Motivated by important applications in image processing, we study a class of second-order geometric quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems associated to gradient flows for energy decaying. In numerical computations, it turns out that the second-order methods are superior to their first-order counter-parts. We concentrate on (i) a damped second-order total variation flow for, e.g., image denoising and (ii) a damped second-order mean curvature flow for level sets of scalar functions. The latter is connected to a nonconvex variational model capable of correcting displacement errors in image data (e.g., dejittering). For the former equation, we prove the existence and uniqueness of the solution and its long time behavior and provide an analytical solution given some simple initial datum. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces and show the existence and uniqueness of the solution for a regularized version of the equation. Finally, some numerical comparisons of the solution behavior for the new equations with first-order flows are presented.

Quantitative magnetic resonance imaging (qMRI) is concerned with estimating (in physical units) values of magnetic and tissue parameters, e.g., relaxation times $T_1$, $T_2$, or proton density $\rho$. Recently, in [Ma et al., Nature, 495 (2013), pp. 187--193], magnetic resonance fingerprinting (MRF) was introduced as a technique being capable of simultaneously recovering such quantitative parameters by using a two-step procedure: (i) given a probe, a series of magnetization maps are computed and then (ii) matched to (quantitative) parameters with the help of a precomputed dictionary which is related to the Bloch manifold. In this paper, we first put MRF and its variants into perspective with optimization and inverse problems to gain mathematical insights concerning identifiability of parameters under noise and interpretation in terms of optimizers. Motivated by the fact that the Bloch manifold is nonconvex and that the accuracy of the MRF-type algorithms is limited by the “discretization size” of the dictionary, a novel physics-based method for qMRI is proposed. In contrast to the conventional two-step method, our model is dictionary-free and is rather governed by a single nonlinear equation, which is studied analytically. This nonlinear equation is efficiently solved via robustified Newton-type methods. The effectiveness of the new method for noisy and undersampled data is shown both analytically and via extensive numerical examples, for which improvement over MRF and its variants is also documented.

Recently, nonconvex regularization models have been introduced in order to provide a better prior for gradient distributions in real images. They are based on using concave energies $\phi$ in the total variation--type functional ${TV}^\phi(u) := \int \phi(|\nabla u(x)|)\,d x$. In this paper, it is demonstrated that for typical choices of $\phi$, functionals of this type pose several difficulties when extended to the entire space of functions of bounded variation, ${BV}(\Omega)$. In particular, if $\phi(t)=t^q$ for $q \in (0, 1)$, and ${TV}^\phi$ is defined directly for piecewise constant functions and extended via weak* lower semicontinuous envelopes to ${BV}(\Omega)$, then it still holds that ${TV}^\phi(u)=\infty$ for $u$ not piecewise constant. If, on the other hand, ${TV}^\phi$ is defined analogously via continuously differentiable functions, then ${TV}^\phi \equiv 0$ (!). We study a way to remedy the models through additional multiscale regularization and area strict convergence, provided that the energy $\phi(t)=t^q$ is linearized for high values. The fact that such energies actually better match reality and improve reconstructions is demonstrated by statistics and numerical experiments.

The minimization of a functional composed of a nonsmooth and nonadditive regularization term and a combined $L^1$ and $L^2$ data-fidelity term is proposed. It is shown analytically and numerically that the new model has noticeable advantages over popular models in image processing tasks. For the numerical minimization of the new objective, subspace correction methods are introduced which guarantee the convergence and monotone decay of the associated energy along the iterates. Moreover, an estimate of the distance between the outcome of the subspace correction method and the global minimizer of the nonsmooth objective is derived. This estimate and numerical experiments for image denoising, inpainting, and deblurring indicate that in practice the proposed subspace correction methods indeed approach the global solution of the underlying minimization problem.

A nonconvex variational model is introduced which contains the $\ell_q$-``norm,” $q\in (0,1)$, of the gradient of the underlying image in the regularization part together with a least squares--type data fidelity term which may depend on a possibly spatially dependent weighting parameter. Hence, the regularization term in this functional is a nonconvex compromise between the minimization of the support of the reconstruction and the classical convex total variation model. In the discrete setting, existence of a minimizer is proved, and a Newton-type solution algorithm is introduced and its global as well as local superlinear convergence toward a stationary point of a locally regularized version of the problem is established. The potential nonpositive definiteness of the Hessian of the objective during the iteration is handled by a trust-region--based regularization scheme. The performance of the new algorithm is studied by means of a series of numerical tests. For the associated infinite dimensional model an existence result based on the weakly lower semicontinuous envelope is established, and its relation to the original problem is discussed.

A two-phase image restoration method based upon total variation regularization combined with an L(1)-data-fitting term for impulse noise removal and deblurring is proposed. In the first phase, suitable noise detectors are used for identifying image pixels contaminated by noise. Then, in the second phase, based upon the information on the location of noise-free pixels, images are deblurred and denoised simultaneously. For efficiency reasons, in the second phase a superlinearly convergent algorithm based upon Fenchel-duality and inexact semismooth Newton techniques is utilized for solving the associated variational problem. Numerical results prove the new method to be a significantly advance over several state-of-the-art techniques with respect to restoration capability and computational efficiency.

Image restoration based on an $\ell^1$-data-fitting term and edge preserving total variation regularization is considered. The associated nonsmooth energy minimization problem is handled by utilizing Fenchel duality and dual regularization techniques. The latter guarantee uniqueness of the dual solution and an efficient way for reconstructing a primal solution, i.e., the restored image, from a dual solution. For solving the resulting primal-dual system, a semismooth Newton solver is proposed and its convergence is studied. The paper ends with a report on restoration results obtained by the new algorithm for salt-and-pepper or random-valued impulse noise including blurring. A comparison with other methods is provided as well.