2,859 research outputs found
Projected Newton Method for noise constrained Tikhonov regularization
Tikhonov regularization is a popular approach to obtain a meaningful solution
for ill-conditioned linear least squares problems. A relatively simple way of
choosing a good regularization parameter is given by Morozov's discrepancy
principle. However, most approaches require the solution of the Tikhonov
problem for many different values of the regularization parameter, which is
computationally demanding for large scale problems. We propose a new and
efficient algorithm which simultaneously solves the Tikhonov problem and finds
the corresponding regularization parameter such that the discrepancy principle
is satisfied. We achieve this by formulating the problem as a nonlinear system
of equations and solving this system using a line search method. We obtain a
good search direction by projecting the problem onto a low dimensional Krylov
subspace and computing the Newton direction for the projected problem. This
projected Newton direction, which is significantly less computationally
expensive to calculate than the true Newton direction, is then combined with a
backtracking line search to obtain a globally convergent algorithm, which we
refer to as the Projected Newton method. We prove convergence of the algorithm
and illustrate the improved performance over current state-of-the-art solvers
with some numerical experiments
Joint Image Reconstruction and Segmentation Using the Potts Model
We propose a new algorithmic approach to the non-smooth and non-convex Potts
problem (also called piecewise-constant Mumford-Shah problem) for inverse
imaging problems. We derive a suitable splitting into specific subproblems that
can all be solved efficiently. Our method does not require a priori knowledge
on the gray levels nor on the number of segments of the reconstruction.
Further, it avoids anisotropic artifacts such as geometric staircasing. We
demonstrate the suitability of our method for joint image reconstruction and
segmentation. We focus on Radon data, where we in particular consider limited
data situations. For instance, our method is able to recover all segments of
the Shepp-Logan phantom from angular views only. We illustrate the
practical applicability on a real PET dataset. As further applications, we
consider spherical Radon data as well as blurred data
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