This work is devoted to image restoration (deblurring-denoising) by constrained and unconstrained optimization models. The image f to be restored is assumed to be the sum of a cartoon component u (a function of bounded variation) and of a texture component v (an oscillatory function in a Sobolev space with negative degree of differentiability). We know that homogeneous Sobolev norms of negative differentiability help capture oscillations in images very well, however, these spaces do not directly provide clear distinction between texture and noise. In order to separate noise from texture in a blurry noisy textured image, we first learn properties that help distinguish these two types of oscillations, using a novel method based on one-dimensional Sobolev norm profiles. It turns out that the two Sobolev norm profiles for texture and noise are distinguishable, which enables us to better separate these two features in the deblurring process.
Joint work with John B. Garnett and Yunho Kim
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