The mathematics of surface reconstruction

This thesis discusses mathematics engineers use to produce computerized three dimensional im- ages of surfaces. It is self-contained in that all background information is included. As a result, mathematicians who know very little about the technology involved in three dimensional imaging should be able to understand the topics herein, and engineers with no differential geometry background will be able to understand the mathematics. The purpose of this thesis is to unify and understand the notation commonly used by engineers, understand their terminology, and appreciate the difficulties faced by engineers in their pursuitsIt is also intended to bridge the gap between mathematics and engineering. This paper proceeds as follows. Chapter one introduces the topic and provides a brief overview of this thesis. Chapter two provides background information on technology and differential geometry. Chapter three discusses various methods by which normal vectors are estimated. In Chapter four, we discuss methods by which curvature is estimatedIn Chapter six, we put it all together to recreate the surface. Finally, in chapter seven, we conclude with a discussion of future research. Each chapter concludes with a comparison of the methods discussed. The study of these reconstruction algorithms originated from various engineering papers on surface reconstruction. The background information was gathered from a thesis and various differential geometry texts. The challenge arises in the nature of the data with which we work. The surface must be recreated based on a set of discrete points. However, the study of surfaces is one of differential geometry which assumes differentiable functions representing the surface. Since we only have a discrete set of points, methods to overcome this shortcoming must be developed. Two categories of surface reconstruction have been developed to overcome this shortcoming. The first category estimates the data by data by smooth functionsThe second reconstructs the surface using the discrete data directly. We found that various aspects of surface reconstruction are very reliable, while others are only marginally so. We found that methods recreating the surface from discrete data directly produce very similar results suggesting that some underlying facts about surfaces represented by discrete information may be influencing the results

Positive Feedbacks in Seagrass Ecosystems – Evidence from Large-Scale Empirical Data

Positive feedbacks cause a nonlinear response of ecosystems to environmental change and may even cause bistability. Even though the importance of feedback mechanisms has been demonstrated for many types of ecosystems, their identification and quantification is still difficult. Here, we investigated whether positive feedbacks between seagrasses and light conditions are likely in seagrass ecosystems dominated by the temperate seagrass Zostera marina. We applied a combination of multiple linear regression and structural equation modeling (SEM) on a dataset containing 83 sites scattered across Western Europe. Results confirmed that a positive feedback between sediment conditions, light conditions and seagrass density is likely to exist in seagrass ecosystems. This feedback indicated that seagrasses are able to trap and stabilize suspended sediments, which in turn improves water clarity and seagrass growth conditions. Furthermore, our analyses demonstrated that effects of eutrophication on light conditions, as indicated by surface water total nitrogen, were on average at least as important as sediment conditions. This suggests that in general, eutrophication might be the most important factor controlling seagrasses in sheltered estuaries, while the seagrass-sediment-light feedback is a dominant mechanism in more exposed areas. Our study demonstrates the potentials of SEM to identify and quantify positive feedbacks mechanisms for ecosystems and other complex systems

Habitat-Mediated Facilitation and Counteracting Ecosystem Engineering Interactively Influence Ecosystem Responses to Disturbance

Recovery of an ecosystem following disturbance can be severely hampered or even shift altogether when a point disturbance exceeds a certain spatial threshold. Such scale-dependent dynamics may be caused by preemptive competition, but may also result from diminished self-facilitation due to weakened ecosystem engineering. Moreover, disturbance can facilitate colonization by engineering species that alter abiotic conditions in ways that exacerbate stress on the original species. Consequently, establishment of such counteracting engineers might reduce the spatial threshold for the disturbance, by effectively slowing recovery and increasing the risk for ecosystem shifts to alternative states. We tested these predictions in an intertidal mudflat characterized by a two-state mosaic of hummocks (humps exposed during low tide) dominated by the sediment-stabilizing seagrass Zostera noltii) and hollows (low-tide waterlogged depressions dominated by the bioturbating lugworm Arenicola marina). In contrast to expectations, seagrass recolonized both natural and experimental clearings via lateral expansion and seemed unaffected by both clearing size and lugworm addition. Near the end of the growth season, however, an additional disturbance (most likely waterfowl grazing and/or strong hydrodynamics) selectively impacted recolonizing seagrass in the largest (1 m2) clearings (regardless of lugworm addition), and in those medium (0.25 m2) clearings where lugworms had been added nearly five months earlier. Further analyses showed that the risk for the disturbance increased with hollow size, with a threshold of 0.24 m2. Hollows of that size were caused by seagrass removal alone in the largest clearings, and by a weaker seagrass removal effect exacerbated by lugworm bioturbation in the medium clearings. Consequently, a sufficiently large disturbance increased the vulnerability of recolonizing seagrass to additional disturbance by weakening seagrass engineering effects (sediment stabilization). Meanwhile, the counteracting ecosystem engineering (lugworm bioturbation) reduced that threshold size. Therefore, scale-dependent interactions between habitat-mediated facilitation, competition and disturbance seem to maintain the spatial two-state mosaic in this ecosystem