Construction of Piecewise Linear Wavelets.

It is well known that in many areas of computational mathematics, wavelet based algorithms are becoming popular for modeling and analyzing data and for providing efficient means for hierarchical data decomposition of function spaces into mutually orthogonal wavelet spaces. Wavelet construction in more than one-dimensional setting is a very challenging and important research topic. In this thesis, we first introduce the method of construction wavelets by using semi-wavelets. Second, we construct piecewise linear wavelets with smaller support over type-2 triangulations. Then, parameterized wavelets are constructed using the orthogonality conditions

A quadratic finite element wavelet Riesz basis

In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in $\mathbb{R}^2$. The wavelets are stable in $H^s$ for $|s|<\frac{3}{2}$ and have two vanishing moments. Each wavelet is a linear combination of 11 or 13 nodal basis functions. Numerically computed condition numbers for $s \in \{-1,0,1\}$ are provided for the unit square.Comment: 13 page

Linear independence and stability of piecewise linear prewavelets on arbitrary triangulations

Abstract: In this paper we establish linear independence and stability of certain piecewise linear prewavelets over arbitrary bounded triangulations. These prewavelets are natural generalizations of the locally supported element constructed by Kotyczka and Oswald for an infinite three-directional mesh

Terrain synthesis: the creation, management, presentation and validation of artificial landscapes

'Synthetic Terrain' is the term used for artificially-composed computer-based Digital Terrain Models (DTMs) created by a combination of techniques and heavily influenced by Earth Sciences applications. The synthetic landscape is created to produce 'geographically acceptable', 'realistic' or 'valid' computer-rendered landscapes, maps and 3D images, which are themselves based on synthetic terrain Digital Elevation Models (OEMs). This thesis examines the way in which mainly physical landscapes can be synthesised, and presents the techniques by which terrain data sets can be managed (created, manipulated, displayed and validated), both for academic reasons and to provide a convenient and cost-effective alternative to expensive 'real world' data sets. Indeed, the latter are collected by ground-based or aerial surveying techniques (e.g. photogrammetry), normally at considerable expense, depending on the scale, resolution and type required. The digital information for a real map could take months to collect, process and reproduce, possibly involving demanding Information Technology (IT) resources and sometimes complicated by differing (or contradictory) formats. Such techniques are invalid if the region lies within an 'unfriendly' or inaccessible part of the globe, where (for example), overflying or ground surveys are forbidden. Previous attempts at synthesising terrain have not necessarily aimed at realism. Digital terrain sets have been created by using fractal mathematical models, as 'special effects' for the entertainment industry (e.g. science fiction 'alien' landscapes' for motion pictures and arcade games) or for artistic reasons. There are no known examples of synthesised DTMs being created with such a wide range of requirements and functionality, and with such a regard to validation and realism. This thesis addresses the whole concept of producing' alternative' landscapes in artificial form - nearly 22 years of research aimed at creating' geographically-sensible' synthetic terrain is described with the emphasis on the last 5 years, when this PhD thesis was conceived. These concepts are based on radical, inexpensive and rapid techniques for synthesising terrain, yet value is also placed on the 'validity', realism and 'fitness for purpose' of such models. The philosophy - or the 'thought processes' - necessary to achieve the development of the algorithms leading to synthesised DTMs is one of the primary achievement of the research. This in turn led to the creation of an interactive software package called GEOFORMA, which requires some manual intervention in the form of preliminary terrain classification. The sequence is thus: the user can choose to create terrain or landform assemblages without reference to any real world area. Alternatively, he can select a real world region or a 'typical' terrain type on a 'dial up' basis, which requires a short period of intensive parametric analysis based on research into established terrain classification techniques (such as fractals and other mathematical routines, process-response models etc.) The creates a composite synthesised terrain model of high quality and realism, a factor examined both qualitatively and quantitatively. Although the physical terrain is the primary concern, similar techniques are applied to the human landscape, noting such attributes as the density, type, nature and distribution of settlements, transport systems etc., and although this thread of the research is limited in scope compared with the physical landscape synthesis, some spectacular results are presented. The system also creates place names based on a simple algorithm. Fluvial landscapes, upland regions and coastlines have been selected from the many possible terrain types for 'treatment', and the thesis gives each of these sample landscapes a separate chapter with appropriate illustrations from this original and extensive research. Finally, and inevitably, the work also poses questions in attempting to provide answers, this is perhaps inevitable in a relatively new genre, encompassing so many disciplines, and with relatively sparse literature on the subject

Terrain synthesis: the creation, management, presentation and validation of artificial landscapes

'Synthetic Terrain' is the term used for artificially-composed computer-based Digital Terrain Models (DTMs) created by a combination of techniques and heavily influenced by Earth Sciences applications. The synthetic landscape is created to produce 'geographically acceptable', 'realistic' or 'valid' computer-rendered landscapes, maps and 3D images, which are themselves based on synthetic terrain Digital Elevation Models (OEMs). This thesis examines the way in which mainly physical landscapes can be synthesised, and presents the techniques by which terrain data sets can be managed (created, manipulated, displayed and validated), both for academic reasons and to provide a convenient and cost-effective alternative to expensive 'real world' data sets. Indeed, the latter are collected by ground-based or aerial surveying techniques (e.g. photogrammetry), normally at considerable expense, depending on the scale, resolution and type required. The digital information for a real map could take months to collect, process and reproduce, possibly involving demanding Information Technology (IT) resources and sometimes complicated by differing (or contradictory) formats. Such techniques are invalid if the region lies within an 'unfriendly' or inaccessible part of the globe, where (for example), overflying or ground surveys are forbidden. Previous attempts at synthesising terrain have not necessarily aimed at realism. Digital terrain sets have been created by using fractal mathematical models, as 'special effects' for the entertainment industry (e.g. science fiction 'alien' landscapes' for motion pictures and arcade games) or for artistic reasons. There are no known examples of synthesised DTMs being created with such a wide range of requirements and functionality, and with such a regard to validation and realism. This thesis addresses the whole concept of producing' alternative' landscapes in artificial form - nearly 22 years of research aimed at creating' geographically-sensible' synthetic terrain is described with the emphasis on the last 5 years, when this PhD thesis was conceived. These concepts are based on radical, inexpensive and rapid techniques for synthesising terrain, yet value is also placed on the 'validity', realism and 'fitness for purpose' of such models. The philosophy - or the 'thought processes' - necessary to achieve the development of the algorithms leading to synthesised DTMs is one of the primary achievement of the research. This in turn led to the creation of an interactive software package called GEOFORMA, which requires some manual intervention in the form of preliminary terrain classification. The sequence is thus: the user can choose to create terrain or landform assemblages without reference to any real world area. Alternatively, he can select a real world region or a 'typical' terrain type on a 'dial up' basis, which requires a short period of intensive parametric analysis based on research into established terrain classification techniques (such as fractals and other mathematical routines, process-response models etc.) The creates a composite synthesised terrain model of high quality and realism, a factor examined both qualitatively and quantitatively. Although the physical terrain is the primary concern, similar techniques are applied to the human landscape, noting such attributes as the density, type, nature and distribution of settlements, transport systems etc., and although this thread of the research is limited in scope compared with the physical landscape synthesis, some spectacular results are presented. The system also creates place names based on a simple algorithm. Fluvial landscapes, upland regions and coastlines have been selected from the many possible terrain types for 'treatment', and the thesis gives each of these sample landscapes a separate chapter with appropriate illustrations from this original and extensive research. Finally, and inevitably, the work also poses questions in attempting to provide answers, this is perhaps inevitable in a relatively new genre, encompassing so many disciplines, and with relatively sparse literature on the subject