arXiv subjects

Christoph Schnörr

Publications and source records attributed to Christoph Schnörr.

At least 55 records · Page 3Linked to original sources

arXiv2016

Symmetry-free SDP Relaxations for Affine Subspace Clustering

We consider clustering problems where the goal is to determine an optimal partition of a given point set in Euclidean space in terms of a collection of affine subspaces. While there is vast literature on heuristics for this kind of problem, such approaches are known to be susceptible to poor initializations and getting trapped in bad local optima. We alleviate these issues by introducing a semidefinite relaxation based on Lasserre's method of moments. While a similiar approach is known for classical Euclidean clustering problems, a generalization to our more general subspace scenario is not straightforward, due to the high symmetry of the objective function that weakens any convex relaxation. We therefore introduce a new mechanism for symmetry breaking based on covering the feasible region with polytopes. Additionally, we introduce and analyze a deterministic rounding heuristic.

math.OC↗

arXiv2016

Joint Recursive Monocular Filtering of Camera Motion and Disparity Map

Monocular scene reconstruction is essential for modern applications such as robotics or autonomous driving. Although stereo methods usually result in better accuracy than monocular methods, they are more expensive and more difficult to calibrate. In this work, we present a novel second order optimal minimum energy filter that jointly estimates the camera motion, the disparity map and also higher order kinematics recursively on a product Lie group containing a novel disparity group. This mathematical framework enables to cope with non-Euclidean state spaces, non-linear observations and high dimensions which is infeasible for most classical filters. To be robust against outliers, we use a generalized Charbonnier energy function in this framework rather than a quadratic energy function as proposed in related work. Experiments confirm that our method enables accurate reconstructions on-par with state-of-the-art.

cs.CV↗

arXiv2016

A Geometric Approach to Color Image Regularization

We present a new vectorial total variation method that addresses the problem of color consistent image filtering. Our approach is inspired from the double-opponent cell representation in the human visual cortex. Existing methods of vectorial total variation regularizers have insufficient (or no) coupling between the color channels and thus may introduce color artifacts. We address this problem by introducing a novel coupling between the color channels related to a pullback-metric from the opponent space to the data (RGB color) space. Our energy is a non-convex, non-smooth higher-order vectorial total variation approach and promotes color consistent image filtering via a coupling term. For a convex variant, we show well-posedness and existence of a solution in the space of vectorial bounded variation. For the higher-order scheme we employ a half-quadratic strategy, which model the non-convex energy terms as the infimum of a sequence of quadratic functions. In experiments, we elaborate on traditional image restoration applications of inpainting, deblurring and denoising. Regarding the latter, we demonstrate state of the art restoration quality with respect to structure coherence and color consistency.

cs.CV↗

arXiv2016

Image Labeling by Assignment

We introduce a novel geometric approach to the image labeling problem. Abstracting from specific labeling applications, a general objective function is defined on a manifold of stochastic matrices, whose elements assign prior data that are given in any metric space, to observed image measurements. The corresponding Riemannian gradient flow entails a set of replicator equations, one for each data point, that are spatially coupled by geometric averaging on the manifold. Starting from uniform assignments at the barycenter as natural initialization, the flow terminates at some global maximum, each of which corresponds to an image labeling that uniquely assigns the prior data. Our geometric variational approach constitutes a smooth non-convex inner approximation of the general image labeling problem, implemented with sparse interior-point numerics in terms of parallel multiplicative updates that converge efficiently.

cs.CV↗

arXiv2016

Multicuts and Perturb & MAP for Probabilistic Graph Clustering

We present a probabilistic graphical model formulation for the graph clustering problem. This enables to locally represent uncertainty of image partitions by approximate marginal distributions in a mathematically substantiated way, and to rectify local data term cues so as to close contours and to obtain valid partitions. We exploit recent progress on globally optimal MAP inference by integer programming and on perturbation-based approximations of the log-partition function, in order to sample clusterings and to estimate marginal distributions of node-pairs both more accurately and more efficiently than state-of-the-art methods. Our approach works for any graphically represented problem instance. This is demonstrated for image segmentation and social network cluster analysis. Our mathematical ansatz should be relevant also for other combinatorial problems.

cs.CV↗

arXiv2015

Partial Optimality by Pruning for MAP-Inference with General Graphical Models

We consider the energy minimization problem for undirected graphical models, also known as MAP-inference problem for Markov random fields which is NP-hard in general. We propose a novel polynomial time algorithm to obtain a part of its optimal non-relaxed integral solution. Our algorithm is initialized with variables taking integral values in the solution of a convex relaxation of the MAP-inference problem and iteratively prunes those, which do not satisfy our criterion for partial optimality. We show that our pruning strategy is in a certain sense theoretically optimal. Also empirically our method outperforms previous approaches in terms of the number of persistently labelled variables. The method is very general, as it is applicable to models with arbitrary factors of an arbitrary order and can employ any solver for the considered relaxed problem. Our method's runtime is determined by the runtime of the convex relaxation solver for the MAP-inference problem.

cs.AI↗

arXiv2015

Second-Order Recursive Filtering on the Rigid-Motion Lie Group SE(3) Based on Nonlinear Observations

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 approximation and assume a Gaussian a posteriori distribution or restrict the state to Euclidean geometry. The physical model is usually also limited to uniform motions. We propose a second-order minimum energy filter with a generalized kinematic model that copes with the full geometry of SE(3) as well as with the nonlinear dependencies between the state space and observations. The derived filter enables reconstructing motions correctly for synthetic and real scenes, e.g. from the KITTI benchmark. Our experiments confirm that the derived minimum energy filter with higher-order state differential equation copes with higher-order kinematics and is also able to minimize model noise. We also show that the proposed filter is superior to state-of-the-art extended Kalman filters on Lie groups in the case of linear observations and that our method reaches the accuracy of modern visual odometry methods.

math.OC↗

arXiv2015

Second Order Minimum Energy Filtering on $\operatorname{SE}_3$ with Nonlinear Measurement Equations

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 the non-linear observer model and for the non-linear structure of the underlying Lie Group $\operatorname{SE}_3$ and have to approximate the unknown posteriori distribution. In this paper we employ a non-linear measurement model for the camera motion estimation problem that incorporates multiple observation equations. We solve the underlying filtering problem using a novel Minimum Energy Filter on $\operatorname{SE}_3$ and give explicit expressions for the optimal state variables. Experiments on the challenging KITTI benchmark show that, although a simple motion model is only employed, our approach improves rotational velocity estimation and otherwise is on par with the state-of-the-art.

math.OC↗

arXiv2014

Globally Optimal Joint Image Segmentation and Shape Matching Based on Wasserstein Modes

A functional for joint variational object segmentation and shape matching is developed. The formulation is based on optimal transport w.r.t. geometric distance and local feature similarity. Geometric invariance and modelling of object-typical statistical variations is achieved by introducing degrees of freedom that describe transformations and deformations of the shape template. The shape model is mathematically equivalent to contour-based approaches but inference can be performed without conversion between the contour and region representations, allowing combination with other convex segmentation approaches and simplifying optimization. While the overall functional is non-convex, non-convexity is confined to a low-dimensional variable. We propose a locally optimal alternating optimization scheme and a globally optimal branch and bound scheme, based on adaptive convex relaxation. Combining both methods allows to eliminate the delicate initialization problem inherent to many contour based approaches while remaining computationally practical. The properties of the functional, its ability to adapt to a wide range of input data structures and the different optimization schemes are illustrated and compared by numerical experiments.

cs.CV↗

arXiv2014

Solving QVIs for Image Restoration with Adaptive Constraint Sets

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 problems 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 poses severe problems. In the present paper we prove uniqueness for a larger class of problems and in particular independent of the image size. Moreover, we provide a numerical algorithm with proved convergence. Experimental results support our theoretical findings.

math.OC↗

arXiv2014

A Comparative Study of Modern Inference Techniques for Structured Discrete Energy Minimization Problems

Szeliski et al. published an influential study in 2006 on energy minimization methods for Markov Random Fields (MRF). This study provided valuable insights in choosing the best optimization technique for certain classes of problems. While these insights remain generally useful today, the phenomenal success of random field models means that the kinds of inference problems that have to be solved changed significantly. Specifically, the models today often include higher order interactions, flexible connectivity structures, large la\-bel-spaces of different cardinalities, or learned energy tables. To reflect these changes, we provide a modernized and enlarged study. We present an empirical comparison of 32 state-of-the-art optimization techniques on a corpus of 2,453 energy minimization instances from diverse applications in computer vision. To ensure reproducibility, we evaluate all methods in the OpenGM 2 framework and report extensive results regarding runtime and solution quality. Key insights from our study agree with the results of Szeliski et al. for the types of models they studied. However, on new and challenging types of models our findings disagree and suggest that polyhedral methods and integer programming solvers are competitive in terms of runtime and solution quality over a large range of model types.

cs.CV↗

arXiv2014

Probabilistic Intra-Retinal Layer Segmentation in 3-D OCT Images Using Global Shape Regularization

With the introduction of spectral-domain optical coherence tomography (OCT), resulting in a significant increase in acquisition speed, the fast and accurate segmentation of 3-D OCT scans has become evermore important. This paper presents a novel probabilistic approach, that models the appearance of retinal layers as well as the global shape variations of layer boundaries. Given an OCT scan, the full posterior distribution over segmentations is approximately inferred using a variational method enabling efficient probabilistic inference in terms of computationally tractable model components: Segmenting a full 3-D volume takes around a minute. Accurate segmentations demonstrate the benefit of using global shape regularization: We segmented 35 fovea-centered 3-D volumes with an average unsigned error of 2.46 $\pm$ 0.22 μm as well as 80 normal and 66 glaucomatous 2-D circular scans with errors of 2.92 $\pm$ 0.53 μm and 4.09 $\pm$ 0.98 μm respectively. Furthermore, we utilized the inferred posterior distribution to rate the quality of the segmentation, point out potentially erroneous regions and discriminate normal from pathological scans. No pre- or postprocessing was required and we used the same set of parameters for all data sets, underlining the robustness and out-of-the-box nature of our approach.

cs.CV↗

arXiv2013

Shape from Texture using Locally Scaled Point Processes

Shape from texture refers to the extraction of 3D information from 2D images with irregular texture. This paper introduces a statistical framework to learn shape from texture where convex texture elements in a 2D image are represented through a point process. In a first step, the 2D image is preprocessed to generate a probability map corresponding to an estimate of the unnormalized intensity of the latent point process underlying the texture elements. The latent point process is subsequently inferred from the probability map in a non-parametric, model free manner. Finally, the 3D information is extracted from the point pattern by applying a locally scaled point process model where the local scaling function represents the deformation caused by the projection of a 3D surface onto a 2D image.

stat.AP↗

arXiv2013

Phase Transitions and Cosparse Tomographic Recovery of Compound Solid Bodies from Few Projections

We study unique recovery of cosparse signals from limited-angle tomographic measurements of two- and three-dimensional domains. Admissible signals belong to the union of subspaces defined by all cosupports of maximal cardinality $\ell$ with respect to the discrete gradient operator. We relate $\ell$ both to the number of measurements and to a nullspace condition with respect to the measurement matrix, so as to achieve unique recovery by linear programming. These results are supported by comprehensive numerical experiments that show a high correlation of performance in practice and theoretical predictions. Despite poor properties of the measurement matrix from the viewpoint of compressed sensing, the class of uniquely recoverable signals basically seems large enough to cover practical applications, like contactless quality inspection of compound solid bodies composed of few materials.

math.NA↗

arXiv2013

Contour Manifolds and Optimal Transport

Describing shapes by suitable measures in object segmentation, as proposed in [24], allows to combine the advantages of the representations as parametrized contours and indicator functions. The pseudo-Riemannian structure of optimal transport can be used to model shapes in ways similar as with contours, while the Kantorovich functional enables the application of convex optimization methods for global optimality of the segmentation functional. In this paper we provide a mathematical study of the shape measure representation and its relation to the contour description. In particular we show that the pseudo-Riemannian structure of optimal transport, when restricted to the set of shape measures, yields a manifold which is diffeomorphic to the manifold of closed contours. A discussion of the metric induced by optimal transport and the corresponding geodesic equation is given.

math.DG↗

arXiv2013

Convex Variational Image Restoration with Histogram Priors

We present a novel variational approach to image restoration (e.g., denoising, inpainting, labeling) that enables to complement established variational approaches with a histogram-based prior enforcing closeness of the solution to some given empirical measure. By minimizing a single objective function, the approach utilizes simultaneously two quite different sources of information for restoration: spatial context in terms of some smoothness prior and non-spatial statistics in terms of the novel prior utilizing the Wasserstein distance between probability measures. We study the combination of the functional lifting technique with two different relaxations of the histogram prior and derive a jointly convex variational approach. Mathematical equivalence of both relaxations is established and cases where optimality holds are discussed. Additionally, we present an efficient algorithmic scheme for the numerical treatment of the presented model. Experiments using the basic total-variation based denoising approach as a case study demonstrate our novel regularization approach.

math.OC↗

arXiv2012

Critical Parameter Values and Reconstruction Properties of Discrete Tomography: Application to Experimental Fluid Dynamics

We analyze representative ill-posed scenarios of tomographic PIV with a focus on conditions for unique volume reconstruction. Based on sparse random seedings of a region of interest with small particles, the corresponding systems of linear projection equations are probabilistically analyzed in order to determine (i) the ability of unique reconstruction in terms of the imaging geometry and the critical sparsity parameter, and (ii) sharpness of the transition to non-unique reconstruction with ghost particles when choosing the sparsity parameter improperly. The sparsity parameter directly relates to the seeding density used for PIV in experimental fluids dynamics that is chosen empirically to date. Our results provide a basic mathematical characterization of the PIV volume reconstruction problem that is an essential prerequisite for any algorithm used to actually compute the reconstruction. Moreover, we connect the sparse volume function reconstruction problem from few tomographic projections to major developments in compressed sensing.

math.NA↗

arXiv2012

Average Case Recovery Analysis of Tomographic Compressive Sensing

The reconstruction of three-dimensional sparse volume functions from few tomographic projections constitutes a challenging problem in image reconstruction and turns out to be a particular instance problem of compressive sensing. The tomographic measurement matrix encodes the incidence relation of the imaging process, and therefore is not subject to design up to small perturbations of non-zero entries. We present an average case analysis of the recovery properties and a corresponding tail bound to establish weak thresholds, in excellent agreement with numerical experiments. Our result improve the state-of-the-art of tomographic imaging in experimental fluid dynamics by a factor of three.

math.NA↗