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Series in Signal and Information Processing, Vol. 34
edited by Hans-Andrea Loeliger








Boxiao Ma


Smoothed-NUV Priors

for Imaging and Beyond


1st Edition 2022. XXVI, 152 pages, € 64,00.

ISBN 978-3-86628-746-4



















Many problems in imaging need to be guided with effective priors or regularizations 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 (including 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. We start the first part by revisiting related work on imaging regularizations/priors and a preview of comparison between SNUV and other priors. We then describe the general form of the SNUV priors and discuss its different variants, including the plain SNUV, the (convex) Huber function, and smoothed Lp norms. The Huber function is from robust statistics and a special case of smoothed Lp norms.


We further show two different genres of algorithms for image estimation with SNUV. All SNUV 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 (IRCD), which has no parameters (in particular, no step size to control) and is empirically robust and efficient. Another style of algorithms is approximate expectation maximization (EM), which can be performed efficiently with the iterative scalar Gaussian message passing (ISGMP) technique. However, IRCD is more reliable than approximate EM and usually yields marginally better results.


In the second part, the described priors and algorithms are demonstrated with different imaging applications. In computed tomography, the results of SNUV exhibit both visually and quantitatively advantages over many others (including TV), especially while reconstructing piecewise smooth objects. We further note that the SNUV priors come with built-in edge detection, which is illustrated by an application to image segmentation (in both 2D and 3D). With the edge detection, we also define a sharpness measure that helps to correct an artifact in tomography. The IRCD algorithm is extended to learn matrices when applied to blind image deblurring, where we also discuss the viability of naive maximum-a-posteriori (MAP) methods.


The best empirical results are usually obtained with nonconvex SNUV priors, which include smoothed versions of the logarithm function (plain SNUV) and smoothed versions of Lp norms with p < 1. In the third and last part of the thesis, we extract the local edge rate of images based on the built-in edge detection of SNUV (or even any other edge detectors) using windowed autonomous linear state space models (LSSMs). The edge rate is obtained while fitting the edge count, and the window localizes the fit. The computational efforts are minimal since efficient recursions are developed to calculate the fit. We verify the effectiveness of the method using a practical example.


Keywords: Image estimation, regularizations, NUV, sparsity, iterative reweighted descent, expectation maximization, tomography, edge detection,

image segmentation, blind image deblurring, edge rate, LSSM.



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