Recursive Motion Estimation on the Essential Manifold

Visual motion estimation can be regarded as estimation of the state of a system of difference equations with unknown inputs defined on a manifold. Such a system happens to be "linear", but it is defined on a space (the so called "Essential manifold") which is not a linear (vector) space. In this paper we will introduce a novel perspective for viewing the motion estimation problem which results in three original schemes for solving it. The first consists in "flattening the space" and solving a nonlinear estimation problem on the flat (euclidean) space. The second approach consists in viewing the system as embedded in a larger euclidean space (the smallest of the embedding spaces), and solving at each step a linear estimation problem on a linear space, followed by a "projection" on the manifold (see fig. 5). A third "algebraic" formulation of motion estimation is inspired by the structure of the problem in local coordinates (flattened space), and consists in a double iteration for solving an "adaptive fixed-point" problem (see fig. 6). Each one of these three schemes outputs motion estimates together with the joint second order statistics of the estimation error, which can be used by any structure from motion module which incorporates motion error [20, 23] in order to estimate 3D scene structure. The original contribution of this paper involves both the problem formulation, which gives new insight into the differential geometric structure of visual motion estimation, and the ideas generating the three schemes. These are viewed within a unified framework. All the schemes have a strong theoretical motivation and exhibit accuracy, speed of convergence, real time operation and flexibility which are superior to other existing schemes [1, 20, 23]. Simulations are presented for real and synthetic image sequences to compare the three schemes against each other and highlight the peculiarities of each one

Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models

This tutorial provides a gentle introduction to the particle Metropolis-Hastings (PMH) algorithm for parameter inference in nonlinear state-space models together with a software implementation in the statistical programming language R. We employ a step-by-step approach to develop an implementation of the PMH algorithm (and the particle filter within) together with the reader. This final implementation is also available as the package pmhtutorial in the CRAN repository. Throughout the tutorial, we provide some intuition as to how the algorithm operates and discuss some solutions to problems that might occur in practice. To illustrate the use of PMH, we consider parameter inference in a linear Gaussian state-space model with synthetic data and a nonlinear stochastic volatility model with real-world data.Comment: 41 pages, 7 figures. In press for Journal of Statistical Software. Source code for R, Python and MATLAB available at: https://github.com/compops/pmh-tutoria

Novel Computational Methods for State Space Filtering

The state-space formulation for time-dependent models has been long used invarious applications in science and engineering. While the classical Kalman filter(KF) provides optimal posterior estimation under linear Gaussian models, filteringin nonlinear and non-Gaussian environments remains challenging.Based on the Monte Carlo approximation, the classical particle filter (PF) can providemore precise estimation under nonlinear non-Gaussian models. However, it suffers fromparticle degeneracy. Drawing from optimal transport theory, the stochastic map filter(SMF) accommodates a solution to this problem, but its performance is influenced bythe limited flexibility of nonlinear map parameterisation. To account for these issues,a hybrid particle-stochastic map filter (PSMF) is first proposed in this thesis, wherethe two parts of the split likelihood are assimilated by the PF and SMF, respectively.Systematic resampling and smoothing are employed to alleviate the particle degeneracycaused by the PF. Furthermore, two PSMF variants based on the linear and nonlinearmaps (PSMF-L and PSMF-NL) are proposed, and their filtering performance is comparedwith various benchmark filters under different nonlinear non-Gaussian models.Although achieving accurate filtering results, the particle-based filters require expensive computations because of the large number of samples involved. Instead, robustKalman filters (RKFs) provide efficient solutions for the linear models with heavy-tailednoise, by adopting the recursive estimation framework of the KF. To exploit the stochasticcharacteristics of the noise, the use of heavy-tailed distributions which can fit variouspractical noises constitutes a viable solution. Hence, this thesis also introduces a novelRKF framework, RKF-SGαS, where the signal noise is assumed to be Gaussian and theheavy-tailed measurement noise is modelled by the sub-Gaussian α-stable (SGαS) distribution. The corresponding joint posterior distribution of the state vector and auxiliaryrandom variables is estimated by the variational Bayesian (VB) approach. Four differentminimum mean square error (MMSE) estimators of the scale function are presented.Besides, the RKF-SGαS is compared with the state-of-the-art RKFs under three kinds ofheavy-tailed measurement noises, and the simulation results demonstrate its estimationaccuracy and efficiency.One notable limitation of the proposed RKF-SGαS is its reliance on precise modelparameters, and substantial model errors can potentially impede its filtering performance. Therefore, this thesis also introduces a data-driven RKF method, referred to asRKFnet, which combines the conventional RKF framework with a deep learning technique. An unsupervised scheduled sampling technique (USS) is proposed to improve theistability of the training process. Furthermore, the advantages of the proposed RKFnetare quantified with respect to various traditional RKFs

The Kalman-Levy filter

The Kalman filter combines forecasts and new observations to obtain an estimation which is optimal in the sense of a minimum average quadratic error. The Kalman filter has two main restrictions: (i) the dynamical system is assumed linear and (ii) forecasting errors and observational noises are taken Gaussian. Here, we offer an important generalization to the case where errors and noises have heavy tail distributions such as power laws and L\'evy laws. The main tool needed to solve this ``Kalman-L\'evy'' filter is the ``tail-covariance'' matrix which generalizes the covariance matrix in the case where it is mathematically ill-defined (i.e. for power law tail exponents $\mu \leq 2$). We present the general solution and discuss its properties on pedagogical examples. The standard Kalman-Gaussian filter is recovered for the case $\mu = 2$. The optimal Kalman-L\'evy filter is found to deviate substantially fro the standard Kalman-Gaussian filter as $\mu$ deviates from 2. As $\mu$ decreases, novel observations are assimilated with less and less weight as a small exponent $\mu$ implies large errors with significant probabilities. In terms of implementation, the price-to-pay associated with the presence of heavy tail noise distributions is that the standard linear formalism valid for the Gaussian case is transformed into a nonlinear matrice equation for the Kalman-L\'evy filter. Direct numerical experiments in the univariate case confirms our theoretical predictions.Comment: 41 pages, 9 figures, correction of errors in the general multivariate cas