Kai Krajsek

A Maximum Likelihood Estimator for Choosing the Regularization Parameters in Global Optical Flow Methods

Global optical flow estimation methods based on variational calculus contain a regularization parameter which controls the tradeoff between the different constraints on the optical flow field. The counterpart to the regularization parameter are the hyper-parameters in the Bayesian framework. These hyper-parameters have distinct physical meanings and thus can be inferred from the observable data. We derive a combined marginal maximum likelihood/maximum a posteriori (MML/MAP) estimator for simultaneously estimating hyper-parameters and optical flow for all differential variational approaches directly from the observed signal without any prior knowledge of the optical flow. Experiments demonstrate the performance of this optimization technique and show that the choice of the regularization parameter is an essential key-point in order to obtain precise motion estimation.

Building Blocks for Computer Vision with Stochastic Partial Differential Equations

We discuss the basic concepts of computer vision with stochastic partial differential equations (SPDEs). In typical approaches based on partial differential equations (PDEs), the end result in the best case is usually one value per pixel, the “expected” value. Error estimates or even full probability density functions PDFs are usually not available. This paper provides a framework allowing one to derive such PDFs, rendering computer vision approaches into measurements fulfilling scientific standards due to full error propagation. We identify the image data with random fields in order to model images and image sequences which carry uncertainty in their gray values, e.g. due to noise in the acquisition process.The noisy behaviors of gray values is modeled as stochastic processes which are approximated with the method of generalized polynomial chaos (Wiener-Askey-Chaos). The Wiener-Askey polynomial chaos is combined with a standard spatial approximation based upon piecewise multi-linear finite elements. We present the basic building blocks needed for computer vision and image processing in this stochastic setting, i.e. we discuss the computation of stochastic moments, projections, gradient magnitudes, edge indicators, structure tensors, etc. Finally we show applications of our framework to derive stochastic analogs of well known PDEs for de-noising and optical flow extraction. These models are discretized with the stochastic Galerkin method. Our selection of SPDE models allows us to draw connections to the classical deterministic models as well as to stochastic image processing not based on PDEs. Several examples guide the reader through the presentation and show the usefulness of the framework.

Bayesian model selection for optical flow estimation

Global optical flow techniques minimize a mixture of two terms: a data term relating the observable signal with the optical flow, and a regularization term imposing prior knowledge/assumptions on the solution. A large number of different data terms have been developed since the first global optical flow estimator proposed by Horn and Schunk [1]. Recently [2], these data terms have been classified with respect to their properties. Thus, for image sequences where certain properties about image as well as motion characteristics are known in advance, the appropriate data term can be chosen from this classification. In this contribution, we deal with the situation where the optimal data term is not known in advance. We apply the Bayesian evidence framework for automatically choosing the optimal relative weight between two data terms as well as the regularization term based only on the given input signal.

The edge preserving wiener filter for scalar and tensor valued images

This contribution presents a variation of the Wiener filter criterion, i.e. minimizing the mean squared error, by combining it with the main principle of normalized convolution, i.e. the introduction of prior information in the filter process via the certainty map. Thus, we are able to optimize a filter according to the signal and noise characteristics while preserving edges in images. In spite of its low computational costs the proposed filter schemes outperforms state of the art filter methods working also in the spatial domain. Furthermore, the Wiener filter paradigm is extended from scalar valued data to tensor valued data.

Diffusion filtering without parameter tuning: Models and inference tools

Relations between deterministic (e.g. variational or PDE based methods) and Bayesian inference have been known for a long time. However, a classification of deterministic approaches into those methods which can be handled within a Bayesian framework and those with no such statistical counterpart is still missing in literature. After providing such taxonomy, we present a Bayesian framework for embedding the former ones into a statistical context allowing to equip them with advantages of probabilistic estimation theory. A stochastic point of view allows us (1) to learn influence functions and derivative filter, (2) adapt diffusion and regularization approaches to changes in the image characteristics (e.g. varying noise levels), and (3) to estimate error bounds on the solution. For the latter ones we present alternative learning schemes also allowing their parameters to be related to the image statistics such that hand tuning becomes dispensable. We demonstrate that a statistical point of view on diffusion and regularization schemes leads to image denoising performances comparable with state of the art Markov random field approaches while being computationally much more effective.

On the equivalence of variational and statistical differential motion estimation

In this contribution, we examine variational based motion estimation techniques, e.g. (B. Horn and B. Schunck, 1981), (A. Bruhn, et al., 2005), from a statistical point of view. The fact that all deterministic motivated methods can be described in a Bayesian framework allows the understanding of the physical meaning of the parameters which occur as free parameters in the deterministic framework. Furthermore, these parameters can directly be estimated from observable data when choosing the statistical point of view. The estimation of the optimal regularization parameter is demonstrated to work successfully on image sequences with known ground truth

Wiener-optimized discrete filters for differential motion estimation

Differential motion estimation is based on detecting brightness changes in local image structures. Filters approximating the local gradient are applied to the image sequence for this purpose. Whereas previous approaches focus on the reduction of the systematical approximation error of filters and motion models, the method presented in this paper is based on the statistical characteristics of the data. We developed a method for adapting separable linear shift invariant filters to image sequences or whole classes of image sequences. Therefore, it is possible to optimize the filters according to the systematical errors as well as to the statistical ones.

View-based robot localization using spherical harmonics: concept and first experimental results

Robot self-localization using a hemispherical camera system can be done without correspondences. We present a view-based approach using view descriptors, which enables us to efficiently compare the image signal taken at different locations. A compact representation of the image signal can be computed using Spherical Harmonics as orthonormal basis functions defined on the sphere. This is particularly useful because rotations between two representations can be found easily. Compact view descriptors stored in a database enable us to compute a likelihood for the current view corresponding to a particular position and orientation in the map.

Riemannian Anisotropic Diffusion for Tensor Valued Images

Tensor valued images, for instance originating from diffusion tensor magnetic resonance imaging (DT-MRI), have become more and more important over the last couple of years. Due to the nonlinear structure of such data it is nontrivial to adapt well-established image processing techniques to them. In this contribution we derive anisotropic diffusion equations for tensor-valued images based on the intrinsic Riemannian geometric structure of the space of symmetric positive tensors. In contrast to anisotropic diffusion approaches proposed so far, which are based on the Euclidian metric, our approach considers the nonlinear structure of positive definite tensors by means of the intrinsic Riemannian metric. Together with an intrinsic numerical scheme our approach overcomes a main drawback of former proposed anisotropic diffusion approaches, the so-called eigenvalue swelling effect. Experiments on synthetic data as well as real DT-MRI data demonstrate the value of a sound differential geometric formulation of diffusion processes for tensor valued data.

Signal and noise adapted filters for differential motion estimation

Differential motion estimation in image sequences is based on measuring the orientation of local structures in spatio-temporal signal volumes. For this purpose, discrete filters which yield estimates of the local gradient are applied to the image sequence. Whereas previous approaches to filter optimization concentrate on the reduction of the systematical error of filters and motion models, the method presented in this paper is based on the statistical characteristics of the data. We present a method for adapting linear shift invariant filters to image sequences or whole classes of image sequences. We show how to simultaneously optimize derivative filters according to the systematical errors as well as to the statistical ones.