient harmed in a public hospital succeeds in a claim against the hospital, public expenditures increase because the State takes part in the litigation process. This poses a problem in a context of increasing national health expenditures and public debt. In Italy, with the aim of increasing patient safety, some regions implemented a monitoring system on medical malpractice claims. However, if properly implemented, this reform shall also allow for a reduction in medical malpractice insurance costs. This thesis is organized as follows. Chapter 1 provides a review of the empirical literature on medical malpractice, where studies on outcomes and merit of claims, costs and defensive medicine are presented. Chapter 2 presents an empirical analysis of medical malpractice claims arriving to the Spanish Supreme Court. The focus is on reversal rates for civil and administrative decisions. Administrative decisions appealed by the plaintiff have the highest reversal rates. The results show a bias in lower administrative courts, which tend to focus on the State side. We provide a detailed explanation for these results, which can rely on the organization of administrative judges career. Chapter 3 assesses predictors of compensation in medical malpractice cases appealed to the Spanish Supreme Court and investigates the amount of damages attributed to patients. The results show horizontal equity between administrative and civil decisions (controlling for observable case characteristics) and vertical inequity (patients suffering more severe injuries tend to receive higher payouts). In order to execute these analyses, a database of medical malpractice decisions appealed to the Administrative and Civil Chambers of the Spanish Supreme Court from 2006 until 2009 (designated by the Spanish Supreme Court Medical Malpractice Dataset (SSCMMD)) has been created. A description of how the SSCMMD was built and of the Spanish legal system is presented as well. Chapter 4 includes an empirical investigation of the effect of a monitoring system for medical malpractice claims on insurance premiums. In Italy, some regions adopted this policy in different years, while others did not. The study uses data on insurance premiums from Italian public hospitals for the years 2001-2008. This is a significant difference as most of the studies use the insurance company as unit of analysis. Although insurance premiums have risen from 2001 to 2008, the increase was lower for regions adopting a monitoring system for medical claims. Possible implications of this system are also provided. Finally, Chapter 5 discusses the main findings, describes possible future research and concludes.

Efficient ultrasonic signal processing techniques for aided medical diagnostics

Ultrasound imaging is widely used in medical diagnostics as it is the fastest, least invasive, and least expensive imaging modality. However, ultrasound images are intrinsically difficult to be interpreted. In this scenario, Computer Aided Detection (CAD) systems can be used to support physicians during diagnosis providing them a second opinion. This thesis discusses efficient ultrasound processing techniques for computer aided medical diagnostics, focusing on two major topics: (i) Ultrasound Tissue Characterization (UTC), aimed at characterizing and differentiating between healthy and diseased tissue; (ii) Ultrasound Image Segmentation (UIS), aimed at detecting the boundaries of anatomical structures to automatically measure organ dimensions and compute clinically relevant functional indices. Research on UTC produced a CAD tool for Prostate Cancer detection to improve the biopsy protocol. In particular, this thesis contributes with: (i) the development of a robust classification system; (ii) the exploitation of parallel computing on GPU for real-time performance; (iii) the introduction of both an innovative Semi-Supervised Learning algorithm and a novel supervised/semi-supervised learning scheme for CAD system training that improve system performance reducing data collection effort and avoiding collected data wasting. The tool provides physicians a risk map highlighting suspect tissue areas, allowing them to perform a lesion-directed biopsy. Clinical validation demonstrated the system validity as a diagnostic support tool and its effectiveness at reducing the number of biopsy cores requested for an accurate diagnosis. For UIS the research developed a heart disease diagnostic tool based on Real-Time 3D Echocardiography. Thesis contributions to this application are: (i) the development of an automated GPU based level-set segmentation framework for 3D images; (ii) the application of this framework to the myocardium segmentation. Experimental results showed the high efficiency and flexibility of the proposed framework. Its effectiveness as a tool for quantitative analysis of 3D cardiac morphology and function was demonstrated through clinical validation.