State-of-the-art image reconstruction often relies on complex, abundantly parameterized deep architectures. We propose an alternative: a data-driven reconstruction method inspired by the classic Tikhonov regularization. Our approach iteratively refines intermediate reconstructions by solving a sequence of quadratic problems. These updates have two key components: (i) learned filters to extract salient image features; and (ii) an attention mechanism that locally adjusts the penalty of the filter responses. Our method matches leading plug-and-play and learned regularizer approaches in performance while offering interpretability, robustness, and convergent behavior. In effect, we bridge traditional regularization and deep learning with a principled reconstruction approach.

We investigate the learning of a convex patch-based synthesis model for the reconstruction of images. In essence, we propose to learn a dictionary via bilevel optimization for denoising. Using implicit differentiation, we find a closed-form formula of the derivative of the minimizer of an objective with respect to the atoms of the dictionary. We also propose a novel way to handle the mean of each patch of the predictions, which improves our model when it is applied to other inverse problems. For minimizing the objective involving the learnt dictionary, we propose an early stopping criterion to further improve the performance of the model for denoising. Finally, we assess our model in a compressed sensing MRI inverse problem and show that, despite being trained on denoising only, our model yields good reconstruction performances.

Lipschitz-constrained neural networks have several advantages over unconstrained ones and can be applied to a variety of problems, making them a topic of attention in the deep learning community. Unfortunately, it has been shown both theoretically and empirically that they perform poorly when equipped with ReLU activation functions. By contrast, neural networks with learnable 1-Lipschitz linear splines are known to be more expressive. In this paper, we show that such networks correspond to global optima of a constrained functional optimization problem that consists of the training of a neural network composed of 1-Lipschitz linear layers and 1-Lipschitz freeform activation functions with second-order total-variation regularization. Further, we propose an efficient method to train these neural networks. Our numerical experiments show that our trained networks compare favorably with existing 1-Lipschitz neural architectures.

In supervised learning, the regularization path is sometimes used as a convenient theoretical proxy for the optimization path of gradient descent initialized from zero. In this paper, we study a modification of the regularization path for infinite-width 2-layer ReLU neural networks with nonzero initial distribution of the weights at different scales. By exploiting a link with unbalanced optimal-transport theory, we show that, despite the non-convexity of the 2-layer network training, this problem admits an infinite-dimensional convex counterpart. We formulate the corresponding functional-optimization problem and investigate its main properties. In particular, we show that, as the scale of the initialization ranges between $0$ and $+\infty$, the associated path interpolates continuously between the so-called kernel and rich regimes. Numerical experiments confirm that, in our setting, the scaling path and the final states of the optimization path behave similarly, even beyond these extreme points.

Abstract. We propose to learn non-convex regularizers with a prescribed upper bound on their weak-convexity modulus. Such regularizers give rise to variational denoisers that minimize a convex energy. They rely on few parameters (less than 15,000) and offer a signal-processing interpretation as they mimic handcrafted sparsity-promoting regularizers. Through numerical experiments, we show that such denoisers outperform convex-regularization methods as well as the popular BM3D denoiser. Additionally, the learned regularizer can be deployed to solve inverse problems with iterative schemes that provably converge. For both CT and MRI reconstruction, the regularizer generalizes well and offers an excellent tradeoff between performance, number of parameters, guarantees, and interpretability when compared to other data-driven approaches.

Recently, variational methods were successfully applied for computing the optical flow in gray and RGB-valued image sequences. A crucial assumption in these models is that pixel-values do not change under transformations. Nowadays, modern image acquisition techniques, such as electron backscatter diffraction (EBSD) used in materials science, can capture images with values in nonlinear spaces. Here, the image values belong to the quotient space ${SO}(3)/ \mathcal S$ of the special orthogonal group modulo the discrete symmetry group of the crystal. For such data, the assumption that pixel-values remain unchanged under transformations appears to be no longer valid. Hence, we propose a variational model for determining the optical flow in ${SO}(3)/\mathcal S$-valued image sequences, taking into account the dependence of pixel-values on the transformation. More precisely, the data is transformed according to the rotation part in the polar decomposition of the Jacobian of the transformation. To model nonsmooth transformations without obtaining so-called staircasing effects, we propose using total generalized variation, such as prior. Then, we prove existence of a minimizer for our model and explain how it can be discretized and minimized by a primal-dual algorithm. Numerical examples illustrate the performance of our method.

Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouvé, Younes, and coworkers [M. I. Miller and L. Younes, Int. J. Comput. Vis., 41 (2001), pp. 61--84; A. Trouvé and L. Younes, Found. Comput. Math., 5 (2005), pp. 173--198] casts this problem in the frame of Riemannian geometry and geodesic paths between images. The associated metric in the space of images incorporates dissipation caused by a viscous flow transporting image intensities and its variations along motion paths. In many applications, images are maps from the image domain into a manifold (e.g., in diffusion tensor imaging (DTI), the manifold of symmetric positive definite matrices with a suitable Riemannian metric). In this paper, we propose a generalized metamorphosis model for manifold-valued images, where the range space is a finite-dimensional Hadamard manifold. A corresponding time discrete version was presented in [S. Neumayer, J. Persch, and G. Steidl, SIAM J. Imaging Sci., 11 (2018), pp. 1898--1930] based on the general variational time discretization proposed in [B. Berkels, A. Effland, and M. Rumpf, SIAM J. Imaging Sci., 8 (2015), pp. 1457--1488]. Here, we prove the Mosco--convergence of the time discrete metamorphosis functional to the proposed manifold-valued metamorphosis model, which implies the convergence of time discrete geodesic paths to a geodesic path in the (time continuous) metamorphosis model. In particular, the existence of geodesic paths is established. In particular, the existence of geodesic paths is established. In fact, images as maps into Hadamard manifold are not only relevant in applications, but it is also shown that the joint convexity of the distance function---which characterizes Hadamard manifolds---is a crucial ingredient to establish existence of the metamorphosis model.

This paper addresses the morphing of manifold-valued images based on the time discrete geodesic paths model of Berkels, Effland, and Rumpf [ SIAM J. Imaging Sci., 8 (2015), pp. 1457--1488]. Although for our manifold-valued setting such an interpretation of the energy functional is not available so far, the model is interesting on its own. We prove the existence of a minimizing sequence within the set of $L^2(\Omega,\mathcal{H})$ images having values in a finite-dimensional Hadamard manifold $\mathcal{H}$ together with a minimizing sequence of admissible diffeomorphisms. To this end, we show that the continuous manifold-valued functions are dense in $L^2(\Omega,\mathcal{H})$. We propose a space discrete model based on a finite difference approach on staggered grids, where we focus on the linearized elastic potential in the regularizing term. The numerical minimization alternates between (i) the computation of a deformation sequence between given images via the parallel solution of certain registration problems for manifold-valued images, and (ii) the computation of an image sequence with fixed first (template) and last (reference) frame based on a given sequence of deformations via the solution of a system of equations arising from the corresponding Euler--Lagrange equation. Numerical examples give a proof of the concept of our ideas.

Failures in fiber-optic networks may be caused by natural disasters, such as floods or earthquakes, as well as other events, such as an Electromagnetic Pulse (EMP) attack. These events occur in specific geographical locations, therefore the geography of the network determines the effect of failure events on the network's connectivity and capacity. In this paper we consider a generalization of the min-cut and max-flow problems under a geographic failure model. Specifically, we consider the problem of finding the minimum number of failures, modeled as circular disks, to disconnect a pair of nodes and the maximum number of failure disjoint paths between pairs of nodes. This model applies to the scenario where an adversary is attacking the network multiple times with intention to reduce its connectivity. We present a polynomial time algorithm to solve the geographic min-cut problem and develop an ILP formulation, an exact algorithm, and a heuristic algorithm for the geographic max-flow problem.

Optical fiber networks consist of fibers that are laid out along physical terrestrial paths. As such, they are vulnerable to geographical physical failures, such as earthquakes and Electromagnetic Pulse (EMP) attacks. Moreover, such disasters can lead to multiple, geographically correlated, failures on the fiber network. Thus, the geographical layout of the fiber infrastructure has a critical impact on the robustness of the network in the face of such geographical physical failures. In this paper, we develop tools to analyze network connectivity after a `random' geographic disaster. The random location of the disaster allows us to model situations where the physical failures are not targeted attacks. In particular, we consider disasters that take the form of a `randomly' located disk in a plane. Using results from geometric probability, we are able to approximate some network performance metrics to such a disaster in polynomial time. We present some numerical results that make clear geographically correlated failures are fundamentally different from independent failures and then discuss network design in the context of random disk-cuts.

Communication networks are vulnerable to natural disasters, such as earthquakes or floods, as well as to physical attacks, such as an electromagnetic pulse (EMP) attack. Such real-world events happen in specific geographical locations and disrupt specific parts of the network. Therefore, the geographical layout of the network determines the impact of such events on the network's connectivity. In this paper, we focus on assessing the vulnerability of (geographical) networks to such disasters. In particular, we aim to identify the most vulnerable parts of the network. That is, the locations of disasters that would have the maximum disruptive effect on the network in terms of capacity and connectivity. We consider graph models in which nodes and links are geographically located on a plane. First, we consider a simplistic bipartite graph model and present a polynomial-time algorithm for finding a worst-case vertical line segment cut. We then generalize the network model to graphs with nodes at arbitrary locations. We model the disaster event as a line segment or a disk and develop polynomial-time algorithms that find a worst-case line segment cut and a worst-case circular cut. Finally, we obtain numerical results for a specific backbone network, thereby demonstrating the applicability of our algorithms to real-world networks. Our novel approach provides a promising new direction for network design to avert geographical disasters or attacks.

Fiber-optic networks are vulnerable to natural disasters, such as tornadoes or earthquakes, as well as to physical failures, such as an anchor cutting underwater fiber cables. Such real-world events occur in specific geographical locations and disrupt specific parts of the network. Therefore, the geography of the network determines the effect of physical events on the network's connectivity and capacity. In this paper, we develop tools to analyze network failures after a `random' geographic disaster. The random location of the disaster allows us to model situations where the physical failures are not targeted attacks. In particular, we consider disasters that take the form of a `random' line in a plane. Using results from geometric probability, we are able to calculate some network performance metrics to such a disaster in polynomial time. In particular, we can evaluate average two-terminal reliability in polynomial time under `random' line-cuts. This is in contrast to the case of independent link failures for which there exists no known polynomial time algorithm to calculate this reliability metric. We also present some numerical results to show the significance of geometry on the survivability of the network and discuss network design in the context of random line-cuts. Our novel approach provides a promising new direction for modeling and designing networks to lessen the effects of geographical disasters or attacks.

Communication networks are vulnerable to natural disasters, such as earthquakes or floods, as well as to physical attacks, such as an Electromagnetic Pulse (EMP) attack. Such real- world events happen in specific geographical locations and disrupt specific parts of the network. Therefore, the geographical layout of the network determines the impact of such events on the network's connectivity. In this paper, we focus on assessing the vulnerability of (geographical) networks to such disasters. In particular, we aim to identify the most vulnerable parts of the network. That is, the locations of disasters that would have the maximum disruptive effect on the network in terms of capacity and connectivity. We consider graph models in which nodes and links are geographically located on a plane, and model the disaster event as a line segment or a circular cut. We develop algorithms that find a worst- case line segment cut and a worst-case circular cut. Then, we obtain numerical results for a specific backbone network, thereby demonstrating the applicability of our algorithms to real-world networks. Our novel approach provides a promising new direction for network design to avert geographical disasters or attacks.