Topics on Numerical Analysis and Applied Mathematics from the Junior Point of View
Image deblurring, i.e., reconstruction of a sharper image from a blurred and noisy one, involves the solution of a large and very ill-conditioned system of linear equations, and regularization is needed in order to compute a stable solution. Krylov subspace methods are often ideally suited for this task: their iterative nature is a natural way to handle such large-scale problems, and the underlying Krylov subspace provides a convenient mechanism to regularized the problem by projecting it onto a low-dimensional "signal subspace" adapted to the particular problem. In this talk we consider the three Krylov subspace methods CGLS, MINRES, and GMRES. We describe their regularizing properties, and we discuss some computational aspects such as preconditioning and stopping criteria.
■ The PHD Movieとは？
大学院生の研究生活をリアルかつユーモラスに描いた"PHD (Piled Higher and Deeper) Comics"の映画版です．
"PHD Comics"はWeb上で連載されており，世界中の大学院生に読まれています。
http://www.phdcomics.com/
2011年9月に，Comicsを実写化した映画が公開され，以降，特に海外のさまざまな大学のキャンパスで上映が行われています。
A fundamental imaging problem in microstructural analysis of metals is the reconstruction of local crystallographic orientations from X-ray diffraction measurements. This work develops a fast, accurate, and robust method for the computation of the 3D orientation distribution function for individual grains of the material in consideration. We study an iterative large-scale reconstruction algorithm, CGLS, and demonstrate that right preconditioning is necessary to provide satisfactory reconstructions. Our right preconditioner is not a traditional one that accelerates convergence; its purpose is to modify the smoothness properties of the reconstruction. We also show that a new stopping criterion, based on the information available in the residual vector, provides a robust choice of the number of iterations for these preconditioned methods.
Total Variation (TV) regularization is a powerful technique for image reconstruction tasks such as denoising, in-painting, and deblurring, because of its ability to produce sharp edges in the images. In this talk we discuss the use of TV regularization for tomographic imaging, where we compute a 2D or 3D reconstruction from noisy projections. We demonstrate that for a small signal-to-noise ratio, this new approach allows us to compute better (i.e., more reliable) reconstructions than those obtained by classical methods. This is possible due to the use of the TV reconstruction model, which incorporates our prior information about the solution and thus compensates for the loss of accuracy in the data. A consequence is that smaller data acquisition times can be used, thus reducing a patient's exposure to X-rays in medical scanning and speeding up non-destructive measurements in
materials science.
There are a number of applications where one wishes to know an acoustic field over an extended domain, whereas in most cases one can only perform point measurements (e.g. with a microphone). Even when few sources are active, it remains a challenging problem due to reverberation, that may be hard to characterize. This is typically a sampling problem, that raises a number of interesting questions: how many sampling points are needed, what are good distributions of sampling points, etc?
In this talk we will review a few test studies, in 2D (plates) and 3D (rooms), with numerical and experimental data, where a physicsbased sparse model for the acoustic wavefield is successfully used for its reconstruction, using significantly less measurements then would be required by classical Shannon sampling.