Frank Lenzen

Heidelberg University ·  Interdisciplinary Center for Scientific Computing

The implicit convex feasibility problem attempts to find a point in the intersection of a finite family of convex sets, some of which are not explicitly determined but may vary. We develop simultaneous and sequential projection methods capable of handling such problems and demonstrate their applicability to image denoising in a specific medical ima...

We propose a variational approach for estimating egomotion and structure of a static scene from a pair of images recorded by a single moving camera. In our approach the scene structure is described by a set of 3D planar surfaces, which are linked to a SLIC superpixel decomposition of the image domain. The continuously parametrized planes are determ...

Camera motion estimation from observed scene features is an important task in image processing to increase the accuracy of many methods, e.g. optical flow and structure-from-motion. Due to the curved geometry of the state space SE 3 and the non-linear relation to the observed optical flow, many recent filtering approaches use a first-order approxim...

We consider solution-driven adaptive variants of Total Variation , in which the adaptivity is introduced as a fixed point problem. We provide existence theory for such fixed points in the continuous domain. For the applications of image denoising, deblurring and inpainting, we provide experiments which demonstrate that our approach in most cases ou...

Accurate camera motion estimation is a fundamental building block for many Computer Vision algorithms. For improved robustness, temporal consistency of translational and rotational camera velocity is often assumed by propagating motion information forward using stochastic filters. Classical stochastic filters, however, use linear approximations for...

In-camera light scattering is a systematic error of Time-of-Flight depth cameras that significantly reduces the accuracy of the systems. A completely new model is presented, based on raw data calibration and only one additional intrinsic camera parameter. It is shown that the approach effectively removes the errors of in-camera light scattering.

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 probl...

We present an approach to jointly estimating camera motion and dense structure of a static scene in terms of depth maps from monocular image sequences in driver-assistance scenarios. At each instant of time, only two consecutive frames are processed as input data of a joint estimator that fully exploits second-order information of the corresponding...

Total variation (TV) regularization, originally introduced by Rudin, Osher and Fatemi in the context of image denoising, has become widely used in the field of inverse problems. Two major directions of modifications of the original approach were proposed later on. The first concerns adaptive variants of TV regularization, the second focuses on high...

We introduce and study Bregman functions as objectives for non-negative sparse compressed sensing problems together with a related first-order iterative scheme employing non-quadratic proximal terms. This scheme yields closed-form multiplicative updates and handles constraints implicitly. Its analysis does not rely on global Lipschitz continuity in...

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 im...

When considering the task of denoising ToF data, two issues arise concerning the optimal strategy. The first one is the choice of an appropriate denoising method and its adaptation to ToF data, the second one is the issue of the optimal positioning of the denoising step within the processing pipeline between acquisition of raw data of the sensor an...

Due to the demand for depth maps of higher quality than possible with a single depth imaging technique today, there has been an increasing interest in the combination of different depth sensors to produce a “super-camera” that is more than the sum of the individual parts. In this survey paper, we give an overview over methods for the fusion of Time...

Current Time-of-Flight approaches mainly incorporate an continuous wave intensity modulation approach. The phase reconstruction is performed using multiple phase images with different phase shifts which is equivalent to sampling the inherent correlation function at different locations. This active imaging approach delivers a very specific set of in...

There was an error in the acknowledgements section of this paper. The correct acknowledgement text is as follows: This work is part of a joint research project with the Filmakademie Baden-Württemberg, Institute of Animation. It is co-funded by the Intel Visual Computing Institute and under grant 2-4225.16/380 of the ministry of economy Baden-Württe...

We introduce a class of adaptive non-smooth convex variational problems for image denoising in terms of a common data fitting term and a support functional as regularizer. Adaptivity is modeled by a set-valued mapping with closed, compact and convex values, that defines and steers the regularizer depending on the variational solution. This extensio...

We propose a generalization of the total variation (TV) minimization method proposed by Rudin, Osher and Fatemi. This generalization allows for adaptive regularization, which depends on the minimizer itself. Existence theory is provided in the framework of quasi-variational inequalities. We demonstrate the usability of our approach by considering a...

We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation methods for finite-dimensional problems. While for the latter case several optimality bounds are known, to our know...

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 parame...

For denoising depth maps from time-of-flight (ToF) cameras we propose an adaptive total variation based approach of first and second order. This approach allows us to take into account the geometric properties of the depth data, such as edges and slopes. To steer adaptivity we utilize a special kind of structure tensor based on both the amplitude a...

In this paper, for imaging applications, we introduce partial differential equations (PDEs), which allow for correcting displacement errors, for dejittering, and for deinterlacing, respectively, in multi-channel data. These equations are derived via semi-groups for non-convex energy functionals. As a particular example, for gray valued data, we fin...

Total variation regularization and anisotropic filtering have been established as standard methods for image denoising because of their ability to detect and keep prominent edges in the data. Both methods, however, introduce artifacts: In the case of anisotropic filtering, the preservation of edges comes at the cost of the creation of additional st...

We study the problem of displacement errors, i.e. errors induced by a sampling process with distorted locations of the sampling points. Starting with a non-convex regularization method, we apply a semi-group concept and derive a partial differential equation, which allows for correcting displacement errors. As main application for correction of dis...

We propose a partial differential equation to be used for interpolating M-channel data, such as digital color images. This equation is derived via a semi-group from a variational regularization method for minimizing displacement errors. For actual image interpolation, the solution of the PDE is projected onto a space of functions satisfying interp...

This book is devoted to the study of variational methods in imaging. The presentation is mathematically rigorous and covers a detailed treatment of the approach from an inverse problems point of view. Key Features: - Introduces variational methods with motivation from the deterministic, geometric, and stochastic point of view - Bridges the gap betw...

We establish a semi-group solution concept for morphological differen-tial equations, such as the mean curvature flow equation. The proposed method consists in generating flows from generalized minimizers of non-convex energy functionals. We use relaxation and convexification to define generalized minimizers. The main part of this work consists in...

In ground based infrared imaging a well-known technique to reduce the influence of thermal and background noise is chopping and nodding, where four different signals of the same object are recorded from which the object is reconstructed numerically. Since noise in the data can severely affect the reconstruction, regularization algorithms have to be...

For image filtering applications, it has been observed recently that both diffusion filtering and associated regularization models provide similar filtering properties. The comparison has been performed for regularization functionals with convex penalization functional. In this paper we discuss the relation between non-convex regularization functio...

Tikhonov initiated the research on stable methods for the numerical solution of inverse and illposed problems. The theory of Tikhonov regularization devel oped systematically. Till the eighties there has been a success in a rigorous and rather complete analysis of regularization methods for solving linear illposed problems. Around 1989 a regulariza...

We present an algorithm developed particularly to detect gravitationally lensed arcs in clusters of galaxies. This algorithm is suited for automated surveys as well as individual arc detections. New methods are used for image smoothing and source detection. The smoothing is performed by so-called anisotropic diffusion, which maintains the shape of...