Research On Fractal-Based Surface Reconstruction From Terrain Image

This project is sponsered by National Natural Science Foundation of China, National Defense Pre-Research Foundation of China, and The Royal Society Foundation of UK.In many applications in the fields of national defense and daily life, terrain model plays an important role. Although plenty of indirect terrain information is available from remote-sensing images, direct 3D terrain surfaces have not been efficiently extracted applying the latest 3D reconstruction techniques. On the other hand, the fractal techniques have been widely applied to terrain simulation but not been explicitly combined with 3D reconstruction techniques. So this thesis researches and discusses the problems of fractal-based 3D terrain surface reconstruction on both models and algorithms. The thesis focuses on Shape-From-Shading(SFS) as one of the most important 3D reconstruction techniques. This not only overcomes the difficulties of applying traditional fractal techniques to the observed images, but also efficiently obtains the terrain surface shape from the observed images within tolerable errors. And the following are the main contributions in this thesis.By reviewing and analyzing the SFS applications with different constraints and prior knowledge we conclude that (1) regardless of the deterministic constraint or given stochastic prior distribution, the corresponding SFS problem can be formed as an optimization problem, (2) the constraints in SFS problems have to meet the requirement of the prior knowledge and algorithm stability, (3) the parameter determining and surface shape extracting iterate one after another or at the same time in the SFS algorithms, and (4) due to the high nonlinearity of the reflectance function, applying the Tailor linearization method to SFS algorithm tends to local minimum state.Based on the analysis of the common features of computing different fractal parameters, we propose a general method to construct an approximated fractal model by using the correlation of fractal feature variables. And then four fractal models with different features are provided based on Variogram Analysis, Power Spectra Analysis, Layered-increment Analysis and Wavelet Analysis respectively. The constructed results show that these four approximating fractal models can give the statistic fractal features and efficiently approximate the nonlinear fractal model. Compared to common fractal tools, the fractal model constructed by our method is more intuitive, easier to control, very convenient to combine with observed data and more amenable to computing.A series of linear MSE sub-problems have to be formed to solve the highly non-linear SFS problem. Generally, Tailor linearization method is adopted and Gaussian noise is assumed, which can easily lead to local minimums. So we provide a modified method in which the nonlinear parts in the SFS problem are transformed as systematic parameters by applying Pentland linearization method, the noise of which has non-Gaussian distribution and can be estimated by our fractal model. The experimental results show that the proposed method can avoid the local minimum state better than the traditional methods. The fractal-based SFS problem is formed as a fractal regularization problem in this thesis and an iterative algorithm is provided by using the regularization theory. An advantage of this method is that the fractal dimension of the recovered surface can be controlled by the number of iterations, which is controllable and flexible. The experimental results show that our method can efficiently recover the surface shape from the terrain image. Compared to traditional SFS methods, the recovered terrain surface by our method keeps the fractal features of the terrain while is not easily affected by the noises.As the non-homogeneously fractal surface being considered, a local fractal regularization SFS algorithm with an inner-outer loop structure is proposed based on the block-Jacobi iteration scheme, the local fractal analysis and our fractal regularization method. The local fractal dimension is controlled by the number of iterations of the corresponding block in the inner loop, while the global surface consistency is determined by the number of iterations of outer loop. The experimental results show that our method can efficiently recover the surface shape from the complicated terrain image and get rid of the topological errors appearing in some applications.As for the SFS problem of the complicated multi-fractal terrain image, a multi-scale fractal regularization method, based on the theories of Multi-Level Optimization (MLO) methods is proposed. The proposed method achieves multi-scale fractal constraints through adjusting the constant vector in the local fractal regularization SFS problem in the smaller scale by the recovered surface obtained in the larger scale. The multi-fractal features of the recovered surface are dominated by the number of iterations relating to the different block in the different scale. The experimental results show that our multi-scale fractal regularization method can efficiently recover the surfhce shape from the complicated terrain image.Compared to the typical SFS methods, our methods are easier to compute, lends itself to parallel computational techniques, and are more suitable for large image problems. Furthermore, our methods can flexibly control the fractal features of the recovered surface and be extended to combine with other models.