Representation Of Covariance Matrix Research Articles

Spectral clustering with anchor graph based on set-to-set distances for large-scale hyperspectral images

ABSTRACT Since labelled samples of hyperspectral images (HSIs) may be unavailable in practical remote sensing applications, large-scale HSI clustering is very important. Due to the huge amount of data brought by the rich spectral and spatial information in large-scale HSIs, HSI clustering is still a challenging task. Among the methods designed for large-scale HSIs clustering, the anchor graph-based methods simultaneously inherit the merits of graph-based clustering and reduce the computational complexity by introducing anchor samples to graph construction. However, the affinity matrix computed by inaccurate distances between anchor samples and other HSI samples can hardly obtain satisfactory clustering performance. To solve this problem, we propose a novel approach for large-scale HSI clustering, namely, spectral clustering with anchor graph based on set-to-set distances (SCAG-SSD) derived from local covariance matrix representation (LCMR). First, superpixels and LCMR features of HSI are obtained via the entropy rate superpixel algorithm and maximum noise fraction, respectively. Second, pure and anomalous samples of each superpixel are distinguished via the distances of LCMR features, and then anchor samples are selected via statistics of the distances between pure samples in each superpixel. In this way, selected anchor samples are representative enough to link all the HSI samples. Third, pure samples in each superpixel, anomalous and anchor samples with their corresponding nearest neighbouring samples are all regarded as different sets. The set-to-set distance is achieved by weighting the LCMR-based distances between samples of two sets. Finally, fast anchor graph clustering is conducted based on set-to-set distances to obtain final clustering maps. Extensive experiments conducted on three publicly available benchmark HSIs demonstrate that the proposed method achieves state-of-the-art clustering accuracy with comparable efficiency.

Non-Gaussian operations in measurement-device-independent quantum key distribution

Non-Gaussian operations in continous variable (CV) quantum key distribution (QKD) have been limited to photon subtraction on squeezed vacuum states only. This is mainly due to the ease of calculating the covariance matrix representation of such states. In this paper we study the effects of general non-Gaussian operations corresponding to photon addition, catalysis and subtraction on squeezed coherent states on CV measurement device independent (MDI) QKD. We find that non-Gaussianity coupled with coherence can yield significantly longer transmission distances than without. Particularly we observe that zero photon catalysis on two mode squeezed coherent state (TMSC) is an optimial choice for CV MDI QKD, while single photon subtraction is also a good candidate; both of them offering nearly 70 km of transmission distances. We also derive a single generalized covariance matrix for the aforementioned states which will be useful in several other aspects of CV quantum information processing.

Locally Homogeneous Covariance Matrix Representation for Hyperspectral Image Classification

Combining spectralandspatial information has been proven to be an effective way for hyperspectral image (HSI) classification. However, making full use of spectral–spatial information of HSI still remains an open problem, especially when only a small number of labeled samples are available. In this article, a new spectral–spatial feature extraction method called locally homogeneous covariance matrix representation (CMR) is proposed for the fusion of spectral and spatial information. Specially, to make use of neighborhood homogeneity of land covers, original HSI is first segmented into many superpixels using modified entropy rate superpixel segmentation. Then, to acquire the most similar pixels, we propose to construct neighborhoods of each pixel from the overlapping areas between the corresponding superpixels and the sliding window centered on it. Subsequently, CMRs of different pixels can be obtained. In the classification stage, we fed the obtained CMRs into SVM with Log-Euclidean-based kernel for classification. Compared to the traditional approach that utilizes neighboring information only within a fixed window, the proposed local homogeneity strategy can absorb more discriminative spectral–spatial features. Experimental results from a series of available HSI datasets show that our proposed method is superior to several state-of-the-art methods, especially when the training set is very limited.

Attention-based Convolutional Autoencoders for 3D-Variational Data Assimilation

We propose a new ‘Bi-Reduced Space’ approach to solving 3D Variational Data Assimilation using Convolutional Autoencoders. We prove that our approach has the same solution as previous methods but has significantly lower computational complexity; in other words, we reduce the computational cost without affecting the data assimilation accuracy. We tested our proposal with data from a real-world application: a pollution model of a site in Elephant and Castle (London, UK) and found that we could (1) reduce the size of the background covariance matrix representation by O(103), and (2) increase our data assimilation accuracy with respect to existing reduced space methods.

SAR Image Speckle Filtering With Context Covariance Matrix Formulation and Similarity Test

Speckle filtering of synthetic aperture radar (SAR) image is a necessary pre-processing for many subsequent applications. The challenge lies in how to adaptively select a sufficient number of similar pixels for an unbiased estimator generation. A novel SAR speckle filter is proposed and the core idea contains two aspects. Firstly, a context covariance matrix representation is developed within a local neighborhood to characterize the contexture information. Then, the Wishart statistic test is extended to examine the similarity of context covariance matrices. The extended similarity test indicator derived from context covariance matrices is verified to be sensitive for similar pixel localization. Thereafter, a sample averaging estimator is adopted based on the similar samples determined by the context covariance matrices similarity test (the proposed method is named as the CCM+SimiTest). Furthermore, a fast similarity test computation scheme is established which can handle large images smoothly even with a normal laptop. Intensive experimental studies with Radarsat-2, MiniSAR and ALOS-2 datasets are carried out. Comparisons with several state-of-the-art methods from both subjective and objective viewpoints demonstrate the superiority of the proposed method.

Uncertainty-Aware Principal Component Analysis.

We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the PCA sample covariance matrix that respects potential uncertainty in each of the inputs, building the mathematical foundation of our new method: uncertainty-aware PCA. In addition to the accuracy and performance gained by our approach over sampling-based strategies, our formulation allows us to perform sensitivity analysis with regard to the uncertainty in the data. For this, we propose factor traces as a novel visualization that enables to better understand the influence of uncertainty on the chosen principal components. We provide multiple examples of our technique using real-world datasets. As a special case, we show how to propagate multivariate normal distributions through PCA in closed form. Furthermore, we discuss extensions and limitations of our approach.

Learning Discriminative Embedding for Hyperspectral Image Clustering Based on Set-to-Set and Sample-to-Sample Distances

Recently, deep learning techniques have been introduced to address hyperspectral image (HSI) classification problems and have achieved the state-of-the-art performances. In this article, we propose a novel clustering algorithm for HSI based on learning embedding using the set-to-set and sample-to-sample distances (LSSDs). This technique consists of four main components: 1) oversegmentation; 2) generation of set-to-set and sample-to-sample distances; 3) learning embedding by training a siamese network; and 4) density-based spectral clustering. First, the HSI is oversegmented into superpixels by using the entropy rate superpixel (ERS) algorithm. Second, the set-to-set distances are obtained by representing the segmented sets of samples as affine hull (AH) models, whereas the sample-to-sample distances are computed by employing the local covariance matrix representation (LCMR) method. Third, sample pairs with the smallest and largest similarities are extracted according to the two distances. Then, these pairs are fed into the siamese multilayer perceptron (MLP) network and discriminative embeddings are learned by training the network with contrastive loss. Finally, density-based spectral clustering is applied to the deep embedding to obtain clustering results. Experimental results on three real HSIs demonstrate that the proposed method can achieve better performance than the considered baseline methods.

Iterated extended Kalman filter for airborne electromagnetic data inversion

ABSTRACTThe iterated extended Kalman filter (IEKF) is a tool within the theory of optimal estimation used for nonlinear problems. The IEKF minimises variance in the estimation error in terms of a probabilistic approach. Despite the special terminology, the Kalman filter algorithm minimises the objective function, representing the normalised squared difference between the measured and calculated vectors for the parameters of a selected model. It works like the weighted least squares method – a conventional method for airborne electromagnetic data inversion. In this article, I describe the essence of the Kalman approach to solving inverse problems. I show how one-dimensional inversion with lateral constraints can be performed in terms of the Kalman filter. The described algorithm takes account of the measurement noise, which is specified as the dispersion of signals in the corresponding measurement channels at high altitude. A specific covariance matrix representation allows use of the corresponding Kalman filter calculation methods. They provide numerical stability of the algorithm. The Kalman approach makes it possible to combine modern techniques used in airborne survey data processing. Some examples of Kalman filter use in frequency-domain airborne data processing are given.

Bootstrap-based maximum multivariate CUSUM control chart

ABSTRACTMaximum multivariate cumulative sum (Max-MCUSUM) is one of the single control charts that plot single statistic as a representation of mean vector and covariance matrix. The Max-MCUSUM statistic has unknown specific distribution. The objective of this paper is to propose bootstrap-based Max-MCUSUM control chart for which reference value is predetermined in Phase I monitoring process. For various numbers of quality characteristics and correlation coefficients, the control limits estimated using bootstrap approach are presented in this paper. Furthermore, the average run lengths of bootstrap-based Max-MCUSUM control chart prove that the proposed control chart tends to be effective for monitoring the small shift in both mean and variance of a process. The illustrative examples are provided to demonstrate the applications of the proposed control chart for both simulation and real data.

Joint 2D direction of arrival estimation using a sparse representation of cross covariance matrix

In this study, the problem of two-dimensional (2D) direction of arrival estimation for uncorrelated signals with an L-shaped array is discussed. The estimated cross covariance matrix (CCM) between the sample data along the x -axis and the z -axis is constructed. The elevation angle of the signal is obtained by finding the sparse coefficients of the estimated CCM based on the discussion of statistical probability distribution of the sample data error. Once the elevation angle is obtained, with the non-coupling between the elevation angle and the azimuth angle, the azimuth angle of each source is estimated by the eigenvalue decomposition of the conjugate transpose matrix of the estimated CCM. The proposed algorithm has better estimation performance in the situation of low signal-to-noise-ratio cases but also the estimated azimuth and elevation angles are paired automatically. The performance of the proposed method is compared with some existing schemes and the effectiveness of the method is demonstrated by computer simulations.

A New Spatial–Spectral Feature Extraction Method for Hyperspectral Images Using Local Covariance Matrix Representation

In this paper, a novel local covariance matrix (CM) representation method is proposed to fully characterize the correlation among different spectral bands and the spatial–contextual information in the scene when conducting feature extraction (FE) from hyperspectral images (HSIs). Specifically, our method first projects the HSI into a subspace, using the maximum noise fraction method. Then, for each test pixel in the subspace, its most similar neighboring pixels (within a local spatial window) are clustered using the cosine distance measurement. The test pixel and its neighbors are used to calculate a local CM for FE purposes. Each nondiagonal entry in the matrix characterizes the correlation between different spectral bands. Finally, these matrices are used as spatial–spectral features and fed to a support vector machine for classification purposes. The proposed method offers a new strategy to characterize the spatial–spectral information in the HSI prior to classification. Experimental results have been conducted using three publicly available hyperspectral data sets for classification, indicating that the proposed method can outperform several state-of-the-art techniques, especially when the training samples available are limited.

Riverine Bathymetry Imaging With Indirect Observations

AbstractBathymetry, i.e., depth, imaging in a river is of crucial importance for shipping operations and flood management. With advancements in sensor technology and plentiful computational resources, various types of indirect measurements can be used to estimate high‐resolution river bed topography. In this work, we image river bed topography using depth‐averaged quasi‐steady velocity observations related to the topography through the 2‐D shallow water equations. The principal component geostatistical approach (PCGA), a fast and scalable variational inverse modeling method powered by low‐rank representation of covariance matrix structure, is presented and applied to two riverine bathymetry identification problems. To compare the efficiency and effectiveness of the proposed method, an ensemble‐based approach is also applied to the test problems. It is demonstrated that PCGA is superior to the ensemble‐based approach in terms of computational effort and accuracy because of the successive linearization of the forward model and the optimal low‐rank representation of the prior covariance matrix. To investigate how different low‐rank covariance matrix representation by the two approaches can affect the solution accuracy, we analyze the direct survey data of the river bottom topography in the test problem and show that PCGA utilizes more efficient and parsimonious choice of the solution basis than the ensemble‐based approach. Geostatistical analysis performed on the direct survey data also confirms the validity of the chosen covariance model and its structural parameters.

Resonance parameter covariance representation: file32 versus file33

Sensitivity and uncertainty analysis in error propagation studies are carried out based on nuclear data uncertainty information available in the basic nuclear data libraries such as ENDF, JEFF, JENDL and others. The uncertainty files (covariance matrices) are generally obtained from analysis of experimental data. In the resonance region, the computer code SAMMY is used for analyses of experimental data and generation of resonance parameters. In addition to resonance parameters evaluation, SAMMY also generates resonance parameter covariance matrices (RPCM). The intent of this paper is to discuss the use of the RPCM in contrast to the use of a cross section covariance matrix (CSCM) representation. For so, the resonance evaluation for 48Ti in the resonance range from 10−5 eV to 400 keV is used. The RPCM is translated into two distinct CSCM by using a coarser and a finer group structures. The results are presented and discussions of the feasibility of using these covariance representations are depicted.

Covariances in Computer Vision and Machine Learning

Abstract Covariance matrices play important roles in many areas of mathematics, statistics, and machine learning, as well as their applications. In computer vision and image processing, they give rise to a powerful data representation, namely the covariance descriptor, with numerous practical applications. In this book, we begin by presenting an overview of the {\it finite-dimensional covariance matrix} representation approach of images, along with its statistical interpretation. In particular, we discuss the various distances and divergences that arise from the intrinsic geometrical structures of the set of Symmetric Positive Definite (SPD) matrices, namely Riemannian manifold and convex cone structures. Computationally, we focus on kernel methods on covariance matrices, especially using the Log-Euclidean distance. We then show some of the latest developments in the generalization of the finite-dimensional covariance matrix representation to the {\it infinite-dimensional covariance operator} representati...

Visualization of and Software for Omnibus Test-Based Change Detected in a Time Series of Polarimetric SAR Data

ABSTRACTBased on an omnibus likelihood ratio test statistic for the equality of several variance-covariance matrices following the complex Wishart distribution and a factorization of this test statistic with associated p-values, change analysis in a time series of multilook polarimetric synthetic aperture radar data in the covariance matrix representation is carried out. The omnibus test statistic and its factorization detect if and when change occurs. Using airborne EMISAR and spaceborne RADARSAT-2 data, this article focuses on change detection based on the p-values, on visualization of change at pixel as well as segment level, and on computer software.

Time series classification with feature covariance matrices

In this work, a novel approach utilizing feature covariance matrices is proposed for time series classification. In order to adapt the feature covariance matrices into time series classification problem, a feature vector is defined for each point in a time series. The feature vector comprises local and global information such as value, derivative, rank, deviation from the mean, the time index of the point and cumulative sum up to the point. Extracted feature vectors for the time instances are concatenated to construct feature matrices for the overlapping subsequences. Covariances of the feature matrices are used to describe the subsequences. Our main purpose in this work is to introduce and evaluate the feature covariance representation for time series classification. Therefore, in classification stage, firstly, 1-NN classifier is utilized. After showing the effectiveness of the representation with 1-NN classifier, the experiments are repeated with SVM classifier. The other novelty in this work is that a novel distance measure is introduced for time series by feature covariance matrix representation. Conducted experiments on UCR time series datasets show that the proposed method mostly outperforms the well-known methods such as DTW, shapelet transform and other state-of-the-art techniques.

An RGB–D based image set classification for robust face recognition from Kinect data

The paper proposes a method for robust face recognition from low quality Kinect acquired images which have a wide range of variations in head pose, illumination, facial expressions, sunglass disguise and occlusions by hand. Multiple Kinect images of a person are considered as an image set and face recognition from these images is formulated as an RGB–D image set classification problem. The Kinect acquired raw depth data is used for pose estimation and an automatic cropping of the face region. Based upon the estimated poses, the face images of a set are divided into multiple image subsets. An efficient block based covariance matrix representation is proposed to model images in an image subset on Riemannian manifold (Lie group). For classification, SVM models are separately learnt for each image subset on the Lie group of Riemannian manifold and a fusion strategy is introduced to combine results from all image subsets. The proposed technique has been evaluated on a combination of three large data sets containing over 35,000 RGB–D images under challenging conditions. The proposed RGB–D based image set classification incurs low computational cost and achieves an identification rate as high as 99.5%.

Error Aware Monocular Visual Odometry using Vertical Line Pairs for Small Robots in Urban Areas

We report a new error-aware monocular visual odometry method that only uses vertical lines, such as vertical edges of buildings and poles in urban areas as landmarks. Since vertical lines are easy to extract, insensitive to lighting conditions/ shadows, and sensitive to robot movements on the ground plane, they are robust features if compared with regular point features or line features. We derive a recursive visual odometry method based on the vertical line pairs. We analyze how errors are propagated and introduced in the continuous odometry process by deriving the closed form representation of covariance matrix. We formulate the minimum variance ego-motion estimation problem and present a method that outputs weights for different vertical line pairs. The resulting visual odometry method is tested in physical experiments and compared with two existing methods that are based on point features and line features, respectively. The experiment results show that our method outperforms its two counterparts in robustness, accuracy, and speed. The relative errors of our method are less than 2% in experiments.

Sampling strategies and square root analysis schemes for the EnKF

The purpose of this paper is to examine how different sampling strategies and implementations of the analysis scheme influence the quality of the results in the EnKF. It is shown that by selecting the initial ensemble, the model noise and the measurement perturbations wisely, it is possible to achieve a significant improvement in the EnKF results, using the same number of members in the ensemble. The results are also compared with a square root implementation of the EnKF analysis scheme where the analyzed ensemble is computed without the perturbation of measurements. It is shown that the measurement perturbations introduce sampling errors which can be reduced using improved sampling schemes in the standard EnKF or fully eliminated when the square root analysis algorithm is used. Further, a new computationally efficient square root algorithm is proposed which allows for the use of a low-rank representation of the measurement error covariance matrix. It is shown that this algorithm in fact solves the full problem at a low cost without introducing any new approximations.

Fully polarimetric classification of the Black Forest MAESTRO 1 AIRSAR data

Abstract In this paper we present a study on the application of polarimetric supervised classification techniques to the case of a forested area where the influence of topography is important. For the experimental part of the study, we use the polarimetric AIRSAR data at C, Land P band, acquired over the Black Forest near Freiburg “Germany” during the 1989 MAESTRO I Campaign. Two classification schemes are considered: one is of the classical Bayes type, while the other is based on the concept of maximum contrast. We show practical results of the use of both classifiers in a classification context based on tree age classes. Also the dependency of the classification accuracy on a number of parameters, such as calibration, feature vector dimensions and covariance matrix representation, is discussed. Finally, we focus on the influence of topography on classification and indicate how we can improve the classification results using a priori knowledge of the terrain slope.