Boundary Tracking in Large Data Sets and Modeling the Evolution of Landscapes

In this work, various aspects of the image segmentation problem are considered along with many applications of interest. The traditional approaches of segmentation are roughly categorized into region-based variational methods, edge-based variational methods, and statistical methods. Each approach can work well with different types of data, but they all have difficulty with large data sets. We consider a new approach, boundary tracking, that combines elements from each of these classes of segmentation methods. By sampling only points along the boundaries of objects, boundary tracking works well in high-dimensional and high-resolution data.;Specific high-dimensional and high-resolution data applications are covered in the form of hyperspectral data and high-resolution coastline data. The processing of high spectral resolution hyperspectral data is a field rich in its possibilities. Several methods for processing the data are mentioned, and boundary tracking is applied to the hyperspectral segmentation problem. A further application is the tracking of coastlines, which have well-known fractal structure, and for which the potential spatial resolution is limitless.;Aside from its use as a segmentation algorithm, boundary tracking has a further distinct application in atomic force microscopy. Boundary tracking can be used actively in the image acquisition process, allowing an atomic force microscope to obtain detailed images that focus on important features. Image inpainting can then be used to recover a representation of the entire field of interest.;Next, the relation of boundary tracking to the optimal search problem is explored, along with other boundary tracking possibilities that can be enhanced with ideas from the optimal search problem and from other ideas in image processing. It is shown that deterministic searches often have tangible advantages over random ones and that in one particular case, straight line searches are optimal.;Lastly, the modeling of landscape evolution is discussed with the idea of finding the factors that provide the greatest influence in the formation of a landscape. Previous models and their strengths and weaknesses are evaluated, and a new model is introduced. The proposed landscape model accounts for many features seen in nature and represents a qualitative approach to landscape modeling based on mathematical laws, rather than empirical laws.