Using wavelet methods to solve noisy Abel-type equations with discontinuous inputs
dc.contributor.author | Hall, Peter | |
dc.contributor.author | Paige, Robert | |
dc.contributor.author | Ruymgaart, Frits | |
dc.date.accessioned | 2015-12-13T22:35:07Z | |
dc.date.available | 2015-12-13T22:35:07Z | |
dc.date.issued | 2003 | |
dc.date.updated | 2015-12-11T09:25:57Z | |
dc.description.abstract | One way of estimating a function from indirect, noisy measurements is to regularise an inverse of its Fourier transformation, using properties of the adjoint of the transform that degraded the function in the first place. It is known that when the function is smooth, this approach can perform well and produce estimators that have optimal convergence rates. When the function is unsmooth, in particular when it suffers jump discontinuities, an analogue of this approach is to invert the wavelet transform and use thresholding to decide whether wavelet terms should be included or excluded in the final approximation. We evaluate the performance of this approach by applying it to a large class of Abel-type transforms, and show that the smoothness of the target function and the smoothness of the transform interact in a particularly subtle way to determine the overall convergence rate. The most serious difficulties arise when the target function has a jump discontinuity at the origin; this has a considerably greater, and deleterious, impact on performance than a discontinuity elsewhere. In the absence of a discontinuity at the origin, the rate of convergence is determined principally by an inequality between the smoothness of the function and the smoothness of the transform. | |
dc.identifier.issn | 0047-259X | |
dc.identifier.uri | http://hdl.handle.net/1885/76445 | |
dc.publisher | Academic Press | |
dc.source | Journal of Multivariate Analysis | |
dc.subject | Keywords: Abel transform; Convergence rate; Fourier inversion; Ill-posed problem; Indirect curve estimation; Integral equation; Mean squared error; Piecewise continuous target; Wicksell's problem | |
dc.title | Using wavelet methods to solve noisy Abel-type equations with discontinuous inputs | |
dc.type | Journal article | |
local.bibliographicCitation.lastpage | 96 | |
local.bibliographicCitation.startpage | 72 | |
local.contributor.affiliation | Hall, Peter, College of Physical and Mathematical Sciences, ANU | |
local.contributor.affiliation | Paige, Robert, Texas Tech University | |
local.contributor.affiliation | Ruymgaart, Frits, College of Physical and Mathematical Sciences, ANU | |
local.contributor.authoremail | repository.admin@anu.edu.au | |
local.contributor.authoruid | Hall, Peter, u7801145 | |
local.contributor.authoruid | Ruymgaart, Frits, u971765 | |
local.description.notes | Imported from ARIES | |
local.description.refereed | Yes | |
local.identifier.absfor | 010405 - Statistical Theory | |
local.identifier.ariespublication | MigratedxPub5250 | |
local.identifier.citationvolume | 86 | |
local.identifier.doi | 10.1016/S0047-259X(02)00043-X | |
local.identifier.scopusID | 2-s2.0-0038454632 | |
local.identifier.uidSubmittedBy | Migrated | |
local.type.status | Published Version |