Regular transformation of image and fast adaptive filtering

An approximate formula for image transformation was proposed. The formula referred to as a Regular-Transformation(RT) is derived from a convolution sum with a locally supported and infinitely differential kernel. According to the law of least squares, a RT-based fast adaptive filter with the under-relaxation iterative scheme is developed. For a N × N image, the computational complexity of the filtering algorithm is O(N²), which is significantly better than O(N³) of the fixed point iterative method for handling LS problem and O(N² log N) of both the preconditioned conjugate gradient iterative algorithm and wavelet-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.