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.