8,896 research outputs found
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
Second order adjoints for solving PDE-constrained optimization problems
Inverse problems are of utmost importance in many fields of science and engineering. In the
variational approach inverse problems are formulated as PDE-constrained optimization problems,
where the optimal estimate of the uncertain parameters is the minimizer of a certain cost
functional subject to the constraints posed by the model equations. The numerical solution
of such optimization problems requires the computation of derivatives of the model output
with respect to model parameters. The first order derivatives of a cost functional (defined
on the model output) with respect to a large number of model parameters can be calculated
efficiently through first order adjoint sensitivity analysis. Second order adjoint models
give second derivative information in the form of matrix-vector products between the Hessian
of the cost functional and user defined vectors. Traditionally, the construction of second
order derivatives for large scale models has been considered too costly. Consequently, data
assimilation applications employ optimization algorithms that use only first order derivative
information, like nonlinear conjugate gradients and quasi-Newton methods.
In this paper we discuss the mathematical foundations of second order adjoint sensitivity
analysis and show that it provides an efficient approach to obtain Hessian-vector products. We
study the benefits of using of second order information in the numerical optimization process
for data assimilation applications. The numerical studies are performed in a twin experiment
setting with a two-dimensional shallow water model. Different scenarios are considered with
different discretization approaches, observation sets, and noise levels. Optimization algorithms
that employ second order derivatives are tested against widely used methods that require
only first order derivatives. Conclusions are drawn regarding the potential benefits and the
limitations of using high-order information in large scale data assimilation problems
Calculation of Relaxation Spectra from Stress Relaxation Measurements
Application of stress on materials increases the energy of the system. After removal of stress,
macromolecules comprising the material shift towards equilibrium to minimize the total
energy of the system. This process occurs through molecular rearrangements or “relaxation”
during which macromolecules attain conformations of a lower energetic state. The time,
however, that is required for these rearrangements can be short or long depending on the
interactions between the macromolecular species that consist the material. When
rearrangements occur faster than the time of observation (experimental timescale) then
molecular motion is observed (flow) and the material is regarded as viscou
A GCV based Arnoldi-Tikhonov regularization method
For the solution of linear discrete ill-posed problems, in this paper we
consider the Arnoldi-Tikhonov method coupled with the Generalized Cross
Validation for the computation of the regularization parameter at each
iteration. We study the convergence behavior of the Arnoldi method and its
properties for the approximation of the (generalized) singular values, under
the hypothesis that Picard condition is satisfied. Numerical experiments on
classical test problems and on image restoration are presented
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
Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells
Adherent cells exert traction forces on to their environment, which allows
them to migrate, to maintain tissue integrity, and to form complex
multicellular structures. This traction can be measured in a perturbation-free
manner with traction force microscopy (TFM). In TFM, traction is usually
calculated via the solution of a linear system, which is complicated by
undersampled input data, acquisition noise, and large condition numbers for
some methods. Therefore, standard TFM algorithms either employ data filtering
or regularization. However, these approaches require a manual selection of
filter- or regularization parameters and consequently exhibit a substantial
degree of subjectiveness. This shortcoming is particularly serious when cells
in different conditions are to be compared because optimal noise suppression
needs to be adapted for every situation, which invariably results in systematic
errors. Here, we systematically test the performance of new methods from
computer vision and Bayesian inference for solving the inverse problem in TFM.
We compare two classical schemes, L1- and L2-regularization, with three
previously untested schemes, namely Elastic Net regularization, Proximal
Gradient Lasso, and Proximal Gradient Elastic Net. Overall, we find that
Elastic Net regularization, which combines L1 and L2 regularization,
outperforms all other methods with regard to accuracy of traction
reconstruction. Next, we develop two methods, Bayesian L2 regularization and
Advanced Bayesian L2 regularization, for automatic, optimal L2 regularization.
Using artificial data and experimental data, we show that these methods enable
robust reconstruction of traction without requiring a difficult selection of
regularization parameters specifically for each data set. Thus, Bayesian
methods can mitigate the considerable uncertainty inherent in comparing
cellular traction forces
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