Many problems in imaging need to be guided with effective priors or reg- ularizations for different reasons. A great variety of regularizations have been proposed that have substantially improved computational imaging and driven the area to a whole new level. The most famous and widely applied among them is L1-regularization and its variations, including total variation (TV) regularization in particular. This thesis presents an alternative class of regularizations for imaging using normal priors with unknown variance (NUV), which produce sharp edges and few staircase artifacts. While many regularizations (includ- ing TV) prefer piecewise constant images, which leads to staricasing, the smoothed-NUV (SNUV) priors have a convex-concave structure and thus prefer piecewise smooth images. We argue that "piecewise smooth" is a more realistic assumption compared to "piecewise constant" and is crucial for good imaging results. The thesis is organized in three parts.

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