A convex minimization model in image restoration via one-dimensional Sobolev norm profiles

We propose a new variational model for image restoration using BV and Sobolev spaces. It is well known that homogeneous Sobolev spaces of negative differentiability can capture oscillatory information very well, however, just one Sobolev space hardly recognizes any difference between texture and noise. By a means of learning a series of Sobolev norms of pure texture and pure noise that will provide us with one dimensional profiles describing different behaviors of texture and noise, we will be able to make a distinction between texture and noise, and use these measurements in restoring a better image. We want to point out that our model is specifically designed to deal with noisy blurred images. Parameter insensitivity is one of the advantages of using a series of Sobolev spaces.