925 research outputs found
A Unifying Approach to Quaternion Adaptive Filtering: Addressing the Gradient and Convergence
A novel framework for a unifying treatment of quaternion valued adaptive
filtering algorithms is introduced. This is achieved based on a rigorous
account of quaternion differentiability, the proposed I-gradient, and the use
of augmented quaternion statistics to account for real world data with
noncircular probability distributions. We first provide an elegant solution for
the calculation of the gradient of real functions of quaternion variables
(typical cost function), an issue that has so far prevented systematic
development of quaternion adaptive filters. This makes it possible to unify the
class of existing and proposed quaternion least mean square (QLMS) algorithms,
and to illuminate their structural similarity. Next, in order to cater for both
circular and noncircular data, the class of widely linear QLMS (WL-QLMS)
algorithms is introduced and the subsequent convergence analysis unifies the
treatment of strictly linear and widely linear filters, for both proper and
improper sources. It is also shown that the proposed class of HR gradients
allows us to resolve the uncertainty owing to the noncommutativity of
quaternion products, while the involution gradient (I-gradient) provides
generic extensions of the corresponding real- and complex-valued adaptive
algorithms, at a reduced computational cost. Simulations in both the strictly
linear and widely linear setting support the approach
Simultaneous diagonalisation of the covariance and complementary covariance matrices in quaternion widely linear signal processing
Recent developments in quaternion-valued widely linear processing have
established that the exploitation of complete second-order statistics requires
consideration of both the standard covariance and the three complementary
covariance matrices. Although such matrices have a tremendous amount of
structure and their decomposition is a powerful tool in a variety of
applications, the non-commutative nature of the quaternion product has been
prohibitive to the development of quaternion uncorrelating transforms. To this
end, we introduce novel techniques for a simultaneous decomposition of the
covariance and complementary covariance matrices in the quaternion domain,
whereby the quaternion version of the Takagi factorisation is explored to
diagonalise symmetric quaternion-valued matrices. This gives new insights into
the quaternion uncorrelating transform (QUT) and forms a basis for the proposed
quaternion approximate uncorrelating transform (QAUT) which simultaneously
diagonalises all four covariance matrices associated with improper quaternion
signals. The effectiveness of the proposed uncorrelating transforms is
validated by simulations on both synthetic and real-world quaternion-valued
signals.Comment: 41 pages, single column, 10 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
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
Deep Quaternion Networks
The field of deep learning has seen significant advancement in recent years.
However, much of the existing work has been focused on real-valued numbers.
Recent work has shown that a deep learning system using the complex numbers can
be deeper for a fixed parameter budget compared to its real-valued counterpart.
In this work, we explore the benefits of generalizing one step further into the
hyper-complex numbers, quaternions specifically, and provide the architecture
components needed to build deep quaternion networks. We develop the theoretical
basis by reviewing quaternion convolutions, developing a novel quaternion
weight initialization scheme, and developing novel algorithms for quaternion
batch-normalization. These pieces are tested in a classification model by
end-to-end training on the CIFAR-10 and CIFAR-100 data sets and a segmentation
model by end-to-end training on the KITTI Road Segmentation data set. These
quaternion networks show improved convergence compared to real-valued and
complex-valued networks, especially on the segmentation task, while having
fewer parametersComment: IJCNN 2018, 8 pages, 1 figur
Employing data fusion & diversity in the applications of adaptive signal processing
The paradigm of adaptive signal processing is a simple yet powerful method for the class of system identification problems. The classical approaches consider standard one-dimensional signals whereby the model can be formulated by flat-view matrix/vector framework. Nevertheless, the rapidly increasing availability of large-scale multisensor/multinode measurement technology has render no longer sufficient the traditional way of representing the data. To this end, the author, who from this point onward shall be referred to as `we', `us', and `our' to signify the author myself and other supporting contributors i.e. my supervisor, my colleagues and other overseas academics specializing in the specific pieces of research endeavor throughout this thesis, has applied the adaptive filtering framework to problems that employ the techniques of data diversity and fusion which includes quaternions, tensors and graphs. At the first glance, all these structures share one common important feature: invertible isomorphism. In other words, they are algebraically one-to-one related in real vector space. Furthermore, it is our continual course of research that affords a segue of all these three data types. Firstly, we proposed novel quaternion-valued adaptive algorithms named the n-moment widely linear quaternion least mean squares (WL-QLMS) and c-moment WL-LMS. Both are as fast as the recursive-least-squares method but more numerically robust thanks to the lack of matrix inversion. Secondly, the adaptive filtering method is applied to a more complex task: the online tensor dictionary learning named online multilinear dictionary learning (OMDL). The OMDL is partly inspired by the derivation of the c-moment WL-LMS due to its parsimonious formulae. In addition, the sequential higher-order compressed sensing (HO-CS) is also developed to couple with the OMDL to maximally utilize the learned dictionary for the best possible compression. Lastly, we consider graph random processes which actually are multivariate random processes with spatiotemporal (or vertex-time) relationship. Similar to tensor dictionary, one of the main challenges in graph signal processing is sparsity constraint in the graph topology, a challenging issue for online methods. We introduced a novel splitting gradient projection into this adaptive graph filtering to successfully achieve sparse topology. Extensive experiments were conducted to support the analysis of all the algorithms proposed in this thesis, as well as pointing out potentials, limitations and as-yet-unaddressed issues in these research endeavor.Open Acces
- …