Adaptive Multiscale Geometric Network: Theory And Application

Multiscale Geometric Analysis (MGA) is a new harmonic analysis tool for giving a more efficient representation of high dimensional data. In the past decade, it has been developed independently in different subjects such as mathematical analysis, pattern recognition, statistics and computer vision. Now it has been successfully applied in regression, signal and image processing.With wider applications of MGA into various kinds of communities, the fixed transformation-based MGA methods are considered to be incapable of dealing with some practical tasks in engineering little by little. The idea of "adaptivity" begins to be adopted by many researches, which helps to solve many related problems. Concretely speaking, there are two kinds of "adaptivity". One is focused on the "adaptive" version of the available MGA methods such as PPR based ridgelet approximation, adaptive curvelet denoising and adaptive selection of contourlet packet and so on. They have better performance than the corresponding non-adaptive version. Another is to construct new MGA tool using adaptive approach such as Bandelet. They mostly have flexible structure and show more adaptivity to inputs. It can be predicted that in quite a long period, MGA will develop in both non-adaptive and adaptive way. The tools for realizing adaptivity are widely distributed in branches of machine learning, which can help to build a more perfect frame of MGA. On the other hand, MGA can help to solve wider range of problems in machine learning. That is, an efficient combination of these two fields can speed up their developments. Consequently, the study on theory and application of adaptive multiscale geometric networks which is based on machine learning is a very important and valuable field.Based on an investigation into MGA and machine learning, this paper proposed several adaptive multiscale geometric networks by using machine learning approaches, such as paralleled multiscale geometric networks with directional neurons, and some MGA recognition systems with neural network. Combined with our projects, the proposed new methods in this paper are applied to practices. Our main work can be summarized as follows:(1) Have an investigation into MGA and its development, and have a study on some MGA tools. An adaptive ridgelet network model is proposed on the foundation of continuous ridgelet approximation theory. The learning algorithm and proof of convergence of this new network are expounded. It is applied into the approximation of high dimensional functions and functions with specific singularities, and better results than that of wavelet neural network are obtained.(2) To improve the generalization ability of adaptive ridgelet network and speed up the convergence, a regularization ridgelet network is constructed based on statistics learning. It can overcome the trend of getting into a local minimum of gradient-based training algorithm, and good generalization results are obtained. Its applications into nonlinear time series predicting and remote sensing image compression prove its feasibility.(3) A directional multiscale ridgelet network is advanced with a binary ridgelet frame being its design foundation. The structure and learning algorithm of the network are given, as well as a theoretical analysis on its approximation property. The superiority of this network is demonstrated by simulation results of function approximation, pattern classification, and applications in 1-D range image based radar target recognition also prove its high efficiency.(4) Aiming at solving especially high dimensional problems, we introduced an efficient linear learning algorithm by inspiration of the traditional kernel smoothing method. It is characteristic of a linear computation complexity in time and space with the number and dimension of input samples. With a small lost in accuracy, it can reduce the complexity in time and space, accordingly improve the real-time processing ability of ridgelet network. The regression of complex high dimensional functions also proves its efficiency.(5) Based on ridgelet theory, kernel and regularization techniques from which we can deduce a regularized kernel regression form, a new ridgelet kernel network is presented to approximate a wide range of multivariate functions. Experiments in the tasks of approximation of a wide range of functions, classification of road signs in visual navigation system and recognition of planes also show its efficiency.(6) Based on the ridgelet-pyrimid-based curvelet transformation, and an immune RBF neural network which can complete a rapid convergence and good generalization, a curvelet network is proposed for pattern recognition. Its application in classification of ships and planes also prove its efficiency.(7) Base on the contourlet transform realized by Laplacian decomposition and directional filter banks, an optimal contourlet packet optimized by quantum genetic algorithm is proposed using wavelet transform. Its application in fingerprint coding proves its superiority. Subsequently we advance an optimal contourlet packet model for recognition of planes. Simulation results of recognition of planes prove its feasibility.