In continuous optimization, we have proposed different algorithms that significantly go beyond the state-of-the-art in continuous optimization:
* In case of total variation minimization, our proposed hybrid algorithms based on dynamic programming and continuous optimization appear as the currently fastest algorithms for such problems. In particular, if only approximate solutions are necessary, the proposed algorithms are of high interest because they deliver good solutions already after a very small number of iterations.
* For non-smooth and non-convex optimization, we have proposed an inertial variant of the proximal alternating linearization method. We have proven convergence of the sequence of iterates in case of semi-algebraic functions. The inertial PALM algorithm appear to be significantly faster compared to the original PALM algorithm and it shows a certain ability to overcome spurious stationary points. The corresponding paper appears to be among the top-20 of the most downloaded papers of the SIAM Imaging journal.
* It has been known for some time that first-order primal-dual algorithms perform significantly better than their theoretical worst-case performance on problems which are only partially strongly convex in either the primal or dual variable. Examples include total generalized variation minimization, image reconstruction involving a linear operator, e.g. image deconvolution or MRI reconstruction. We have shown that we can partially accelerate first-order primal-dual algorithms and hence can get fast convergence at least at the block of variables corresponding to the strongly convex part of the problem.
On the variational modeling side, we have achieved the following most striking results:
* We have proposed a new convex relaxation of curvature minimizing variational models that represent the gradient of an image in the so-called roto-translation space, which is the 3D product space of the 2D image domain and the domain of the orientation of the image gradient. By means of the representation, we have found a general framework to represent the curvature of the level lines of an image as a convex energy in terms of a vector-valued measure. For numerical simulation, we found a discrete approximation of the continuous model, that can be solved by means of a first-order primal-dual algorithm.
* We have proposed variational networks, which can be seen as a gradient descent on a variational model, where the parameters of the model (filers, potential functions) are allowed to change in each iteration. Hence, variational models combine two important advantages: On the one hand, they offer a very strong model and on the other hand, they are extremely efficient. Moreover, variational networks show strong connections to state-of-the-art deep-learning architectures such as residual networks.
* We have successfully combined efficient algorithms for solving image labeling problems with deep-learning resulting in end-to-end learnable algorithms which yield state-of-the art results.
* Recently, we have proposed algorithms for learning highly accurate discretizations of the total variation and we have shown consistency of the learned discretization in the framework of Gamma-convergence.
