The geometry of proper quaternion random variables

Second order circularity, also called properness, for complex random variables is a well known and studied concept. In the case of quaternion random variables, some extensions have been proposed, leading to applications in quaternion signal processing (detection, filtering, estimation). Just like in the complex case, circularity for a quaternion-valued random variable is related to the symmetries of its probability density function. As a consequence, properness of quaternion random variables should be defined with respect to the most general isometries in $4D$, i.e. rotations from $SO(4)$. Based on this idea, we propose a new definition of properness, namely the $(\mu_1,\mu_2)$-properness, for quaternion random variables using invariance property under the action of the rotation group $SO(4)$. This new definition generalizes previously introduced properness concepts for quaternion random variables. A second order study is conducted and symmetry properties of the covariance matrix of $(\mu_1,\mu_2)$-proper quaternion random variables are presented. Comparisons with previous definitions are given and simulations illustrate in a geometric manner the newly introduced concept.Comment: 14 pages, 3 figure

Adaptive signal processing algorithms for noncircular complex data

The complex domain provides a natural processing framework for a large class of signals encountered in communications, radar, biomedical engineering and renewable energy. Statistical signal processing in C has traditionally been viewed as a straightforward extension of the corresponding algorithms in the real domain R, however, recent developments in augmented complex statistics show that, in general, this leads to under-modelling. This direct treatment of complex-valued signals has led to advances in so called widely linear modelling and the introduction of a generalised framework for the differentiability of both analytic and non-analytic complex and quaternion functions. In this thesis, supervised and blind complex adaptive algorithms capable of processing the generality of complex and quaternion signals (both circular and noncircular) in both noise-free and noisy environments are developed; their usefulness in real-world applications is demonstrated through case studies. The focus of this thesis is on the use of augmented statistics and widely linear modelling. The standard complex least mean square (CLMS) algorithm is extended to perform optimally for the generality of complex-valued signals, and is shown to outperform the CLMS algorithm. Next, extraction of latent complex-valued signals from large mixtures is addressed. This is achieved by developing several classes of complex blind source extraction algorithms based on fundamental signal properties such as smoothness, predictability and degree of Gaussianity, with the analysis of the existence and uniqueness of the solutions also provided. These algorithms are shown to facilitate real-time applications, such as those in brain computer interfacing (BCI). Due to their modified cost functions and the widely linear mixing model, this class of algorithms perform well in both noise-free and noisy environments. Next, based on a widely linear quaternion model, the FastICA algorithm is extended to the quaternion domain to provide separation of the generality of quaternion signals. The enhanced performances of the widely linear algorithms are illustrated in renewable energy and biomedical applications, in particular, for the prediction of wind profiles and extraction of artifacts from EEG recordings

Asymptotic regime for impropriety tests of complex random vectors

Impropriety testing for complex-valued vector has been considered lately due to potential applications ranging from digital communications to complex media imaging. This paper provides new results for such tests in the asymptotic regime, i.e. when the vector dimension and sample size grow commensurately to infinity. The studied tests are based on invariant statistics named impropriety coefficients. Limiting distributions for these statistics are derived, together with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in the Gaussian case. This characterization in the asymptotic regime allows also to identify a phase transition in Roy's test with potential application in detection of complex-valued low-rank subspace corrupted by proper noise in large datasets. Simulations illustrate the accuracy of the proposed asymptotic approximations.Comment: 11 pages, 8 figures, submitted to IEEE TS

Feature Selection and Non-Euclidean Dimensionality Reduction: Application to Electrocardiology.

Heart disease has been the leading cause of human death for decades. To improve treatment of heart disease, algorithms to perform reliable computer diagnosis using electrocardiogram (ECG) data have become an area of active research. This thesis utilizes well-established methods from cluster analysis, classification, and localization to cluster and classify ECG data, and aims to help clinicians diagnose and treat heart diseases. The power of these methods is enhanced by state-of-the-art feature selection and dimensionality reduction. The specific contributions of this thesis are as follows. First, a unique combination of ECG feature selection and mixture model clustering is introduced to classify the sites of origin of ventricular tachycardias. Second, we apply a restricted Boltzmann machine (RBM) to learn sparse representations of ECG signals and to build an enriched classifier from patient data. Third, a novel manifold learning algorithm is introduced, called Quaternion Laplacian Information Maps (QLIM), and is applied to visualize high-dimensional ECG signals. These methods are applied to design of an automated supervised classification algorithm to help a physician identify the origin of ventricular arrhythmias (VA) directed from a patient's ECG data. The algorithm is trained on a large database of ECGs and catheter positions collected during the electrophysiology (EP) pace-mapping procedures. The proposed algorithm is demonstrated to have a correct classification rate of over 80% for the difficult task of classifying VAs having epicardial or endocardial origins.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113303/1/dyjung_1.pd

Temporal and Spatial Independent Component Analysis for fMRI Data Sets Embedded in the AnalyzeFMRI R Package

For statistical analysis of functional magnetic resonance imaging (fMRI) data sets, we propose a data-driven approach based on independent component analysis (ICA) implemented in a new version of the AnalyzeFMRI R package. For fMRI data sets, spatial dimension being much greater than temporal dimension, spatial ICA is the computationally tractable approach generally proposed. However, for some neuroscientific applications, temporal independence of source signals can be assumed and temporal ICA becomes then an attractive exploratory technique. In this work, we use a classical linear algebra result ensuring the tractability of temporal ICA. We report several experiments on synthetic data and real MRI data sets that demonstrate the potential interest of our R package

Simultaneous Source Localization and Polarization Estimation via Non-Orthogonal Joint Diagonalization with Vector-Sensors

Joint estimation of direction-of-arrival (DOA) and polarization with electromagnetic vector-sensors (EMVS) is considered in the framework of complex-valued non-orthogonal joint diagonalization (CNJD). Two new CNJD algorithms are presented, which propose to tackle the high dimensional optimization problem in CNJD via a sequence of simple sub-optimization problems, by using LU or LQ decompositions of the target matrices as well as the Jacobi-type scheme. Furthermore, based on the above CNJD algorithms we present a novel strategy to exploit the multi-dimensional structure present in the second-order statistics of EMVS outputs for simultaneous DOA and polarization estimation. Simulations are provided to compare the proposed strategy with existing tensorial or joint diagonalization based methods

Image analysis using visual saliency with applications in hazmat sign detection and recognition

Visual saliency is the perceptual process that makes attractive objects stand out from their surroundings in the low-level human visual system. Visual saliency has been modeled as a preprocessing step of the human visual system for selecting the important visual information from a scene. We investigate bottom-up visual saliency using spectral analysis approaches. We present separate and composite model families that generalize existing frequency domain visual saliency models. We propose several frequency domain visual saliency models to generate saliency maps using new spectrum processing methods and an entropy-based saliency map selection approach. A group of saliency map candidates are then obtained by inverse transform. A final saliency map is selected among the candidates by minimizing the entropy of the saliency map candidates. The proposed models based on the separate and composite model families are also extended to various color spaces. We develop an evaluation tool for benchmarking visual saliency models. Experimental results show that the proposed models are more accurate and efficient than most state-of-the-art visual saliency models in predicting eye fixation.^ We use the above visual saliency models to detect the location of hazardous material (hazmat) signs in complex scenes. We develop a hazmat sign location detection and content recognition system using visual saliency. Saliency maps are employed to extract salient regions that are likely to contain hazmat sign candidates and then use a Fourier descriptor based contour matching method to locate the border of hazmat signs in these regions. This visual saliency based approach is able to increase the accuracy of sign location detection, reduce the number of false positive objects, and speed up the overall image analysis process. We also propose a color recognition method to interpret the color inside the detected hazmat sign. Experimental results show that our proposed hazmat sign location detection method is capable of detecting and recognizing projective distorted, blurred, and shaded hazmat signs at various distances.^ In other work we investigate error concealment for scalable video coding (SVC). When video compressed with SVC is transmitted over loss-prone networks, the decompressed video can suffer severe visual degradation across multiple frames. In order to enhance the visual quality, we propose an inter-layer error concealment method using motion vector averaging and slice interleaving to deal with burst packet losses and error propagation. Experimental results show that the proposed error concealment methods outperform two existing methods

Adaptive filtering algorithms for quaternion-valued signals

Advances in sensor technology have made possible the recoding of three and four-dimensional signals which afford a better representation of our actual three-dimensional world than the ``flat view'' one and two-dimensional approaches. Although it is straightforward to model such signals as real-valued vectors, many applications require unambiguous modeling of orientation and rotation, where the division algebra of quaternions provides crucial advantages over real-valued vector approaches. The focus of this thesis is on the use of recent advances in quaternion-valued signal processing, such as the quaternion augmented statistics, widely-linear modeling, and the HR-calculus, in order to develop practical adaptive signal processing algorithms in the quaternion domain which deal with the notion of phase and frequency in a compact and physically meaningful way. To this end, first a real-time tracker of quaternion impropriety is developed, which allows for choosing between strictly linear and widely-linear quaternion-valued signal processing algorithms in real-time, in order to reduce computational complexity where appropriate. This is followed by the strictly linear and widely-linear quaternion least mean phase algorithms that are developed for phase-only estimation in the quaternion domain, which is accompanied by both quantitative performance assessment and physical interpretation of operations. Next, the practical application of state space modeling of three-phase power signals in smart grid management and control systems is considered, and a robust complex-valued state space model for frequency estimation in three-phase systems is presented. Its advantages over other available estimators are demonstrated both in an analytical sense and through simulations. The concept is then expanded to the quaternion setting in order to make possible the simultaneous estimation of the system frequency and its voltage phasors. Furthermore, a distributed quaternion Kalman filtering algorithm is developed for frequency estimation over power distribution networks and collaborative target tracking. Finally, statistics of stable quaternion-valued random variables, that include quaternion-valued Gaussian random variables as a special case, is investigated in order to develop a framework for the modeling and processing of heavy-tailed quaternion-valued signals.Open Acces

Registration of 3D Face Scans with Average Face Models

The accuracy of a 3D face recognition system depends on a correct registration that aligns the facial surfaces and makes a comparison possible. The best results obtained so far use a costly one-to-all registration approach, which requires the registration of each facial surface to all faces in the gallery. We explore the approach of registering the new facial surface to an average face model (AFM), which automatically establishes correspondence to the pre-registered gallery faces. We propose a new algorithm for constructing an AFM, and show that it works better than a recent approach. Extending the single-AFM approach, we propose to employ category-specific alternative AFMs for registration, and evaluate the effect on subsequent classification. We perform simulations with multiple AFMs that correspond to different clusters in the face shape space and compare these with gender and morphology based groupings. We show that the automatic clustering approach separates the faces into gender and morphology groups, consistent with the other race effect reported in the psychology literature. We inspect thin-plate spline and iterative closest point based registration schemes under manual or automatic landmark detection prior to registration. Finally, we describe and analyse a regular re-sampling method that significantly increases the accuracy of registration