Analysis of flow and visibility on triangulated terrains

Landscapes and their morphology have been widely studied for predicting physical phenomena, such as floods or erosion, but also for planning human activities effectively, such as building prominent fortifications and watchtowers. Nowadays, the study of terrains is done in a computer-based environment; terrains are modelled by digital representations, and algorithms are used to simulate physical processes like water flow and to compute attributes like visibility from certain locations. In the current thesis we focus on designing new algorithms for computing structures related to water ow and visibility on digital terrain representations. Most specifically, the terrain representations that we considered are the so-called Triangulated Irregular Networks (tins), that is, piecewise linear surfaces that consist of triangles. One of the problems that are considered is the effect of noise on the worst-case complexity of visibility structures on tins. The view that a person can have from a point on the surface of a tin can be very complex, since in the worst case thin obstacles in the foreground may appear to fragment many long terrain edges in the background into visible and invisible pieces. In our analysis we considered tins whose triangles have some well-defined properties that terrains in practice are expected to have. Although complex visibility structures can be induced on such tins as well, we proved formally that slight perturbations on the elevations of the tin vertex set will always get rid of the high complexity. Another key problem that is studied is to design efficient algorithms that compute flow-related structures on tins. So far it was known that, in the case of tins, drainage structures that were computed using a consistent flow-model could have high complexity for specific input instances. We managed to develop a mechanism that can extract important information on flow paths and other drainage structures without computing those structures explicitly. This mechanism can be used as a basis for designing a variety of efficient algorithms, such as for computing the area measure of drainage structures or for computing structures that represent the terrain topology. The last part of the presented work involves the implementation of a software package that computes drainage structures on tins. In this package flow is modelled as following strictly the direction of steepest descent on the tin surface. Existing software for related applications either constrain flow on the edge set of the tin, or use inexact arithmetic, both of which introduces imprecise and/or incorrect results in the output. Our implementation is the first one that, at the same time, follows a robust flow model and uses exact arithmetic. We have used this implementation as a point of reference for evaluating experimentally the quality of the output of other flow models which are used in many hydrological applications. We have also used our software for conducting experiments on extracting watersheds on imprecise tins, that is, tins where the elevation values of the vertices are not exactly defined but are subject to noise from some given interval. Based on the results of these experiments, we have designed a novel method for extracting watersheds on imprecise terrains that produces high quality output

Fast generation of multiple resolution instances of raster data sets

In many GIS applications it is important to study the characteristics of a raster data set at multiple resolutions. Often this is done by generating several coarser resolution rasters from a fine resolution raster. In this paper we describe efficient algorithms for different variants of this problem. Given a raster G of vN × vN cells we first consider the problem of computing for every 2 = µ = vN a raster Gµ of vN/µ × vN/µ cells such that each cell of Gµ stores the average of the values of µ × µ cells of G. We describe an algorithm that solves this problem in T(N) time when the handled data fit in the main memory of the computer. We also provide two algorithms that solve this problem in external memory, that is when the input raster is larger than the main memory. The first external algorithm is very easy to implement and requires O(sort(N)) data block transfers from/to the external memory, and the second algorithm requires only O(scan(N)) transfers, where sort(N) and scan(N) are the number of transfers needed to sort and scan N elements, respectively. We also study a variant of the problem where instead of the full input raster we handle only a connected subregion of arbitrary shape. For this variant we describe an algorithm that runs in T(U log N) time in internal memory, where U is the size of the output. We show how this algorithm can be adapted to perform efficiently in the external memory using O(sort(U)) data transfers from the disk. We have also implemented two of the presented algorithms, the O(sort(N)) external memory algorithm for full rasters, and the internal memory algorithm that handles connected subregions, and we demonstrate their efficiency in practice

Implicit flow routing on triangulated terrains

Flow-related structures on terrains are defined in terms of paths of steepest descent (or ascent). A steepest descent path on a polyhedral terrain T with n vertices can have T(n2) complexity, since at worst case the path can cross T(n) triangles for T(n) times each. We present a technique for tracing a path of steepest descent on T in O(n log n) time implicitly, without computing all the intersection points of the path with the terrain triangles