Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics

We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian manifolds, where non-analytic ordinary differential equations are involved in virtually all computations. The probabilistic formulation permits marginalising the uncertainty of the numerical solution such that statistics are less sensitive to inaccuracies. This leads to new Riemannian algorithms for mean value computations and principal geodesic analysis. Marginalisation also means results can be less precise than point estimates, enabling a noticeable speed-up over the state of the art. Our approach is an argument for a wider point that uncertainty caused by numerical calculations should be tracked throughout the pipeline of machine learning algorithms.Comment: 11 page (9 page conference paper, plus supplements

Lie algebra-based kinematic prior for 3D human pose tracking

We propose a novel kinematic prior for 3D human pose tracking that allows predicting the position in subsequent frames given the current position. We first define a Riemannian manifold that models the pose and extend it with its Lie algebra to also be able to represent the kinematics. We then learn a joint Gaussian mixture model of both the human pose and the kinematics on this manifold. Finally by conditioning the kinematics on the pose we are able to obtain a distribution of poses for subsequent frames that which can be used as a reliable prior in 3D human pose tracking. Our model scales well to large amounts of data and can be sampled at over 100,000 samples/second. We show it outperforms the widely used Gaussian diffusion model on the challenging Human3.6M dataset.Peer ReviewedPostprint (author's final draft

On-Manifold Preintegration for Real-Time Visual-Inertial Odometry

Current approaches for visual-inertial odometry (VIO) are able to attain highly accurate state estimation via nonlinear optimization. However, real-time optimization quickly becomes infeasible as the trajectory grows over time, this problem is further emphasized by the fact that inertial measurements come at high rate, hence leading to fast growth of the number of variables in the optimization. In this paper, we address this issue by preintegrating inertial measurements between selected keyframes into single relative motion constraints. Our first contribution is a \emph{preintegration theory} that properly addresses the manifold structure of the rotation group. We formally discuss the generative measurement model as well as the nature of the rotation noise and derive the expression for the \emph{maximum a posteriori} state estimator. Our theoretical development enables the computation of all necessary Jacobians for the optimization and a-posteriori bias correction in analytic form. The second contribution is to show that the preintegrated IMU model can be seamlessly integrated into a visual-inertial pipeline under the unifying framework of factor graphs. This enables the application of incremental-smoothing algorithms and the use of a \emph{structureless} model for visual measurements, which avoids optimizing over the 3D points, further accelerating the computation. We perform an extensive evaluation of our monocular \VIO pipeline on real and simulated datasets. The results confirm that our modelling effort leads to accurate state estimation in real-time, outperforming state-of-the-art approaches.Comment: 20 pages, 24 figures, accepted for publication in IEEE Transactions on Robotics (TRO) 201

Correction of joint angles from kinect for balance exercising and assessment

[EN] The new generation of videogame interfaces such as Microsoft's Kinect opens the possibility of implementing exercise programs for physical training, and of evaluating and reducing the risks of elderly people falling. However, applications such as these might require measurements of joint kinematics that are more robust and accurate than the standard output given by the available middleware. This article presents a method based on particle filters for calculating joint angles from the positions of the anatomical points detected by PrimeSense's NITE software. The application of this method to the measurement of lower limb kinematics reduced the error by one order of magnitude, to less than 10 degrees, except for hip axial rotation, and it was advantageous over inverse kinematic analysis, in ensuring a robust and smooth solution without singularities, when the limbs are out-stretched and anatomical landmarks are aligned.This work has been undertaken within the framework of the iStoppFalls project, which has received funding from the European Community (grant agreement FP7-ICT-2011-7-287361) and the Australian Government.De Rosario Martínez, H.; Belda Lois, JM.; Fos Ros, F.; Medina Ripoll, E.; Poveda Puente, R.; Kroll, M. (2014). Correction of joint angles from kinect for balance exercising and assessment. Journal of Applied Biomechanics. 30(2):294-299. https://doi.org/10.1123/jab.2013-0062S29429930

Vision-based and marker-less surgical tool detection and tracking: a review of the literature

In recent years, tremendous progress has been made in surgical practice for example with Minimally Invasive Surgery (MIS). To overcome challenges coming from deported eye-to-hand manipulation, robotic and computer-assisted systems have been developed. Having real-time knowledge of the pose of surgical tools with respect to the surgical camera and underlying anatomy is a key ingredient for such systems. In this paper, we present a review of the literature dealing with vision-based and marker-less surgical tool detection. This paper includes three primary contributions: (1) identification and analysis of data-sets used for developing and testing detection algorithms, (2) in-depth comparison of surgical tool detection methods from the feature extraction process to the model learning strategy and highlight existing shortcomings, and (3) analysis of validation techniques employed to obtain detection performance results and establish comparison between surgical tool detectors. The papers included in the review were selected through PubMed and Google Scholar searches using the keywords: “surgical tool detection”, “surgical tool tracking”, “surgical instrument detection” and “surgical instrument tracking” limiting results to the year range 2000 2015. Our study shows that despite significant progress over the years, the lack of established surgical tool data-sets, and reference format for performance assessment and method ranking is preventing faster improvement

Model Transport: Towards Scalable Transfer Learning on Manifolds

We consider the intersection of two research fields: transfer learning and statistics on manifolds. In particular, we consider, for manifold-valued data, transfer learning of tangent-space models such as Gaussians distributions, PCA, regression, or classifiers. Though one would hope to simply use ordinary Rn-transfer learning ideas, the manifold structure prevents it. We overcome this by basing our method on inner-product-preserving parallel transport, a well-known tool widely used in other problems of statistics on manifolds in computer vision. At first, this straightforward idea seems to suffer from an obvious shortcoming: Transporting large datasets is prohibitively expensive, hindering scalability. Fortunately, with our approach, we never transport data. Rather, we show how the statistical models themselves can be transported, and prove that for the tangent-space models above, the transport “commutes” with learning. Consequently, our compact framework, applicable to a large class of manifolds, is not restricted by the size of either the training or test sets. We demonstrate the approach by transferring PCA and logistic-regression models of real-world data involving 3D shapes and image descriptors