Asymptotic Analysis of Inpainting via Universal Shearlet Systems

Recently introduced inpainting algorithms using a combination of applied harmonic analysis and compressed sensing have turned out to be very successful. One key ingredient is a carefully chosen representation system which provides (optimally) sparse approximations of the original image. Due to the common assumption that images are typically governed by anisotropic features, directional representation systems have often been utilized. One prominent example of this class are shearlets, which have the additional benefitallowing faithful implementations. Numerical results show that shearlets significantly outperform wavelets in inpainting tasks. One of those software packages, www.shearlab.org, even offers the flexibility of usingdifferent parameter for each scale, which is not yet covered by shearlet theory. In this paper, we first introduce universal shearlet systems which are associated with an arbitrary scaling sequence, thereby modeling the previously mentioned flexibility. In addition, this novel construction allows for a smooth transition between wavelets and shearlets and therefore enables us to analyze them in a uniform fashion. For a large class of such scaling sequences, we first prove that the associated universal shearlet systems form band-limited Parseval frames for $L^2(\mathbb{R}^2)$ consisting of Schwartz functions. Secondly, we analyze the performance for inpainting of this class of universal shearlet systems within a distributional model situation using an $\ell^1$-analysis minimization algorithm for reconstruction. Our main result in this part states that, provided the scaling sequence is comparable to the size of the (scale-dependent) gap, nearly-perfect inpainting is achieved at sufficiently fine scales

Large-Scale Light Field Capture and Reconstruction

This thesis discusses approaches and techniques to convert Sparsely-Sampled Light Fields (SSLFs) into Densely-Sampled Light Fields (DSLFs), which can be used for visualization on 3DTV and Virtual Reality (VR) devices. Exemplarily, a movable 1D large-scale light field acquisition system for capturing SSLFs in real-world environments is evaluated. This system consists of 24 sparsely placed RGB cameras and two Kinect V2 sensors. The real-world SSLF data captured with this setup can be leveraged to reconstruct real-world DSLFs. To this end, three challenging problems require to be solved for this system: (i) how to estimate the rigid transformation from the coordinate system of a Kinect V2 to the coordinate system of an RGB camera; (ii) how to register the two Kinect V2 sensors with a large displacement; (iii) how to reconstruct a DSLF from a SSLF with moderate and large disparity ranges. To overcome these three challenges, we propose: (i) a novel self-calibration method, which takes advantage of the geometric constraints from the scene and the cameras, for estimating the rigid transformations from the camera coordinate frame of one Kinect V2 to the camera coordinate frames of 12-nearest RGB cameras; (ii) a novel coarse-to-fine approach for recovering the rigid transformation from the coordinate system of one Kinect to the coordinate system of the other by means of local color and geometry information; (iii) several novel algorithms that can be categorized into two groups for reconstructing a DSLF from an input SSLF, including novel view synthesis methods, which are inspired by the state-of-the-art video frame interpolation algorithms, and Epipolar-Plane Image (EPI) inpainting methods, which are inspired by the Shearlet Transform (ST)-based DSLF reconstruction approaches

Applied microlocal analysis of deep neural networks for inverse problems

Deep neural networks have recently shown state-of-the-art performance in different imaging tasks. As an example, EfficientNet is today the best image classifier on the ImageNet challenge. They are also very powerful for image reconstruction, for example, deep learning currently yields the best methods for CT reconstruction. Most imaging problems, such as CT reconstruction, are ill-posed inverse problems, which hence require regularization techniques typically based on a-priori information. Also, due to the human visual system, singularities such as edge-like features are the governing structures of images. This leads to the question of how to incorporate such information into a solver of an inverse problem in imaging and how deep neural networks operate on singularities. The main research theme of this thesis is to introduce theoretically founded approaches to use deep neural networks in combination with model-based methods to solve inverse problems from imaging science. We do this by heavily exploring the singularity structure of images as a-priori information. We then develop a comprehensive analysis of how neural networks act on singularities using predominantly methods from the microlocal analysis. For analyzing the interaction of deep neural networks with singularities, we introduce a novel technique to compute the propagation of wavefront sets through convolutional residual neural networks (conv-ResNet). This is achieved in a two-fold manner: We first study the continuous case where the neural network is defined in an infinite-dimensional continuous space. This problem is tackled by using the structure of these networks as a sequential application of continuous convolutional operators and ReLU non-linearities and applying microlocal analysis techniques to track the propagation of the wavefront set through the layers. This then leads to the so-called \emph{microcanonical relation} that describes the propagation of the wavefront set under the action of such a neural network. Secondly, for studying real-world discrete problems, we digitize the necessary microlocal analysis methods via the digital shearlet transform. The key idea is the fact that the shearlet transform optimally represents Fourier integral operators hence such a discretization decays rapidly, allowing a finite approximation. Fourier integral operators play an important role in microlocal analysis, since it is well known that they preserve singularities on functions, and, in addition, they have a closed form microcanonical relation. Also, based on the newly developed theoretical analysis, we introduce a method that uses digital shearlet coefficients to compute the digital wavefront set of images by a convolutional neural network. Our approach is then used for a similar analysis of the microlocal behavior of the learned-primal dual architecture, which is formed by a sequence of conv-ResNet blocks. This architecture has shown state-of-the-art performance in inverse problem regularization, in particular, computed tomography reconstruction related to the Radon transform. Since the Radon operator is a Fourier integral operator, our microlocal techniques can be applied. Therefore, we can study with high precision the singularities propagation of this architecture. Aiming to empirically analyze our theoretical approach, we focus on the reconstruction of X-ray tomographic data. We approach this problem by using a task-adapted reconstruction framework, in which we combine the task of reconstruction with the task of computing the wavefront set of the original image as a-priori information. Our numerical results show superior performance with respect to current state-of-the-art tomographic reconstruction methods; hence we anticipate our work to also be a significant contribution to the biomedical imaging community.Tiefe neuronale Netze haben in letzter Zeit bei verschiedenen Bildverarbeitungsaufgaben Spitzenleistungen gezeigt. Zum Beispiel ist AlexNet heute der beste Bildklassifikator bei der ImageNet-Challenge. Sie sind auch sehr leistungsfaehig fue die Bildrekonstruktion, zum Beispiel liefert Deep Learning derzeit die besten Methoden fuer die CT-Rekonstruktion. Die meisten Bildgebungsprobleme wie die CT-Rekonstruktion sind schlecht gestellte inverse Probleme, die daher Regularisierungstechniken erfordern, die typischerweise auf vorherigen Informationen basieren. Auch aufgrund des menschlichen visuellen Systems sind Singularitaeten wie kantenartige Merkmale die bestimmenden Strukturen von Bildern. Dies fuehrt zu der Frage, wie man solche Informationen in einen Loeser eines inversen Problems in der Bildverarbeitung einbeziehen kann und wie tiefe neuronale Netze mit Singularitaeten arbeiten. Das Hauptforschungsthema dieser Arbeit ist die Einfuehrung theoretisch fundierter konzeptioneller Ansaetze zur Verwendung von tiefen neuronalen Netzen in Kombination mit modellbasierten Methoden zur Loesung inverser Probleme aus der Bildwissenschaft. Wir tun dies, indem wir die Singularitaetsstruktur von Bildern als Vorinformation intensiv erforschen. Dazu entwickeln wir eine umfassende Analyse, wie neuronale Netze auf Singularitaeten wirken, indem wir vorwiegend Methoden aus der mikrolokalen Analyse verwenden. Um die Interaktion von tiefen neuronalen Netzen mit Singularitaeten zu analysieren, fuehren wir eine neuartige Technik ein, um die Ausbreitung von Wellenfrontsaetzen mit Hilfe von Convolutional Residual neuronalen Netzen (Conv-ResNet) zu berechnen. Dies wird auf zweierlei Weise erreicht: Zunaechst untersuchen wir den kontinuierlichen Fall, bei dem das neuronale Netz in einem unendlich dimensionalen kontinuierlichen Raum definiert ist. Dieses Problem wird angegangen, indem wir die besondere Struktur dieser Netze als sequentielle Anwendung von kontinuierlichen Faltungsoperatoren und ReLU-Nichtlinearitaeten nutzen und mikrolokale Analyseverfahren anwenden, um die Ausbreitung einer Wellenfrontmenge durch die Schichten zu verfolgen. Dies fuehrt dann zu einer mikrokanonischen Beziehung, die die Ausbreitung der Wellenfrontmenge unter ihrer Wirkung beschreibt. Zweitens digitalisieren wir die notwendigen mikrolokalen Analysemethoden ueber die digitale Shearlet-Transformation, wobei die Digitalisierung fuer die Untersuchung realer Probleme notwendig ist. Die Schluesselidee ist die Tatsache, dass die Shearlet-Transformation Fourier-Integraloperatoren optimal repraesentiert, so dass eine solche Diskretisierung schnell abklingt und eine endliche Approximation ermoeglicht. Nebenbei stellen wir auch eine Methode vor, die digitale Shearlet-Koeffizienten verwendet, um den digitalen Wellenfrontsatz von Bildern durch ein Faltungsneuronales Netzwerk zu berechnen. Unser Ansatz wird dann fuer eine aehnliche Analyse fuer die gelernte primale-duale Architektur verwendet, die durch eine Sequenz von conv-ResNet-Bloecken gebildet wird. Diese Architektur hat bei der Rekonstruktion inverser Probleme, insbesondere bei der Rekonstruktion der Computertomographie im Zusammenhang mit der Radon-Transformation, Spitzenleistungen gezeigt. Da der Radon-Operator ein Fourier-Integraloperator ist, koennen unsere mikrolokalen Techniken angewendet werden. Um unseren theoretischen Ansatz numerisch zu analysieren, konzentrieren wir uns auf die Rekonstruktion von Roentgentomographiedaten. Wir naehern uns diesem Problem mit Hilfe eines aufgabenangepassten Rekonstruktionsrahmens, in dem wir die Aufgabe der Rekonstruktion mit der Aufgabe der Berechnung der Wellenfrontmenge des Originalbildes als Vorinformation kombinieren. Unsere numerischen Ergebnisse zeigen eine ueberragende Leistung, daher erwarten wir, dass dies auch ein interessanter Beitrag fuer die biomedizinische Bildgebung sein wird

Densely-sampled light field reconstruction

In this chapter, we motivate the use of densely-sampled light fields as the representation which can bring the required density of light rays for the correct recreation of 3D visual cues such as focus and continuous parallax and can serve as an intermediary between light field sensing and light field display. We consider the problem of reconstructing such a representation from few camera views and approach it in a sparsification framework. More specifically, we demonstrate that the light field is well structured in the set of so-called epipolar images and can be sparsely represented by a dictionary of directional and multi-scale atoms called shearlets. We present the corresponding regularization method, along with its main algorithm and speed-accelerating modifications. Finally, we illustrate its applicability for the cases of holographic stereograms and light field compression.acceptedVersionPeer reviewe