Connecting spatial and frequency domains for the quaternion Fourier transform

The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency domains: the convolution of two quaternion signals does not map to the pointwise product of their qFT images. The recently defined ‘Mustard’ convolution behaves nicely in the frequency domain, but complicates the corresponding spatial domain analysis. The present paper analyses in detail the correspondence between classical convolution and the new Mustard convolution. In particular, an expression is derived that allows one to write classical convolution as a finite linear combination of suitable Mustard convolutions. This result is expected to play a major role in the further development of quaternion image processing, as it yields a formula for the qFT spectrum of the classical convolution

Convolution theorems associated with quaternion linear canonical transform and applications

Novel types of convolution operators for quaternion linear canonical transform (QLCT) are proposed. Type one and two are defined in the spatial and QLCT spectral domains, respectively. They are distinct in the quaternion space and are consistent once in complex or real space. Various types of convolution formulas are discussed. Consequently, the QLCT of the convolution of two quaternionic functions can be implemented by the product of their QLCTs, or the summation of the products of their QLCTs. As applications, correlation operators and theorems of the QLCT are derived. The proposed convolution formulas are used to solve Fredholm integral equations with special kernels. Some systems of second-order partial differential equations, which can be transformed into the second-order quaternion partial differential equations, can be solved by the convolution formulas as well. As a final point, we demonstrate that the convolution theorem facilitates the design of multiplicative filters

Hyperbolic linear canonical transforms of quaternion signals and uncertainty

*The final version is published in Applied Mathematics and Computation (450), 2023, Article 127971. It as available via the website https://doi.org/10.1016/j.amc.2023.127971Acknowledgements: The first author’s work was supported by the Asociaci´on Mexicana de Cultura, A. C.. The work of M. Ferreira was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT – Fundação para a Ciência e a Tecnologia, within projects UIDB/04106/2020 and UIDP/04106/202.This paper is concerned with Linear Canonical Transforms (LCTs) associated with two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure, which we call Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). These transforms are defined by replacing the Euclidean plane wave with a corresponding hyperbolic relativistic plane wave in one dimension multiplied by quadratic modulations in both the hyperbolic spatial and frequency domains, giving the hyperbolic counterpart of the Euclidean LCTs. We prove the fundamental properties of the partial QHLCTs and the right-sided QHLCT by employing hyperbolic geometry tools and establish main results such as the Riemann-Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. The analysis is carried out in terms of novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHLCTs. The results are applied to establish two quaternionic versions of the Heisenberg uncertainty principle for the right-sided QHLCT. These uncertainty principles prescribe a lower bound on the product of the effective widths of quaternion-valued signals in the hyperbolic spatial and frequency domains. It is shown that only hyperbolic Gaussian quaternion functions minimize the uncertainty relations.info:eu-repo/semantics/publishedVersio

Color graph based wavelet transform with perceptual information

International audienceIn this paper, we propose a numerical strategy to define a multiscale analysis for color and multicomponent images based on the representation of data on a graph. Our approach consists in computing the graph of an image using the psychovisual information and analysing it by using the spectral graph wavelet transform. We suggest introducing color dimension into the computation of the weights of the graph and using the geodesic distance as a means of distance measurement. We thus have defined a wavelet transform based on a graph with perceptual information by using the CIELab color distance. This new representation is illustrated with denoising and inpainting applications. Overall, by introducing psychovisual information in the graph computation for the graph wavelet transform we obtain very promising results. Therefore results in image restoration highlight the interest of the appropriate use of color information

A Coupled Map Lattice Model for Rheological Chaos in Sheared Nematic Liquid Crystals

A variety of complex fluids under shear exhibit complex spatio-temporal behaviour, including what is now termed rheological chaos, at moderate values of the shear rate. Such chaos associated with rheological response occurs in regimes where the Reynolds number is very small. It must thus arise as a consequence of the coupling of the flow to internal structural variables describing the local state of the fluid. We propose a coupled map lattice (CML) model for such complex spatio-temporal behaviour in a passively sheared nematic liquid crystal, using local maps constructed so as to accurately describe the spatially homogeneous case. Such local maps are coupled diffusively to nearest and next nearest neighbours to mimic the effects of spatial gradients in the underlying equations of motion. We investigate the dynamical steady states obtained as parameters in the map and the strength of the spatial coupling are varied, studying local temporal properties at a single site as well as spatio-temporal features of the extended system. Our methods reproduce the full range of spatio-temporal behaviour seen in earlier one-dimensional studies based on partial differential equations. We report results for both the one and two-dimensional cases, showing that spatial coupling favours uniform or periodically time-varying states, as intuitively expected. We demonstrate and characterize regimes of spatio-temporal intermittency out of which chaos develops. Our work suggests that such simplified lattice representations of the spatio-temporal dynamics of complex fluids under shear may provide useful insights as well as fast and numerically tractable alternatives to continuum representations.Comment: 32 pages, single column, 20 figure

Gait rehabilitation monitor

This work presents a simple wearable, non-intrusive affordable mobile framework that allows remote patient monitoring during gait rehabilitation, by doctors and physiotherapists. The system includes a set of 2 Shimmer3 9DoF Inertial Measurement Units (IMUs), Bluetooth compatible from Shimmer, an Android smartphone for collecting and primary processing of data and persistence in a local database. Low computational load algorithms based on Euler angles and accelerometer, gyroscope and magnetometer signals were developed and used for the classification and identification of several gait disturbances. These algorithms include the alignment of IMUs sensors data by means of a common temporal reference as well as heel strike and stride detection algorithms to help segmentation of the remotely collected signals by the System app to identify gait strides and extract relevant features to feed, train and test a classifier to predict gait abnormalities in gait sessions. A set of drivers from Shimmer manufacturer is used to make the connection between the app and the set of IMUs using Bluetooth. The developed app allows users to collect data and train a classification model for identifying abnormal and normal gait types. The system provides a REST API available in a backend server along with Java and Python libraries and a PostgreSQL database. The machine-learning type is Supervised using Extremely Randomized Trees method. Frequency, time and time-frequency domain features were extracted from the collected and processed signals to train the classifier. To test the framework a set of gait abnormalities and normal gait were used to train a model and test the classifier.Este trabalho apresenta uma estrutura móvel acessível, simples e não intrusiva, que permite a monitorização e a assistência remota de pacientes durante a reabilitação da marcha, por médicos e fisioterapeutas que monitorizam a reabilitação da marcha do paciente. O sistema inclui um conjunto de 2 IMUs (Inertial Mesaurement Units) Shimmer3 da marca Shimmer, compatíveís com Bluetooth, um smartphone Android para recolha, e pré-processamento de dados e armazenamento numa base de dados local. Algoritmos de baixa carga computacional baseados em ângulos Euler e sinais de acelerómetros, giroscópios e magnetómetros foram desenvolvidos e utilizados para a classificação e identificação de diversas perturbações da marcha. Estes algoritmos incluem o alinhamento e sincronização dos dados dos sensores IMUs usando uma referência temporal comum, além de algoritmos de detecção de passos e strides para auxiliar a segmentação dos sinais recolhidos remotamente pelaappdestaframeworke identificar os passos da marcha extraindo as características relevantes para treinar e testar um classificador que faça a predição de deficiências na marcha durante as sessões de monitorização. Um conjunto de drivers do fabricante Shimmer é usado para fazer a conexão entre a app e o conjunto de IMUs através de Bluetooth. A app desenvolvida permite aos utilizadores recolher dados e treinar um modelo de classificação para identificar os tipos de marcha normais e patológicos. O sistema fornece uma REST API disponível num servidor backend recorrendo a bibliotecas Java e Python e a uma base de dados PostgreSQL. O tipo de machine-learning é Supervisionado usando Extremely Randomized Trees. Features no domínio do tempo, da frequência e do tempo-frequência foram extraídas dos sinais recolhidos e processados para treinar o classificador. Para testar a estrutura, um conjunto de marchas patológicas e normais foram utilizadas para treinar um modelo e testar o classificador

Data-driven time-frequency analysis of multivariate data

Empirical Mode Decomposition (EMD) is a data-driven method for the decomposition and time-frequency analysis of real world nonstationary signals. Its main advantages over other time-frequency methods are its locality, data-driven nature, multiresolution-based decomposition, higher time-frequency resolution and its ability to capture oscillation of any type (nonharmonic signals). These properties have made EMD a viable tool for real world nonstationary data analysis. Recent advances in sensor and data acquisition technologies have brought to light new classes of signals containing typically several data channels. Currently, such signals are almost invariably processed channel-wise, which is suboptimal. It is, therefore, imperative to design multivariate extensions of the existing nonlinear and nonstationary analysis algorithms as they are expected to give more insight into the dynamics and the interdependence between multiple channels of such signals. To this end, this thesis presents multivariate extensions of the empirical mode de- composition algorithm and illustrates their advantages with regards to multivariate non- stationary data analysis. Some important properties of such extensions are also explored, including their ability to exhibit wavelet-like dyadic filter bank structures for white Gaussian noise (WGN), and their capacity to align similar oscillatory modes from multiple data channels. Owing to the generality of the proposed methods, an improved multi- variate EMD-based algorithm is introduced which solves some inherent problems in the original EMD algorithm. Finally, to demonstrate the potential of the proposed methods, simulations on the fusion of multiple real world signals (wind, images and inertial body motion data) support the analysis

Semiclassical resolvent estimates for Schroedinger operators with Coulomb singularities

Consider the Schroedinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator's resolvent at a positive energy E are bounded by O(1/h) if and only if the associated Hamilton flow is non-trapping at energy E. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.Comment: 39 pages, no figures, corrected versio