Regularization matrices determined by matrix nearness problems

This paper is concerned with the solution of large-scale linear discrete ill-posed problems with error-contaminated data. Tikhonov regularization is a popular approach to determine meaningful approximate solutions of such problems. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. This matrix should be chosen to promote the recovery of known important features of the desired solution, such as smoothness and monotonicity. We describe a novel approach to determine regularization matrices with desired properties by solving a matrix nearness problem. The constructed regularization matrix is the closest matrix in the Frobenius norm with a prescribed null space to a given matrix. Numerical examples illustrate the performance of the regularization matrices so obtained

ON THE CORRECTION OF DOSE PROFILE DISCREPANCIES BY INTRODUCING AIR IN THE DERIVATION OF AN ELECTRON SPECTRUM

Knowledge of the energy spectrum of an electron beam is relevant for accurate dose calculation in radiotherapy. In previous works, it has been possible to reconstruct the electron spectrum of various clinical energies (6, 9, 12 and 15 MeV) within the typical percentage of clinical acceptance (Pa > 95 %) according to the gamma index (GI) (2 %/2mm), for both depth dose percentages (PDD) and dose profiles (DP), except for 6 MeV profiles. Therefore, the purpose of this work was to introduce air between the radiation source and the phantom surface to simulate both the monoenergetic PDDs necessary in the reconstruction of the spectrum of a 6 MeV beam and to obtain the PDD of this spectrum. Validation was performed using the gamma index with the typical threshold for clinical acceptance. The results showed that the PDP of the vacuum spectrum had a better agreement than the PDP of the air spectrum (Pa = 100 %), with respect to the measured PDD (Pa = 97 %). Regarding the PD, the introduction of air improved the agreement in clinical interest but not enough to reach the acceptance percentage. It is concluded that this technique does not seem to be a good alternative to correct the discrepancies in the field edges between the DP of an inversely reconstructed spectrum and the measured DP.Knowledge of the energy spectrum of an electron beam is relevant for accurate dose calculation in radiotherapy. In previous works, it has been possible to reconstruct the electron spectrum of various clinical energies (6, 9, 12 and 15 MeV) within the typical percentage of clinical acceptance (Pa > 95 %) according to the gamma index (GI) (2 %/2mm), for both depth dose percentages (PDD) and dose profiles (DP), except for 6 MeV profiles. Therefore, the purpose of this work was to introduce air between the radiation source and the phantom surface to simulate both the monoenergetic PDDs necessary in the reconstruction of the spectrum of a 6 MeV beam and to obtain the PDD of this spectrum. Validation was performed using the gamma index with the typical threshold for clinical acceptance. The results showed that the PDP of the vacuum spectrum had a better agreement than the PDP of the air spectrum (Pa = 100 %), with respect to the measured PDD (Pa = 97 %). Regarding the PD, the introduction of air improved the agreement in clinical interest but not enough to reach the acceptance percentage. It is concluded that this technique does not seem to be a good alternative to correct the discrepancies in the field edges between the DP of an inversely reconstructed spectrum and the measured DP

Some matrix nearness problems suggested by Tikhonov regularization

The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for problems of small to moderate size are Tikhonov regularization and truncated singular value decomposition (TSVD). By considering matrix nearness problems related to Tikhonov regularization, several novel regularization methods are derived. These methods share properties with both Tikhonov regularization and TSVD, and can give approximate solutions of higher quality than either one of these methods

Fractional regularization matrices for linear discrete ill-posed problems

The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore the use of fractional powers of the matrices {Mathematical expression} (for Tikhonov regularization) and A (for Lavrentiev regularization) as regularization matrices, where A is the matrix that defines the linear discrete ill-posed problem. Both small- and large-scale problems are considered. © 2013 Springer Science+Business Media Dordrecht

"Plug-and-Play" Edge-Preserving Regularization

In many inverse problems it is essential to use regularization methods that preserve edges in the reconstructions, and many reconstruction models have been developed for this task, such as the Total Variation (TV) approach. The associated algorithms are complex and require a good knowledge of large-scale optimization algorithms, and they involve certain tolerances that the user must choose. We present a simpler approach that relies only on standard computational building blocks in matrix computations, such as orthogonal transformations, preconditioned iterative solvers, Kronecker products, and the discrete cosine transform -- hence the term "plug-and-play." We do not attempt to improve on TV reconstructions, but rather provide an easy-to-use approach to computing reconstructions with similar properties.Comment: 14 pages, 7 figures, 3 table

Inversion of spinning sound fields

A method is presented for the reconstruction of rotating monopole source distributions using acoustic pressures measured on a sideline parallel to the source axis. The method requires no \textit{a priori} assumptions about the source other than that its strength at the frequency of interest vary sinusoidally in azimuth on the source disc so that the radiated acoustic field is composed of a single circumferential mode. When multiple azimuthal modes are present, the acoustic field can be decomposed into azimuthal modes and the method applied to each mode in sequence. The method proceeds in two stages, first finding an intermediate line source derived from the source distribution and then inverting this line source to find the radial variation of source strength. A far-field form of the radiation integrals is derived, showing that the far field pressure is a band-limited Fourier transform of the line source, establishing a limit on the quality of source reconstruction which can be achieved using far-field measurements. The method is applied to simulated data representing wind-tunnel testing of a ducted rotor system (tip Mach number~0.74) and to control of noise from an automotive cooling fan (tip Mach number~0.14), studies which have appeared in the literature of source identification.Comment: Revised version of paper submitted to JASA; five more figures; expanded content with more discussion of error behaviour and relation to Nearfield Acoustical Holograph

A projection method for general form linear least-squares problems

One of the possible approaches for the solution of underdetermined linear least-squares problems in general form, for a chosen regularization operator L, projects the problem in the null space of L and in its orthogonal complement. In this paper, we show that the projected problem cannot be solved by the generalized singular value decomposition, and propose some approaches to overcome this issue. Numerical experiments ascertain the stability of the new procedures