Variations of L1 -regularization including, in particular, total variation regularization, have hugely improved computational imaging. However, sharper edges and fewer staircase artifacts can be achieved with convex-concave regularizers. We present a new class of such regularizers using normal priors with unknown variance (NUV), which include smoothed versions of the logarithm function and smoothed versions of Lp norms with p ≤ 1 . All NUV priors allow variational representations that lead to efficient algorithms for image reconstruction by iterative reweighted descent. A preferred such algorithm is iterative reweighted coordinate descent, which has no parameters (in particular, no step size to control) and is empirically robust and efficient. The proposed priors and algorithms are demonstrated with applications to tomography. We also note that the proposed priors come with built-in edge detection, which is demonstrated by an application to image segmentation.

A new method for blind image deblurring is proposed that relies on a smoothed-NUV (normal with unknown variance) prior for images, which promotes piecewise smooth images with crisp edges. The proposed method can use multiple blurred versions of the same image. The variational representation of the prior allows the joint estimation of the image and the blurring kernel(s) to be decomposed into descent steps in reweighted least-squares problems and nonlinear scalar updates of the individual variances of the prior. Specifically, we propose an iteratively reweighted coordinate descent algorithm that has no parameters. Simulation results demonstrate that the proposed approach compares favorably to state-of-the-art methods.

Most of the algorithms for tomographic reconstruction face the same problem: high computational complexity. In order to tackle this problem, this paper proposes a general multi-resolution approach that enables a flexible choice of reconstruction focus and thus saves computational power in reconstructions. The approach is demonstrated in this paper based on a reconstruction algorithm using a (improper) Markov random field prior with sparsifying NUV terms (nor-mal with unknown variance), where the unknown variances are learned by approximate EM (expectation maximization). The experimental and practical results show that both for simulated and real-world objects the proposed framework yields satisfying results with much lower computational cost.