We consider a class of quasi-variational inequalities (QVIs) for adaptive image restoration, where the adaptivity is described via solution-dependent constraint sets. In previous work we studied both theoretical and numerical issues. While we were able to show the existence of solutions for a relatively broad class of problems, we encountered difficulties concerning uniqueness of the solution as well as convergence of existing algorithms for solving QVIs. In particular, it seemed that with increasing image size the growing condition number of the involved differential operator posed severe problems. In the present paper we prove uniqueness for a larger class of problems, particularly independent of the image size. Moreover, we provide a numerical algorithm with proved convergence. Experimental results support our theoretical findings.

A new approach for edge detection in Time-of-Flight (ToF) depth images is presented. Especially for depth images, accurate edge detection can facilitate many image processing tasks, but rarely any methods for ToF data exist. The proposed algorithm yields highly accurate results through combining edge information both from the intensity and depth image acquired by the imager. The applicability and advantage of the new approach is demonstrated on several recorded scenes and through ToF denoising using adaptive total variation as an application. It is shown that results improve considerably compared to another state-of-the art edge detection algorithm adapted for ToF depth images.

We present an approach to jointly estimating camera motion and dense scene structure in terms of depth maps from monocular image sequences in driver-assistance scenarios. For two consecutive frames of a sequence taken with a single fast moving camera, the approach combines numerical estimation of egomotion on the Euclidean manifold of motion parameters with variational regularization of dense depth map estimation. Embedding this online joint estimator into a recursive framework achieves a pronounced spatio-temporal filtering effect and robustness. We report the evaluation of thousands of images taken from a car moving at speed up to 100 km/h. The results compare favorably with two alternative settings that require more input data: stereo based scene reconstruction and camera motion estimation in batch mode using multiple frames. The employed benchmark dataset is publicly available.

Abstract In this article the modulation of intensity matrices arising in cancer radiation therapy using multileaf collimators (MLC) is investigated. It is shown that the problem is equivalent to decomposing a given integer matrix into a positive linear combination of (0, 1) matrices. These matrices, called shape matrices, must have the strict consecutive‐1‐property, together with another property derived from the technological restrictions of the MLC equipment. Various decompositions can be evaluated by their beam‐on time (time during which radiation is applied to the patient) or the treatment time (beam‐on time plus time for setups). We focus on the former, and develop a nonlinear mixed‐integer programming formulation of the problem. This formulation can be decomposed to yield a column generation formulation: a linear program with a large number of variables that can be priced by solving a subproblem. We then develop a network model in which paths in the network correspond to feasible shape matrices. As a consequence, we deduce that the column generation subproblem can be solved as a shortest path problem. Furthermore, we are able to develop two alternative models of the problem as side‐constrained network flow formulations, and so obtain our main theoretical result that the problem is solvable in polynomial time. Finally, a numerical comparison of our exact solutions with those of well‐known heuristic methods shows that the beam‐on time can be reduced by a considerable margin. © 2004 Wiley Periodicals, Inc.