Recursive Update Equations Research Articles

Recursive Local Summation of RX Detection for Hyperspectral Image Using Sliding Windows

Anomaly detection has received considerable interest for hyperspectral data exploitation due to its high spectral resolution. Fast processing and good detection performance are practically significant in real world problems. Aiming at these requirements, this paper develops a recursive local summation RX anomaly detection approach by virtue of sliding windows. This paper develops a recursive local summation RX anomaly detection approach by virtue of sliding windows. A causal sample covariance/correlation matrix is derived for local window background. As for the real-time sliding windows, the W o o d b u r y identity is used in recursive update equations, which could avoid the calculation of historical information and thus speed up the processing. Furthermore, a background suppression algorithm is also proposed in this paper, which removes the current under test pixel from the recursively update processing. Experiments are implemented on a real hyperspectral image. The experiment results demonstrate that the proposed anomaly detector outperforms the traditional real-time local background detector and has a significant speed-up effect on calculation time compared with the traditional detectors.

Progressive line processing of global and local real-time anomaly detection in hyperspectral images

Hyperspectral imaging, which is characterized by its abundant spectral and spatial information, can effectively identify and detect ground objects. In order to detect moving targets and relieve the stress of big data storage, real-time processing of anomaly detection is greatly desired. This paper investigates both global and local real-time implementations of the most widely used RX detector in a line-by-line fashion. Firstly, global and local causal frameworks are designed to meet the causality, which is one requirement of real-time character. Secondly, taking advantage of the Woodbury matrix identity, recursive update equations of the inverse covariance matrix and background data estimate mean are derived, thereby achieving very low computational complexity. As for local real-time architecture, multiple local semi-windows are designed to simultaneously detect all pixels of a data line. This designation has an advantage that it is very beneficial for the implementation of real-time anomaly detection on graphics processing units. The proposed global and local real-time strategies have been deeply analyzed summarizing that the computational complexity is greatly reduced under the comparable detection accuracy. This is finally validated by experimental results.

Hierarchical linear dynamical systems for unsupervised musical note recognition

Abstract In this paper we develop a new framework for time series segmentation based on a Hierarchical Linear Dynamical System (HLDS), and test its performance on monophonic and polyphonic musical note recognition. The center piece of our approach is the inclusion of constraints in the filter topology, instead of on the cost function as normally done in machine learning. Just by slowing down the dynamics of the top layer of an augmented (multilayer) state model, which is still compatible with the recursive update equation proposed originally by Kalman, the system learns directly from data all the musical notes, without labels, effectively creating a time series clustering algorithm that does not require segmentation. We analyze the HLDS properties and show that it provides better classification accuracy compared to current state-of-the-art approaches.

Anomaly Detection Using Causal Sliding Windows

Anomaly detection using sliding windows is not new but using causal sliding windows has not been explored in the past. The need for causality arises from real-time processing where the used sliding windows should not include future data samples that have not been visited, i.e., those data sample vectors come in after the currently being processed data sample vector. This chapter presents an approach developed by Chang et al. (2015) to anomaly detection using causal sliding windows, which has the capability of being implemented in real time. In doing so, two types of causal windows are defined, causal sliding matrix windows including square matrix windows and rectangular matrix windows and causal sliding array windows, each of which derives a causal sample covariance/correlation matrix for causal anomaly detection. As for the causal sliding array windows, recursive update equations are also derived and, thus, can speed up real-time processing.

3D normal distributions transform occupancy maps: An efficient representation for mapping in dynamic environments

In order to enable long-term operation of autonomous vehicles in industrial environments numerous challenges need to be addressed. A basic requirement for many applications is the creation and maintenance of consistent 3D world models. This article proposes a novel 3D spatial representation for online real-world mapping, building upon two known representations: normal distributions transform (NDT) maps and occupancy grid maps. The proposed normal distributions transform occupancy map (NDT-OM) combines the advantages of both representations; compactness of NDT maps and robustness of occupancy maps. One key contribution in this article is that we formulate an exact recursive updates for NDT-OMs. We show that the recursive update equations provide natural support for multi-resolution maps. Next, we describe a modification of the recursive update equations that allows adaptation in dynamic environments. As a second key contribution we introduce NDT-OMs and formulate the occupancy update equations that allow to build consistent maps in dynamic environments. The update of the occupancy values are based on an efficient probabilistic sensor model that is specially formulated for NDT-OMs. In several experiments with a total of 17 hours of data from a milk factory we demonstrate that NDT-OMs enable real-time performance in large-scale, long-term industrial setups.

Multi-target state extraction for the particle probability hypothesis density filter

The probability hypothesis density (PHD) filter has emerged as a promising tool for dealing with the multi-target tracking problem in recent years. However, except in some special situations, closed-form recursive update equations for the PHD filter do not exist and the particle filter approaches have to be used. The output of the particle filter at each step is the particle clouds approximation of the PHD. Thus, some special algorithms are needed to extract the target states from those particles. Utilising the information of both particles’ weight and their spatial distribution, an improved algorithm named C-Clean is proposed in this study. This algorithm is comprised of two steps. First, clustering techniques are used to exploit the spatial distribution of particles. Then, within those clusters whose corresponding PHD weight is beyond some predefined threshold, the peak extraction procedure modified from the CLEAN technique is taken to extract the multi-target state. Simulation results demonstrate that its performance is better than those algorithms using the information of particles’ spatial distribution or weight only.

Sparsity regularised recursive least squares adaptive filtering

The authors propose a new approach for the adaptive identification of sparse systems. This approach improves on the recursive least squares (RLS) algorithm by adding a sparsity inducing weighted l 1 norm penalty to the RLS cost function. Subgradient analysis is utilised to develop the recursive update equations for the calculation of the optimum system estimate, which minimises the regularised cost function. Two new algorithms are introduced by considering two different weighting scenarios for the l 1 norm penalty. These new l 1 relaxation-based RLS algorithms emphasise sparsity during the adaptive filtering process, and they allow for faster convergence than standard RLS when the system under consideration is sparse. The authors test the performance of the novel algorithms and compare it with standard RLS and other adaptive algorithms for sparse system identification.

Particle filtering in the Dempster–Shafer theory

► A particle filter algorithm is presented within the Dempster–Shafer framework. ► Ignorance can be modeled with respect to an underlying hidden Markov process. ► Recursive filtering equations are derived for belief functions. ► Importance sampling is used to efficiently incorporate new observations. ► The resulting algorithm forms a strict generalization of Bayesian filtering. This paper derives a particle filter algorithm within the Dempster–Shafer framework. Particle filtering is a well-established Bayesian Monte Carlo technique for estimating the current state of a hidden Markov process using a fixed number of samples. When dealing with incomplete information or qualitative assessments of uncertainty, however, Dempster–Shafer models with their explicit representation of ignorance often turn out to be more appropriate than Bayesian models. The contribution of this paper is twofold. First, the Dempster–Shafer formalism is applied to the problem of maintaining a belief distribution over the state space of a hidden Markov process by deriving the corresponding recursive update equations, which turn out to be a strict generalization of Bayesian filtering. Second, it is shown how the solution of these equations can be efficiently approximated via particle filtering based on importance sampling, which makes the Dempster–Shafer approach tractable even for large state spaces. The performance of the resulting algorithm is compared to exact evidential as well as Bayesian inference.

Fast robust quasi-Newton algorithm for adaptive arrays

A stable, fast (order-of-N) quasi-Newton (QN) algorithm (FRQN) applicable to arbitrary array signals is presented. Its complexity is only twice that of the normalised least mean squares (NLMS) algorithm, yet its convergence is faster and smoother, proceeding along the optimal QN path as that of the order N2 recursive least squares (RLS) algorithm, albeit more slowly. The development of its recursive update equation is outlined and compared to those of the NLMS and RLS algorithms, as well as variations of the conjugate gradient (CG) algorithm. Analytic expressions for its excess mean squared error in stationary and non-stationary scenarios are presented and compared with those of the LMS algorithm. Simulations of an adaptive array in a mobile wireless base-station show that the FRQN performance falls between that of the NLMS and RLS algorithms for a wide range of signal scenarios.

Efficient covariance ladder algorithms for finite arithmetic applications

This paper is concerned with the problem of constructing numerically robust covariance ladder estimation algorithms for finite arithmetic applications. Conventional least-squares (LS) ladder algorithms suffer from mixed time and order recursive update equations resulting in a poor numerical accuracy when implemented with finite arithmetic. In this paper, a more ‘direct’ and numerically robust computation approach of the ladder update recursions using the most recently introduced algebraic method of generalized residual energies (GRE's) is presented. The new algorithms separate time and order recursions in two independent subalgorithms with a highly modular structure. Besides the general framework, five algorithms of this type are presented. Based on these algorithms, a VLSI ladder chip-set is proposed. Fixed-point simulations of the new VLSI structures are performed for several types of input data. The experimental analysis shows that the new algorithms are superior over conventional techniques and can operate at a multiplier wordlength as low as 8 bits. Dieser Aufsatz behandelt das Problem des Entwurfs numerisch robuster Kovarianz Ladder Algorithmen für die Anwendung in Echtzeitrechnern mit beschränkter numerischer Genauigkeit. Konventionelle Least-Squares (LS) Ladder Algorithmen basieren auf gemischt zeit- und ordnungsrekursiven Aktualisierungsgleichungen welche auf eine schlechte numerische Genauigkeit der Algorithmen führen. Eine erhebliche Verbesserung der numerischen Eigenschaften von Ladder Algorithmen läβt sich mit Hilfe der neuen algebraischen Methode der verallgemeinerten Residualenergien (GRE's) erreichen, welche in diesem Aufsatz vorgestellt wird. Die aus dem neuen Ansatz resultierenden Verfahren erlauben die Trennung von Zeit- und Ordnungsrekursionen in zwei unabhängige Teilalgorithmen mit höchst modularer Struktur. Neben der allgemeinen Herleitung werden insgesamt fünf numerisch robuste Ladder Algorithmen dieses Typs vorgestellt. Basierend auf diesen Verfahren wurde eine VLSI Ladder Prozessorfamilie entwickelt. Diese Prozessoren wurden in Festkommaarithmetik implementiert und mit verschiedenen Signalen getestet. Die experimentelle Analyse zeigt, daβ die vorgestellten Verfahren den bisher behannten Methoden überlegen sind und selbst in Rechnern mit niedrigsten Wortbreiten bis 8 Bit eine ausreichende Schätzgenauigkeit erzielen. Cet article concerne par le problème de construction d'algorithmes numériquement robustes d'estimation des treillis de covariance pour des application à arithmétique finie. Les algorithmes conventionnels de treillis moindres-carrés (LS) souffrent des équations de mise à jour recursive mixte en temps et en ordre qui conduisent à une mauvaise précision numérique quand ils sont implantés avec une arithmétique finie. Dans cet article, on présente une approche plus directe et numériquement robuste de la mise à jour récursive des treillis, en utilisant la méthode algébrique des énergies résiduelles généralisées, très récemment introduite (GRE). Le nouvel algorithme separe les récursions temporelles et sur l'ordre en deux sous-algorithmes indépendants avec une structure très modulaire. Cinq algorithmes de ce type sont présentés, basés sur le contexte général. Sur la base de ces algorithmes un ensemble de puces de treillis VLSI est proposé. Les simulations à virgule fixe des nouvelles structures VLSI sont effectuées pour plusieurs types de données d'entrée. L'analyse expérimentale montre que les nouveaux algorithmes sont supérieurs aux techniques conventionnelles et peuvent fonctionner avec une longueur de multiplieur aussi faible que 8 bits.