| In image space, denoising, compressing, feature detection and fusion are equivalent to approximation in continuous space. So the performance in image processing is controlled by smoothness, sparity, similarity to feature of the functions, which are used to approximate the image. Generally, these problems are all ill-posed, and cannot be resolved by the traditional methods. In other words, the base functions are very important for the whole system.The problem of approximating multi-dimension function is the focus of theoretic research in recent years. The dissertation begins with the relations among neural network, functional network, ridgelet analysis, and wavelet analysis in the sight of function approximation. Especially for the approximation in multi-dimension space, the sparity of sample data for learning (nearly empty) results in that the traditional tools, which offer the good performance in one-dimension space, cannot be applied. Fortunately, many new methods are presented these years, which can give the super performance for its good property in the multi-dimension space. For example, wavelet analysis is good at point-singularity while ridgelet at hyperplane-singularity representation. Neural network is always used to approximate the single function and functional network prefers to approximate the functional model.SAR images get the rapid development and wide application because they can be obtained any-time, any-weather, and with high resolution. Unfortunately, SAR images are always polluted by the multiplicative noise, which does not satisfy the Gaussian distribution and is difficult to remove. For this reason, the traditional methods of image processing do not work well.Combined with our projects, a series of new methods are proposed and applied to SAR image donoising, edge detection, singularity detection, compressing and fusion. And good performances are achieved.The main innovative points follow:1. A method to construct multiwavelet based on complex wavelet is presented. For the absence of symmetry in real wavelet analysis, its application is limited. Complex wavelet has good symmetry, but it's too complicate to analyze the real signa: and the imaginary part of the processed result is truncated. This dissertation separates the real part and the imaginary part from the complex wavelet to construct multiwavelet with the good properties, such as short supported, orthogonality andsymmetry. The muitiwavelet via this method need not pre/post filter and its symmetry of filter coefficients reduces the computation complication. The new kind multiwavelet, named CLi, is applied to denoise and compress the SAR image and get good performance.2. A general theory for the construction of wavelet network is given. It deals with the existence of minimum number of hidden layer's units when the error is given. Also, the construction theory and algorithm are discussed here. The result is guidance to the construction of the network structure.3. Functional network is a very new field, which has been applied in many fields. In this dissertation, functional network and chaotic communication are discussed. And a functional network-based method is proposed for extracting a useful signal hidden in chaotic background.4. An important character of functional network, commutability, is discussed in this dissertation, and the method to construct three special models of the commutable and separable functional network is introduced here. The simulate examples demonstrate its good theoretical result.5. An improved Mean of Least Variance filter is presented to remove the multiplicative noise, such as the noise in SAR image. It changes the multiplicative noise to the additive noise, and then uses the MLV-like to remove the additive noise. Its performance is better than Minimum Coefficient of Variation (MCV) filter and Mean Least of Variation (MLV) filters. Both one-dimension and image experiments demonstrate its theoretical validity.6. This dissertation apply ridgelet functional network to approximate...
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