Introduction

Artifacts degrading the quality of recorded images are mainly caused by blurring and random noise that are intrinsic to the detection process. To alleviate these effects, image restoration can serve as a desirable pre-processing technique. To obtain a reasonable estimate of the source image, a common approach is to form an objective function which quantifies the quality of a given estimate by measuring the consistency between the estimation and the measurements and by penalizing solutions that do not satisfy certain properties of the image. In this variational framework, a parameter that needs special consideration is the form of the regularizer to be chosen as it significantly affects the quality of the restored image.

Main Contribution

Our main contributions are twofold.

• Second-order extensions of TV
• Restoration based on second-order regularization

The definition of second-order extensions of the well-known TV regularization functional is based on an alternative interpretation of TV that we derive and that relies on mixed norms of the first directional derivative. The obtained second-order functionals turn out to be well-suited to the task of image restoration, especially for biomedical images. In particular, we prove that they preserve some of the most favorable properties of TV, namely, convexity, homogeneity, rotation, and translation invariance.

Our unifying computational framework for image restoration is based on second-order regularization. More specifically, we develop several efficient algorithms for the minimization of the corresponding objective functions.