Geometric Variational Models for Inverse Problems in Imaging

This dissertation develops geometric variational models for different inverse problems in imaging that are ill-posed, designing at the same time efficient numerical algorithms to compute their solutions. Variational methods solve inverse problems by the following two steps: formulation of a variational model as a minimization problem, and design of a minimization algorithm to solve it. This dissertation is organized in the same manner. It first formulates minimization problems associated with geometric models for different inverse problems in imaging, and it then designs efficient minimization algorithms to compute their solutions. The minimization problem summarizes both the data available from the measurements and the prior knowledge about the solution in its objective functional; this naturally leads to the combination of a measurement or data term and a prior term. Geometry can play a role in any of these terms, depending on the properties of the data acquisition system or the object being imaged. In this context, each chapter of this dissertation formulates a variational model that includes geometry in a different manner in the objective functional, depending on the inverse problem at hand. In the context of compressed sensing, the first chapter exploits the geometric properties of images to include an alignment term in the sparsity prior of compressed sensing; this additional prior term aligns the normal vectors of the level curves of the image with the reconstructed signal, and it improves the quality of reconstruction. A two-step recovery method is designed for that purpose: first, it estimates the normal vectors to the level curves of the image; second, it reconstructs an image matching the compressed sensing measurements, the geometric alignment of normals, and the sparsity constraint of compressed sensing. The proposed method is extended to non-local operators in graphs for the recovery of textures. The harmonic active contours of Chapter 2 make use of differential geometry to interpret the segmentation of an image as a minimal surface manifold. In this case, geometry is exploited in both the measurement term, by coupling the different image channels in a robust edge detector, and in the prior term, by imposing smoothness in the segmentation. The proposed technique generalizes existing active contours to higher dimensional spaces and non-flat images; in the plane, it improves the segmentation of images with inhomogeneities and weak edges. Shape-from-shading is investigated in Chapter 3 for the reconstruction of a silicon wafer from images of printed circuits taken with a scanning electron microscope. In this case, geometry plays a role in the image acquisition system, that is, in the measurement term of the objective functional. The prior term involves a smoothness constraint on the surface and a shape prior on the expected pattern in the circuit. The proposed reconstruction method also estimates a deformation field between the ideal pattern design and the reconstructed surface, substituting the model of shape variability necessary in shape priors with an elastic deformation field that quantifies deviations in the manufacturing process. Finally, the techniques used for the design of efficient numerical algorithms are explained with an example problem based on the level set method. To this purpose, Chapter 4 develops an efficient algorithm for the level set method when the level set function is constrained to remain a signed distance function. The distance function is preserved by the introduction of an explicit constraint in the minimization problem, the minimization algorithm is efficient by the adequate use of variable-splitting and augmented Lagrangian techniques. These techniques introduce additional variables, constraints, and Lagrange multipliers in the original minimization problem, and they decompose it into sub-optimization problems that are simple and can be efficiently solved. As a result, the proposed algorithm is five to six times faster than the original algorithm for the level set method

Surface analysis and visualization from multi-light image collections

Multi-Light Image Collections (MLICs) are stacks of photos of a scene acquired with a fixed viewpoint and a varying surface illumination that provides large amounts of visual and geometric information. Over the last decades, a wide variety of methods have been devised to extract information from MLICs and have shown its use in different application domains to support daily activities. In this thesis, we present methods that leverage a MLICs for surface analysis and visualization. First, we provide background information: acquisition setup, light calibration and application areas where MLICs have been successfully used for the research of daily analysis work. Following, we discuss the use of MLIC for surface visualization and analysis and available tools used to support the analysis. Here, we discuss methods that strive to support the direct exploration of the captured MLIC, methods that generate relightable models from MLIC, non-photorealistic visualization methods that rely on MLIC, methods that estimate normal map from MLIC and we point out visualization tools used to do MLIC analysis. In chapter 3 we propose novel benchmark datasets (RealRTI, SynthRTI and SynthPS) that can be used to evaluate algorithms that rely on MLIC and discusses available benchmark for validation of photometric algorithms that can be also used to validate other MLIC-based algorithms. In chapter 4, we evaluate the performance of different photometric stereo algorithms using SynthPS for cultural heritage applications. RealRTI and SynthRTI have been used to evaluate the performance of (Neural)RTI method. Then, in chapter 5, we present a neural network-based RTI method, aka NeuralRTI, a framework for pixel-based encoding and relighting of RTI data. In this method using a simple autoencoder architecture, we show that it is possible to obtain a highly compressed representation that better preserves the original information and provides increased quality of virtual images relighted from novel directions, particularly in the case of challenging glossy materials. Finally, in chapter 6, we present a method for the detection of crack on the surface of paintings from multi-light image acquisitions and that can be used as well on single images and conclude our presentation

Adapted Compressed Sensing: A Game Worth Playing

Despite the universal nature of the compressed sensing mechanism, additional information on the class of sparse signals to acquire allows adjustments that yield substantial improvements. In facts, proper exploitation of these priors allows to significantly increase compression for a given reconstruction quality. Since one of the most promising scopes of application of compressed sensing is that of IoT devices subject to extremely low resource constraint, adaptation is especially interesting when it can cope with hardware-related constraint allowing low complexity implementations. We here review and compare many algorithmic adaptation policies that focus either on the encoding part or on the recovery part of compressed sensing. We also review other more hardware-oriented adaptation techniques that are actually able to make the difference when coming to real-world implementations. In all cases, adaptation proves to be a tool that should be mastered in practical applications to unleash the full potential of compressed sensing

NormalNet: Learning based Guided Normal Filtering for Mesh Denoising

Mesh denoising is a critical technology in geometry processing, which aims to recover high-fidelity 3D mesh models of objects from noise-corrupted versions. In this work, we propose a deep learning based face normal filtering scheme for mesh denoising, called \textit{NormalNet}. Different from natural images, for mesh, it is difficult to collect enough examples to build a robust end-to-end training scheme for deep networks. To remedy this problem, we propose an iterative framework to generate enough face-normal pairs, based on which a convolutional neural networks (CNNs) based scheme is designed for guidance normal learning. Moreover, to facilitate the 3D convolution operation in CNNs, for each face in mesh, we propose a voxelization strategy to transform irregular local mesh structure into regular 4D-array form. Finally, guided normal filtering is performed to obtain filtered face normals, according to which denoised positions of vertices are derived. Compared to the state-of-the-art works, the proposed scheme can generate accurate guidance normals and remove noise effectively while preserving original features and avoiding pseudo-features

Single-pixel imaging 12 years on: a review

Modern cameras typically use an array of millions of detector pixels to capture images. By contrast, single-pixel cameras use a sequence of mask patterns to filter the scene along with the corresponding measurements of the transmitted intensity which is recorded using a single-pixel detector. This review considers the development of single-pixel cameras from the seminal work of Duarte et al. up to the present state of the art. We cover the variety of hardware configurations, design of mask patterns and the associated reconstruction algorithms, many of which relate to the field of compressed sensing and, more recently, machine learning. Overall, single-pixel cameras lend themselves to imaging at non-visible wavelengths and with precise timing or depth resolution. We discuss the suitability of single-pixel cameras for different application areas, including infrared imaging and 3D situation awareness for autonomous vehicles

Measuring and modelling lung microstructure with hyperpolarised gas MRI

This thesis is concerned with the development of new techniques for measuring and modelling lung microstructure with hyperpolarised gas magnetic resonance imaging (MRI). This aim was pursued in the following five chapters: Development of a framework for lobar comparison of lung microstructure measurements derived from computed tomography (CT) and 3He diffusion-weighted MRI evaluated in an asthmatic cohort. Statistically significant linear correlations were obtained between 3He diffusion-weighted MRI and CT lung microstructure metrics in all lobar regions. Implementation of compressed sensing (CS) to facilitate the acquisition of 3D multiple b-value 3He diffusion-weighted MRI in a single breath-hold for whole lung morphometry mapping. Good agreement between CS-derived and fully-sampled whole lung morphometry maps demonstrates that CS undersampled 3He diffusion-weighted MRI is suitable for clinical lung imaging studies. Acquisition of whole lung morphometry maps with 129Xe diffusion-weighted MRI and CS. An empirically-optimised 129Xe diffusion time (8.5 ms) was derived and 129Xe lung morphometry values demonstrated strong agreement with 3He equivalent measurements. This indicates that 129Xe diffusion-weighted MRI is a viable alternative to 3He for whole lung morphometry mapping. Implementation of an in vivo comparison of the stretched exponential and cylinder theoretical gas diffusion models with both 3He and 129Xe diffusion-weighted MRI. Stretched exponential model diffusive length scale was related to cylinder model mean chord length in a non-linear power relationship; while the cylinder model mean alveolar diameter demonstrated excellent agreement with diffusive length scale. Investigation of clinical and physiological changes in lung microstructure with 3He and 129Xe diffusion-weighted MRI. Longitudinal studies with 3He and 129Xe diffusion-weighted MRI were used investigate changes in lung microstructure in cystic fibrosis and idiopathic pulmonary fibrosis. Lung inflation mechanisms at the acinar level were also investigated with 3He and 129Xe diffusion-weighted MRI acquired at two different lung volumes