The Problem Of Super Resolution With Gaussian Low Pass Filter

With the progress and development of science and technology,digital image has become the most commonly used information carrier in human activities,and it is also an important way to obtain external information.It contains a large amount of information,and its quality plays an import role in the process of visual perception and communication.However,in the process of image acquisition,storage,etc,the quality of the image will be affected by the interference of the noise,and when the image is pre-processed,different algorithms will also affect the final result.In recent years,image super-resolution analysis plays an important role in image process-ing because of its wide application.It can overcome the problem of insufficient information and is to restore the information in the high frequency part of the signal through the low frequency part of the signal.In the super-resolution problem,The technique is to restore the information in the high frequency part of the signal through the low frequency part of the signal.for a given point source,an accurate recovery of the signal can be obtained by solving l1 convex optimization model.The minimum separation ?(T)needs to satisfy the condition ?(T)? 2/fc.For the signal which is convoluted by Gaussian low-pass filters,the high frequency information of the signal can be recovered stably by solving a convex optimization model.The stability of the problem is proved theoretically.The proof depends on the duality certification in compressed sensing.A significant difference is that Gaussian low-pass filters do not satisfy the conditions of measurement matrix in compressive sensing,such as mutual coherence,restricted isometry property,etc.Constructing a conditional dual polynomial guarantees the existence of a unique solution to the optimization model.This paper constructs a dual polynomial to interpolate the signal to ensure that the matrix can satisfy the null-space property.Thus,the gap between the original signal and the recovered signal and the stability analysis of the convex optimization model can also be obtained.