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Second Order Minimum Energy Filtering of Joint Variational Camera Motion and Depth Map Reconstructions

Berger, Johannes Peter

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Camera motion estimation and dense scene reconstruction are essential for modern computer vision applications such as autonomous driving, robot navigation and virtual reality. State-of-the-art methods are usually based on stereo camera systems that use the information about the distance between the two cameras to uniquely estimate the depth map. However, these systems need to be calibrated and are too expensive for some special industrial applications. Thus, we focus in this work on monocular camera systems that consist of a single moving camera. To increase the robustness of the method we use temporal information in terms of filters. These use temporal consistency to improve the accuracy of the estimation of the current state of a system, e.g. the unknown camera motion or the depth map. Instead of using established stochastic filters such as extended Kalman filters, unscented Kalman filters or particle filters we use novel minimum energy filters that do not base on a stochastic model but on the minimization of an energy function. In a first step we derive the minimum energy filter and provide differential equations for the optimal state and the corresponding second-order operator. We demonstrate that this filter is as exact as state-of-the-art stochastic filters for most problems and, in addition to it, superior in more involved scenarios. Then we consider a simple geometric setting for the reconstruction of the camera motion within a static scene based on stereo image data. There we formulate a non-linear filtering problem on the special Euclidean group based on non-linear observations of optical flow and depth map. In experiments we show that the underlying camera motion can be reconstructed with minimum energy filters as accurate as in other state-of-the-art stereo methods. Finally, we present an approach for the joint reconstruction of camera motion and disparity map (inverse of depth map) in a monocular approach by means of minimum energy filters. By introducing a novel disparity group we can derive the filter without additional constraints or barrier functions. Further we generalize the used energy function to a Charbonnier penalty function which is robuster against outliers in the optical flow. We also demonstrate that additional regularizers can be easily integrated within the overall filtering problem providing a rich basis for many applications. From the mathematical point of view we solve by means of minimum energy filters a non-linear filtering problem on a Lie group for a high dimensional problem -- thus a problem which is infeasible for most stochastic filter.

Item Type: Dissertation
Supervisor: Schnörr, Prof. Dr. Christoph
Date of thesis defense: 20 December 2016
Date Deposited: 26 Jan 2017 08:38
Date: 2017
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Applied Mathematics
Subjects: 500 Natural sciences and mathematics
Controlled Keywords: Minimum Energy Filter, Lie Groups, Special Euclidean Group, Visual Odometry, Depth Map
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