| Résumé : |
(Auteur) The laser scanning provides very dense information about the surface to be modelled in the form of a irregularly distributed points cloud {x, y, z} c R3. Such dense information makes possible an efficient modelling of characteristic structures of the terrain surface like discontinuities, which are necessary for high-qualitative description of the surface. Simultaneously, the points not belonging to the modelled surface (for example: reflexes from buildings, trees etc.) stand a very important influence on the obtained data. During the modelling process, such data should be effectively filtered from the whole data set.
The laser scanning data can be efficiently elaborated by the use of deformable models of curves and surfaces. These models base on the physical principle of energy-minimizing and are presented as the solution of variational problem. The total energy consists both of internal and external energy. The external energy, depending on a context is generated by the data; in most cases it describes a deviation between the data and a model. The internal energy describes geometrical properties of curves and is characterized by elasticity and viscosity. Both terms are mutually weighted by the local control parameters a and 0. Varying the parameters makes it possible to stretch the curves to a geometrical shapes. The snake-approximation is used for a profiled modelling of surfaces. Due to that, a formulation of external energy was proposed making possible a robust modelling of profiles: during an iterative process, gross errors can be filtered, measuring errors can be smoothed and discontinuities can be preserved. Fitting the snakes-models to the data runs iteratively, however the control parameters depending on the data are being spread.
By generalizing the snakes, the model is introduced by sufficient smoothes, energy-charged pieces of a surface and furthermore described by flakes. The internal energy within the flakes model consists of a membrane and a thin-plate kernel which describes the inclination and curvature properties of the modelled terrain surface. The energy pieces will furthermore be weighted by the local control parameters a and fl. A minimizing of the total flakes energy leads again to the variational problem which had been differently solved. By formulate the Euler equations and their further diskretizing by finite differences, the flakes-model stands for regular data. The previous variational problem will also be solved by the use of so-called Ritz method. The improved flakes model was developed for regular data by using a linear base function. However, for the irregular data the flakes model was modified by the use of a Gaussian function. The modelling of the data by flakes runs iteratively. By using the flakes model for regular data it is possible to reject the gross errors, also to smooth the noise by simultaneous preserving the form of edges.
In many applications the information about spatial location of terrain edges is needed. To present such spatial location description of edges in a vector format based on irregular points cloud {x, y, z} obtained during laser scanning, it was proposed to describe a gross-errors-free data by surface functions and to average the edges as a intersection of two surfaces zi = fi(x, y) ; i = 1, 2. To this purpose, all the data should furthermore be ordered in separate pieces of the surface. This problem can be solved by using the standard methods of image processing. The projection {x(s), y(s)} of edges is found in the xy coordinate plane and the z-coordinates consist of fi (x(s), y(s)). To the intersection of two surfaces relates: f1 (x(s), y(s)) = f2 (x(s), y(s)) = 0. Based on this condition, two approaches of intersection curve identification were developed. The line-tracking algorithm relies on numerical integration of differential equations relative to the particular problem. For the numerical integration there is a starting point needed. Due to that, a seeking-approach was proposed. Opposite to the local algorithm it was presented a global approach using a snakes-method with a proper definition of external energy. Both algorithms make it possible a reliable, high-accurate identification of terrain edges basing on irregular points cloud.
The algorithms and approaches developed in this work have been tested on real data sets obtained by a laser scanning. Furthermore, a qualitative consideration of a modelling has been given. Finally, some hints for user according the steering and operating of the approaches have been presented. |
Modellierung von Oberflächen mit Diskontinuitäten (2004)
