Spatial and environmental data approximation

Continuous surfaces represent 2-D phenomena that have values at every point across their extent. The values at an infinite number of points across the surface are derived from the surface represen- tation. Surface generation from point cloud data is in essence to go from a discrete to a continuous representation by enhancing the data with structure and in addition pursuing a more convenient representation format. A data set can be raw, i.e., it has not been subject to any processing op- erations, or it can be processed by, for instance, thinning, removal of outliers, or created from merging several initial data sets. A data set is typically equipped with x-, y-, and z -coordinates and can be enriched with other sensor data. Airborne acquisition provides 2.5D data, where a single height value is defined for each point in the plane. In very steep areas and in areas with fully 3D shapes, the data set will be generally incomplete. Other methods, for instance mobile mapping systems, provide more complete information in smaller areas (Chapter 1). Misalign-ment is generally visible as an incorrect registration of acquired data. Missing data correspond to unassembled regions of the surface, e.g., from occlusions during the acquisition process, different absorption of regions or limits of the sensor components. A data set may be quite uniformly dis- tributed, but can also consist of a set of scan lines. Figure 2.1 shows two data sets obtained from sea bottom consisting of scan lines. Figure 2.1a shows data resulting from one data acquisition (one survey) that consists of several disjoint pieces. Different surveys are likely to be obtained at different dates and possibly with different equipments. In Fig.2.1b, several surveys have been merged to create a block of data, where a misalignment may have occurred. In this chapter, we will start by looking into the expected input data and define some criteria for a good surface generation method (Sec.2.1), and describe some surface formats and concepts used in the context of geographical information systems (GIS). Then, we will continue with more detailed information on some surface formats and generation methods. The emphasis will be on splines (Sec. 2.2), in particular locally refined splines and meshless methods (Sec.2.3) for surface generation.