A Study On Theory And Technique Of Geometric Modeling Based On Neural Networks And Sparse Representation

The random weights learning algorithm of neural network has become one of important research content in the field of machine learning. Sparse representation and learning theory are the hot research topics among the current signal processing, image analysis and pattern recognition. So exploring novel theory and approach of geometric modeling based on neural network and sparse representation has significant theory value and practical meaning.We takes neural network, sparse representation technique and compressed sensing as basis, deeply analyzes their applications in geometric modeling, and then focuses on the random weights learning algorithm of neural network, makes an exploratory and some innovative researches of curve approximation, curve blending on sphere, and surface reconstruction combined with sparse representation techniques. The main contents of the thesis include:An algorithm of curve approximation on surface based on random weights network(RWN) is proposed. During the process of learning, the network’s inner weights and bias are selected randomly in some domains, the output weights of network are determined through the Moore-Penrose pseudo inverse of hidden output matrix. For enhancing the stability of solution, a regularization technique is applied in obtaining the Moore-Penrose pseudo inverse of hidden output matrix. This scheme improves the approximation accuracy of curve.An algorithm concerning curve blending on sphere is presented. The C0,C1, and C2 curves blending algorithm on sphere are realized through constructing appropriate blending basis functions. Furthermore, improved C1 and G1 algorithms of curve blending on sphere are given. Compared with other method, we only use one degree blending basis function to achieve C1 and G1 curve blending on sphere.A surface reconstruction algorithm based on random weights neural network is proposed. For further improve the approximation accuracy and the smoothness of the reconstructed surface, a polychromic random weights network(PRWN) is established by adding a low degree polynomial to the classical random weights neural network. Compared with RWN, the proposed PRWN is superior to RWNN in the aspects of approximation accuracy and surface’s smoothness.For the issue of high noise data surface reconstruction, a novel surface reconstruction algorithm based on elastic random weights network(ERWN) is developed. It is unlike the traditional disposal scheme that the data are first disposed by removing the noise, the proposed ERWN can reconstruct the surface without removing the noise contained in data. The elastic penalty item is added into the optimization objective function for obtaining stabile economical RWN model, reducing the influence induced by noise to real data. Compressed sensing algorithm is used to obtain the output weights, which enhances the solution’s stability. Experimental results show that ERWN still complete surface reconstruction even under the condition of high noise.A reduction algorithm of point clouds based on sparse representation that can retain key character points is presented. According to the data, implicit random weights neural network with the technique of unit decomposition is proposed to perform complex surface reconstruction. This method not only has excellent global approximation capability, but also cares for local approximation performance.A hypersurface reconstruction method on the sphere based on interpolation is presented. The profile curve of spherical interpolation basis function is designed by using the technique of spherical triangulation, through which the basis functions with local support property are constructed in the neighborhood of interpolation points. Finally, the surface function on the sphere is constructed by a suitable combination of the basis functions and the function values corresponding to the sample points. This method can utilize more function values of adjacent points to compute the function values.For the problem of hypersurface reconstruction in the case of nonuniform data distribution and noise disturbance, the polyharmonic random weights neural network is employed to carry out the hypersurface reconstruction on the sphere. In the phase of model training, the inner weights are randomly selected on the unit sphere, while the output weights are determined via the regularization Moore-Penrose pseudo inverse of hidden output matrix. Experimental tests illustrate that the proposed method outperform the method of classical spherical triangulation.