A Surface Interpolation for Large-scale Representation of Terrain Surface in An Urban Area
Huag Shaobo,R.Shibasaki
Institute of Industrial Science, University of Tokyo
7-22, Roppongi Minatoku, Tokyo, Japan
Abstract:
The 3D representation of the urban space is essential in managing and utilizing urban information in geographic information systems. The 2.5-d surface based boundary representation (SR) model seems to be an suitable 3D representation model of the urban space. In order to efficiently implement SR model, an improved surface interpolation method is proposed in the paper. Through an example of the representation of terrain surface, which have large volume of data and complex shape, it is demonstrated that the proposed surface interpolation method can be successfully and efficiently applied.
Introduction:
Nowadays, with the considerable growth in geographic information systems to meet two dimensional mapping needs, more and more demands are turning to the design and development of three dimensional mapping and modeling systems in a range of application areas. These demands are much stronger for those applications related to urban planning, infrastructure construction and facility management due to the complicated and dense utilization of urban space from the underground to the air.
The conventional maps and map-drove data such as digital models (DTMs) can not represent 3D geographic objects. To handle 3D geographic data, solid modeling methods have been developed for computer aided design(CAD)/computer graphics(CG) systems. In geological information management, solid modeling methods and related surface interpolation techniques such as NURBS [Fisher and Wales, 1991) are applied for the 3D representation of geological objects [Raper et.al, 1988], [Jones, 1989], [Raper and Kelk, 1991]. However, these solid modeling methods could not be directly applied to the 3D representation of urban space because urban space contains & a much wider variety and larger amount of geographical objects.
The basic requirement for 3D representation model of urban space can be summarized as follows (Shibasaki et.al. 1990) : (1) ease of representation of a variety of geographic objects in urban space; (2) Efficiency in building and updating 3D spatial database; (3) Ease of spatial query and analysis; (4) Compatibility with existing map data and CAD/CG systems. The surface based boundary representation(SR) model proposed by [Shibasaki & Huang 1992] can satisfy the requirement of 3D representation in urban space.
The BR model fulfills both the compatibility with 2D map data and the ease of the representation of urban space, but the input and update of 3D spatial data with the BR model is severely labour-demanding. Although an algorithm was proposed for uncovered both planar polygons(faces) in edges and solids in polygons, the identification of non-planar polygons(faces) in edges still remains difficult. The polygon identification in the 2.5D surface(which is represented by a single-valued and continuous function), whether planar or non-planar, can be easily done by applying conventional algorithm to its projected plane. Surface interpolation algorithms for 3D data can be directly applied to the interpolation of their surface. By combining 2.5D surfaces into the BR model 3D spatial database can be easily developed.
The 2.5D surface such as the terrain surface serves as a basis of SR model. The suitable surface interpolation for the 2.5 surface will be important in automatically and efficiently generating a 2.5D surface, especially the terrain surface.
The existing Surface interpolation methods:
A large amount of elevation points are necessary to represent 3D spatial object with a SR model. Especially the representation of terrain surfaces requires many reliable elevation points. This is not only because terrain surfaces have complicated shaped but also because the elevation of other spatial objects such as underground structures have to be determined based on the elevations of terrain surfaces.
However it is not easy task to assign elevation data to so many points manually. For example, it is very labor-demanding to obtain elevation data from conventional maps in urban areas because contour lines are usually cut in pieces due to buildings and other man-made features. With aerial surveying techniques, it is not so easy to obtain enough number of elevation points due to occasions. Only roads and the roofs of buildings are exceptionally easy places for 3D measurement. A method of surface interpolation in urban areas is indispensable to meet the requirement of elevation data and to give a sound basis of elevation to a SR model.
The conversional surface interpolation methods were developed for digital terrain models (DTMs). Since the conventional surface interpolation methods are mostly used for the small scale representation of rural terrain area, they can not be directly applied into the large scale representation of urban terrain surface because the discontinuities of slopes and elevation are often the cases in an urban area. It is necessary to improve existing surface interpolation method to represent the characteristic of urban terrain surface.
The outline of a surface interpolation method:
Existing terrain surface interpolation methods usually assume that terrain surfaces are smooth although the discontinuities of slopes and elevations are often the cases in urban areas. To make larger-scale representations of terrain surfaces and related spatial objects in urban areas, the following geometric conditions(Figure 1) must be considered, which characterize terrain surfaces in urban areas.
Figure 1 Examples of geometric constraint conditions in surface interpolation
Break lines : The steepness of slopes shows discontinuities on a break line. Break lines are often to be seen in the boundaries of man-made objects such as roads and levees.
Step lines : Elevation shows a abrupt change (like steps) on a step lines. Retaining walls and the side walls of buildings are generated by step lines. Sometimes, step also can seen in the boundaries of the buildings, especially when the building is constructed in a steep slope. It should be noted that a step line must be derived into two lines in triangulation.
Horizontal planes : Every points in a horizontal plane has the same elevation value. Floors of buildings are the examples.
Under these geometric conditions, surface are represented by TIN to easily integrate the interpolated surface with the SR model. At places where these geometric conditions do not hold, elevations are interpolated under the assumption that a terrain surface is smooth. Smooth terrain surfaces are obtained to maximize the sum of the square of inner-products of unit normal vectors of neighbouring triangular planes. Moreover, some lines such as road boundaries sometimes have to be interpolated smoothly. The "smoothness" of lines is evaluated in terms of the sum of the squares of vertical changes of unit vectors along the lines. Thus elevations are interpolated so as to maximize to smoothness of terrain surface and lines under the above geometric constraint conditions.
The basic process of the surface interpolation is shown in figure 2. The surface interpolation process consists of four parts as following:
(1) 3D coordinates (x, y, z) of selected elevation points(known points) and 2D coordinates (x, y) of the rest(unknown pints) are given. The geometric conditions or structural features such as break lines and step lines are also specified.
Figure 2. The basic process of the surface interpolation
