Given a blurry image, the goal is to find the most clear image. There are many methods to solve this inverse problem in the case of cartoon images containing rather piecewise smooth objects. However, in the presence of oscillations the blurring process removes those oscillations in the images and that makes this inverse problem harder to solve. We approach this problem by minimizing a convex functional whose domain is the product of the space of functions of bounded variation and the homogeneous Sobolev space. As we will see, the homogeneous Sobolev space turns out to be a good space to capture oscillations. We will talk about the existence of a minimizer and characterization of the minimizers and PDE based numerical scheme and then briefly discuss a noisy case. If time permits, we will also talk about a medical image denoising application.