图像的正则变换与快速自适应滤波

Regular Transformation of Image and Fast Adaptive Filtering

本文给出了图像的一类卷积逼近公式，称之为图像的正则变换，其变换核是无穷次连续可微的二元函数，并具有局部支撑性质．以此变换为基础，以最小二乘为准则，构造了基于低松弛迭代格式的反卷积快速自适应滤波算法．对于一个Ｎ×Ｎ输入图像，由于变换核的局部支撑性质以及低松弛迭代算法的引入，使得最小二乘滤波算法的计算复杂度降为Ｏ（Ｎ２），比不动点（ＦＰ）迭代算法的Ｏ（Ｎ３）及预处理共轭梯度（ＰＣＧ）算法或小波重构算法的Ｏ（Ｎ２·ｌｏｇＮ）都好，从而使最小二乘滤波算法真正成为高维信号处理中的一类实用有效的自适应滤波算法．大量的实验结果表明，该滤波算法对Ｇａｕｓ白噪声及均匀分布的噪声都有良好的抑制特性．

In this paper,an approximate formula for image transformation is proposed.The formula referred to as a Regular Transformation(RT) is derived from a convolution sum with a locally supported and infinitely differentiable kernel.According to the law of least squares,a RT based fast adaptive filter with the underralaxation iterative scheme is developed.For a N×N image,the computational complexity of the filtering algorithm is O(N 2) ,which is significantly better than O(N 3 ) of the fixed point iterative method for handling LS problem and O( N 2 log N ) of both the preconditioned conjugate gradient iterative algorithm and wavelet transform based denoising algorithms.Consequently,the filter may be used in computer vision and real time signal processing.The numerical results show that the filter is suitable for the reduction of both Gaussian noise and noise with uniform distribution.