The Boundedness of the Hardy-Littlewood Maximal Function and the Strong Maximal Function on the Space BMO

In this thesis, we present the space BMO, the one-parameter Hardy-Littlewood maximal function, and the two-parameter strong maximal function. We use the John-Nirenberg inequality, the relation between Muckenhoupt weights and BMO, and the Coifman-Rochberg proposition on constructing A1 weights with the Hardy- Littlewood maximal function to show the boundedness of the Hardy-Littlewood maximal function on BMO. The analogous statement for the strong maximal function is not yet understood. We begin our exploration of this problem by discussing an equivalence between the boundedness of the strong maximal function on rectangular BMO and the fact that the strong maximal function maps A∞ weights into the A1 class. We then extend a multiparameter counterexample to the Coifman-Rochberg proposition proposed by Soria (1987) and discuss the difficulties in modifying it into an A∞ counterexample that would disapprove the boundedness of the strong maximal function

2D and 3D computer vision analysis of gaze, gender and age

Human-Computer Interaction (HCI) has been an active research area for over four decades. Research studies and commercial designs in this area have been largely facilitated by the visual modality which brings diversified functionality and improved usability to HCI interfaces by employing various computer vision techniques. This thesis explores a number of facial cues, such as gender, age and gaze, by performing 2D and 3D based computer vision analysis. The ultimate aim is to create a natural HCI strategy that can fulfil user expectations, augment user satisfaction and enrich user experience by understanding user characteristics and behaviours. To this end, salient features have been extracted and analysed from 2D and 3D face representations; 3D reconstruction algorithms and their compatible real-world imaging systems have been investigated; case study HCI systems have been designed to demonstrate the reliability, robustness, and applicability of the proposed method.More specifically, an unsupervised approach has been proposed to localise eye centres in images and videos accurately and efficiently. This is achieved by utilisation of two types of geometric features and eye models, complemented by an iris radius constraint and a selective oriented gradient filter specifically tailored to this modular scheme. This approach resolves challenges such as interfering facial edges, undesirable illumination conditions, head poses, and the presence of facial accessories and makeup. Tested on 3 publicly available databases (the BioID database, the GI4E database and the extended Yale Face Database b), and a self-collected database, this method outperforms all the methods in comparison and thus proves to be highly accurate and robust. Based on this approach, a gaze gesture recognition algorithm has been designed to increase the interactivity of HCI systems by encoding eye saccades into a communication channel similar to the role of hand gestures. As well as analysing eye/gaze data that represent user behaviours and reveal user intentions, this thesis also investigates the automatic recognition of user demographics such as gender and age. The Fisher Vector encoding algorithm is employed to construct visual vocabularies as salient features for gender and age classification. Algorithm evaluations on three publicly available databases (the FERET database, the LFW database and the FRCVv2 database) demonstrate the superior performance of the proposed method in both laboratory and unconstrained environments. In order to achieve enhanced robustness, a two-source photometric stereo method has been introduced to recover surface normals such that more invariant 3D facia features become available that can further boost classification accuracy and robustness. A 2D+3D imaging system has been designed for construction of a self-collected dataset including 2D and 3D facial data. Experiments show that utilisation of 3D facial features can increase gender classification rate by up to 6% (based on the self-collected dataset), and can increase age classification rate by up to 12% (based on the Photoface database). Finally, two case study HCI systems, a gaze gesture based map browser and a directed advertising billboard, have been designed by adopting all the proposed algorithms as well as the fully compatible imaging system. Benefits from the proposed algorithms naturally ensure that the case study systems can possess high robustness to head pose variation and illumination variation; and can achieve excellent real-time performance. Overall, the proposed HCI strategy enabled by reliably recognised facial cues can serve to spawn a wide array of innovative systems and to bring HCI to a more natural and intelligent state

Analysis and improvement of sports industry development and public health strategy under low-carbon economic structure

With the continuous development of society, various industries are rising and developing rapidly. Against this background, the energy crisis has come quietly. Therefore, to improve the quality of life of residents and promote the comprehensive and sustainable development of society, it is essential to enhance the development of the sports industry and formulate public health strategies under the background of a low carbon economy (LCE). Based on this, to promote the low-carbon development of the sports industry and optimize the formulation of social public health strategies, firstly, this paper introduces the low-carbon economic structure and its role in society. Then, it discusses the development of the sports industry and the necessity of perfecting public health strategy. Finally, based on LCE’s development background, the sports industry’s development situation in the whole society and M enterprises is analyzed, and suggestions are put forward to improve the public health strategy. The research results show that the current development prospect of the sports industry is extensive, and the added value of the sports industry will be 1,124.81 billion yuan in 2020, up by 11.6% year-on-year, accounting for 1.14% of Gross Domestic Product (GDP). Although industrial development declined in 2021, the added value of the sports industry accounts for an increasing proportion of GDP yearly, which shows that the sports industry is playing an increasingly important role in economic growth. And through the analysis of the development of M enterprise sports industry as a whole and in different directions, this paper shows that enterprises should reasonably control the development of various industries to provide impetus for the overall development of enterprises. The innovation of this paper lies in the innovative use of the sports industry as the primary research object, and its development under LCE is studied. This paper not only supports the sustainable development of sports industry in the future, but also contributes to improving public health strategy

Rectangle expansion A∗ pathfinding for grid maps

AbstractSearch speed, quality of resulting paths and the cost of pre-processing are the principle evaluation metrics of a pathfinding algorithm. In this paper, a new algorithm for grid-based maps, rectangle expansion A∗ (REA∗), is presented that improves the performance of A∗ significantly. REA∗ explores maps in units of unblocked rectangles. All unnecessary points inside the rectangles are pruned and boundaries of the rectangles (instead of individual points within those boundaries) are used as search nodes. This makes the algorithm plot fewer points and have a much shorter open list than A∗. REA∗ returns jump and grid-optimal path points, but since the line of sight between jump points is protected by the unblocked rectangles, the resulting path of REA∗ is usually better than grid-optimal. The algorithm is entirely online and requires no offline pre-processing. Experimental results for typical benchmark problem sets show that REA∗ can speed up a highly optimized A∗ by an order of magnitude and more while preserving completeness and optimality. This new algorithm is competitive with other highly successful variants of A∗