Søren Hauberg

Manifold valued statistics, exact principal, geodesic analysis and the effect of linear, approximations

Manifolds are widely used to model non-linearity arising in a range of computer vision applications. This paper treats statistics on manifolds and the loss of accuracy occurring when linearizing the manifold prior to performing statistical operations. Using recent advances in manifold computations, we present a comparison between the non-linear analog of Principal Component Analysis, Principal Geodesic Analysis, in its linearized form and its exact counterpart that uses true intrinsic distances. We give examples of datasets for which the linearized version provides good approximations and for which it does not. Indicators for the differences between the two versions are then developed and applied to two examples of manifold valued data: outlines of vertebrae from a study of vertebral fractures and spacial coordinates of human skeleton end-effectors acquired using a stereo camera and tracking software.

Unscented Kalman Filtering on Riemannian Manifolds

In recent years there has been a growing interest in problems, where either the observed data or hidden state variables are confined to a known Riemannian manifold. In sequential data analysis this interest has also been growing, but rather crude algorithms have been applied: either Monte Carlo filters or brute-force discretisations. These approaches scale poorly and clearly show a missing gap: no generic analogues to Kalman filters are currently available in non-Euclidean domains. In this paper, we remedy this issue by first generalising the unscented transform and then the unscented Kalman filter to Riemannian manifolds. As the Kalman filter can be viewed as an optimisation algorithm akin to the Gauss-Newton method, our algorithm also provides a general-purpose optimisation framework on manifolds. We illustrate the suggested method on synthetic data to study robustness and convergence, on a region tracking problem using covariance features, an articulated tracking problem, a mean value optimisation and a pose optimisation problem.

Gaussian-like spatial priors for articulated tracking

We present an analysis of the spatial covariance structure of an articulated motion prior in which joint angles have a known covariance structure. From this, a well-known, but often ignored, deficiency of the kinematic skeleton representation becomes clear: spatial variance not only depends on limb lengths, but also increases as the kinematic chains are traversed. We then present two similar Gaussian-like motion priors that are explicitly expressed spatially and as such avoids any variance coming from the representation. The resulting priors are both simple and easy to implement, yet they provide superior predictions.

Natural metrics and least-committed priors for articulated tracking

In articulated tracking, one is concerned with estimating the pose of a person in every frame of a film. This pose is most often represented as a kinematic skeleton where the joint angles are the degrees of freedom. Least-committed predictive models are then phrased as a Brownian motion in joint angle space. However, the metric of the joint angle space is rather unintuitive as it ignores both bone lengths and how bones are connected. As Brownian motion is strongly linked with the underlying metric, this has severe impact on the predictive models. We introduce the spatial kinematic manifold of joint positions, which is embedded in a high dimensional Euclidean space. This Riemannian manifold inherits the metric from the embedding space, such that distances are measured as the combined physical length that joints travel during movements. We then develop a least-committed Brownian motion model on the manifold that respects the natural metric. This model is expressed in terms of a stochastic differential equation, which we solve using a novel numerical scheme. Empirically, we validate the new model in a particle filter based articulated tracking system. Here, we not only outperform the standard Brownian motion in joint angle space, we are also able to specialise the model in ways that otherwise are both difficult and expensive in joint angle space.

Geodesic exponential kernels: When curvature and linearity conflict

We consider kernel methods on general geodesic metric spaces and provide both negative and positive results. First we show that the common Gaussian kernel can only be generalized to a positive definite kernel on a geodesic metric space if the space is flat. As a result, for data on a Riemannian manifold, the geodesic Gaussian kernel is only positive definite if the Riemannian manifold is Euclidean. This implies that any attempt to design geodesic Gaussian kernels on curved Riemannian manifolds is futile. However, we show that for spaces with conditionally negative definite distances the geodesic Laplacian kernel can be generalized while retaining positive definiteness. This implies that geodesic Laplacian kernels can be generalized to some curved spaces, including spheres and hyperbolic spaces. Our theoretical results are verified empirically.

Means in spaces of tree-like shapes

The mean is often the most important statistic of a dataset as it provides a single point that summarizes the entire set. While the mean is readily defined and computed in Euclidean spaces, no commonly accepted solutions are currently available in more complicated spaces, such as spaces of tree-structured data. In this paper we study the notion of means, both generally in Gromovs CAT(0)-spaces (metric spaces of non-positive curvature), but also specifically in the space of tree-like shapes. We prove local existence and uniqueness of means in such spaces and discuss three different algorithms for computing means. We make an experimental evaluation of the three algorithms through experiments on three different sets of data with tree-like structure: a synthetic dataset, a leaf morphology dataset from images, and a set of human airway subtrees from medical CT scans. This experimental study provides great insight into the behavior of the different methods and how they relate to each other. More importantly, it also provides mathematically well-founded, tractable and robust “average trees”. This statistic is of utmost importance due to the ever-presence of tree-like structures in human anatomy, e.g., airways and vascularization systems.

Predicting Articulated Human Motion from Spatial Processes

We present a probabilistic interpretation of inverse kinematics and extend it to sequential data. The resulting model is used to estimate articulated human motion in visual data. The approach allows us to express the prior temporal models in spatial limb coordinates, which is in contrast to most recent work where prior models are derived in terms of joint angles. This approach has several advantages. First of all, it allows us to construct motion models in low dimensional spaces, which makes motion estimation more robust. Secondly, as many types of motion are easily expressed in spatial coordinates, the approach allows us to construct high quality application specific motion models with little effort. Thirdly, the state space is a real vector space, which allows us to use off-the-shelf stochastic processes as motion models, which is rarely possible when working with joint angles. Fourthly, we avoid the problem of accumulated variance, where noise in one joint affects all joints further down the kinematic chains. All this combined allows us to more easily construct high quality motion models. In the evaluation, we show that an activity independent version of our model is superior to the corresponding state-of-the-art model. We also give examples of activity dependent models that would be hard to phrase directly in terms of joint angles.

Stick it articulated tracking using spatial rigid object priors

Articulated tracking of humans is a well-studied field, but most work has treated the humans as being independent of the environment. Recently, Kjellstrom et al. [1] showed how knowledge of interaction with a known rigid object provides constraints that lower the degrees of freedom in the model. While the phrased problem is interesting, the resulting algorithm is computationally too demanding to be of practical use. We present a simple and elegant model for describing this problem. The resulting algorithm is computationally much more efficient, while it at the same time produces superior results

Probabilistic Shortest Path Tractography in DTI Using Gaussian Process ODE Solvers

Tractography in diffusion tensor imaging estimates connectivity in the brain through observations of local diffusivity. These observations are noisy and of low resolution and, as a consequence, connections cannot be found with high precision. We use probabilistic numerics to estimate connectivity between regions of interest and contribute a Gaussian Process tractography algorithm which allows for both quantification and visualization of its posterior uncertainty. We use the uncertainty both in visualization of individual tracts as well as in heat maps of tract locations. Finally, we provide a quantitative evaluation of different metrics and algorithms showing that the adjoint metric (8] combined with our algorithm produces paths which agree most often with experts.

GPU accelerated likelihoods for stereo-based articulated tracking

For many years articulated tracking has been an active research topic in the computer vision community. While working solutions have been suggested, computational time is still problematic. We present a GPU implementation of a ray-casting based likelihood model that is orders of magnitude faster than a traditional CPU implementation. We explain the non-intuitive steps required to attain an optimized GPU implementation, where the dominant part is to hide the memory latency effectively. Benchmarks show that computations which previously required several minutes, are now performed in few seconds.