Deep Neural Networks for Inverse Problems with Pseudodifferential Operators: An Application to Limited-Angle Tomography

We propose a novel convolutional neural network (CNN), called \Psi DONet, designed for learning pseudodifferential operators (\Psi DOs) in the context of linear inverse problems. Our starting point is the iterative soft thresholding algorithm (ISTA), a well-known algorithm to solve sparsity-promoting minimization problems. We show that, under rather general assumptions on the forward operator, the unfolded iterations of ISTA can be interpreted as the successive layers of a CNN, which in turn provides fairly general network architectures that, for a specific choice of the parameters involved, allow us to reproduce ISTA, or a perturbation of ISTA for which we can bound the coefficients of the filters. Our case study is the limited-angle X-ray transform and its application to limited-angle computed tomography (LA-CT). In particular, we prove that, in the case of LA-CT, the operations of upscaling, downscaling, and convolution, which characterize our \Psi DONet and most deep learning schemes, can be exactly determined by combining the convolutional nature of the limited-angle Xray transform and basic properties defining an orthogonal wavelet system. We test two different implementations of \Psi DONet on simulated data from limited-angle geometry, generated from the ellipse data set. Both implementations provide equally good and noteworthy preliminary results, showing the potential of the approach we propose and paving the way to applying the same idea to other convolutional operators which are \Psi DOs or Fourier integral operators

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

Compressive Wave Computation

This paper considers large-scale simulations of wave propagation phenomena. We argue that it is possible to accurately compute a wavefield by decomposing it onto a largely incomplete set of eigenfunctions of the Helmholtz operator, chosen at random, and that this provides a natural way of parallelizing wave simulations for memory-intensive applications. This paper shows that L1-Helmholtz recovery makes sense for wave computation, and identifies a regime in which it is provably effective: the one-dimensional wave equation with coefficients of small bounded variation. Under suitable assumptions we show that the number of eigenfunctions needed to evolve a sparse wavefield defined on N points, accurately with very high probability, is bounded by C log(N) log(log(N)), where C is related to the desired accuracy and can be made to grow at a much slower rate than N when the solution is sparse. The PDE estimates that underlie this result are new to the authors' knowledge and may be of independent mathematical interest; they include an L1 estimate for the wave equation, an estimate of extension of eigenfunctions, and a bound for eigenvalue gaps in Sturm-Liouville problems. Numerical examples are presented in one spatial dimension and show that as few as 10 percents of all eigenfunctions can suffice for accurate results. Finally, we argue that the compressive viewpoint suggests a competitive parallel algorithm for an adjoint-state inversion method in reflection seismology.Comment: 45 pages, 4 figure