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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
Abstract
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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