A Multiscale Guide to Brownian Motion

We revise the Levy's construction of Brownian motion as a simple though still rigorous approach to operate with various Gaussian processes. A Brownian path is explicitly constructed as a linear combination of wavelet-based "geometrical features" at multiple length scales with random weights. Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths. This formula elucidates many classical results about Brownian motion (e.g., non-differentiability of its path), providing intuitive feeling for non-mathematicians. The illustrative character of the wavelet representation, along with the simple structure of the underlying probability space, is different from the usual presentation of most classical textbooks. Similar concepts are discussed for fractional Brownian motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional Gaussian fields. Wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate Gaussian processes and fields, including Brownian motion and other diffusive processes in confining domains

Fast retrieval of weather analogues in a multi-petabyte meteorological archive

The European Centre for Medium-Range Weather Forecasts (ECMWF) manages the largest archive of meteorological data in the world. At the time of writing, it holds around 300 petabytes and grows at a rate of 1 petabyte per week. This archive is now mature, and contains valuable datasets such as several reanalyses, providing a consistent view of the weather over several decades. Weather analogue is the term used by meteorologists to refer to similar weather situations. Looking for analogues in an archive using a brute force approach requires data to be retrieved from tape and then compared to a user-provided weather pattern, using a chosen similarity measure. Such an operation would be very long and costly. In this work, a wavelet-based fingerprinting scheme is proposed to index all weather patterns from the archive, over a selected geographical domain. The system answers search queries by computing the fingerprint of the query pattern and looking for close matched in the index. Searches are fast enough that they are perceived as being instantaneous. A web-based application is provided, allowing users to express their queries interactively in a friendly and straightforward manner by sketching weather patterns directly in their web browser. Matching results are then presented as a series of weather maps, labelled with the date and time at which they occur. The system has been deployed as part of the Copernicus Climate Data Store and allows the retrieval of weather analogues from ERA5, a 40-years hourly reanalysis dataset. Some preliminary results of this work have been presented at the International Conference on Computational Science 2018 (Raoult et al. (2018))

Topological inference for EEG and MEG

Neuroimaging produces data that are continuous in one or more dimensions. This calls for an inference framework that can handle data that approximate functions of space, for example, anatomical images, time--frequency maps and distributed source reconstructions of electromagnetic recordings over time. Statistical parametric mapping (SPM) is the standard framework for whole-brain inference in neuroimaging: SPM uses random field theory to furnish $p$-values that are adjusted to control family-wise error or false discovery rates, when making topological inferences over large volumes of space. Random field theory regards data as realizations of a continuous process in one or more dimensions. This contrasts with classical approaches like the Bonferroni correction, which consider images as collections of discrete samples with no continuity properties (i.e., the probabilistic behavior at one point in the image does not depend on other points). Here, we illustrate how random field theory can be applied to data that vary as a function of time, space or frequency. We emphasize how topological inference of this sort is invariant to the geometry of the manifolds on which data are sampled. This is particularly useful in electromagnetic studies that often deal with very smooth data on scalp or cortical meshes. This application illustrates the versatility and simplicity of random field theory and the seminal contributions of Keith Worsley (1951--2009), a key architect of topological inference.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS337 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Adaptive Haar wavelets for the angular discretisation of spectral wave models

A new framework for applying anisotropic angular adaptivity in spectral wave modelling is presented. The angular dimension of the action balance equation is discretised with the use of Haar wavelets, hierarchical piecewise-constant basis functions with compact support, and an adaptive methodology for anisotropically adjusting the resolution of the angular mesh is proposed. This work allows a reduction of computational effort in spectral wave modelling, through a reduction in the degrees of freedom required for a given accuracy, with an automated procedure and minimal cost

Map online system using internet-based image catalogue

Digital maps carry along its geodata information such as coordinate that is important in one particular topographic and thematic map. These geodatas are meaningful especially in military field. Since the maps carry along this information, its makes the size of the images is too big. The bigger size, the bigger storage is required to allocate the image file. It also can cause longer loading time. These conditions make it did not suitable to be applied in image catalogue approach via internet environment. With compression techniques, the image size can be reduced and the quality of the image is still guaranteed without much changes. This report is paying attention to one of the image compression technique using wavelet technology. Wavelet technology is much batter than any other image compression technique nowadays. As a result, the compressed images applied to a system called Map Online that used Internet-based Image Catalogue approach. This system allowed user to buy map online. User also can download the maps that had been bought besides using the searching the map. Map searching is based on several meaningful keywords. As a result, this system is expected to be used by Jabatan Ukur dan Pemetaan Malaysia (JUPEM) in order to make the organization vision is implemented

An investigation into the design of a satellite based stereo imaging sensor and the use of automatic image matching in the production of digital elevation models

Bibliography: leaves 106-110.Two problems are addressed in this dissertation. They are the design of a micro-satellite based stereo imaging sensor and the automatic matching of digital stereo images for automatic cartography applications. The two problems are related; they are both components of a stereo vision system. The research was initially motivated by the decision of the Department of Electrical and Electronic Engineering of the University of Stellenbosch to develop and build an experimental micro-satellite, SUNSAT. The proposed payload included a high resolution multi-spectral stereo imaging sensor. The second problem was motivated by the desire to use an automatic matching system to process the images produced by the sensor. The investigation, into the sensor design, was divided into two parts. The first part investigated the feasibility of the sensor and the second part dealt with the development of a design specification. The investigation, into automatic matching, dealt with the degree to which a set of requirements could be met. These requirements relate to the accuracy, reliability, generality, predictability and complexity of the matching system. The effect of scene characteristics was also investigated. The results showed that it is possible to build a micro-satellite based stereo imaging sensor. The recommended sensor design included three spectral bands, an 8 bit analogue-to-digital converter and a focal length of 535 mm. Furthermore, it was found that a sub-pixel accuracy matching requirement can be met and that a matching reliability of 89.6 can be achieved. Finally, it was found that the best matching results are obtained in areas of high image variance and low disparity variance