By Topic

On the approximation power of convolution-based least squares versus interpolation

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$31 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

2 Author(s)
Unser, M. ; Nat. Center for Res. Resources, Nat. Inst. of Health, Bethesda, MD, USA ; Daubechies, I.

There are many signal processing tasks for which convolution-based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine-based methods. The coefficients of the corresponding signal approximations are typically obtained by direct sampling (interpolation or quasi-interpolation) or by using least squares techniques that apply a prefilter prior to sampling. We compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h. Specifically, we review several results in approximation theory with a special emphasis on the Strang-Fix (1971) conditions, which relate the general O(hL ) behavior of the error to the ability of the representation to reproduce polynomials of degree n=L-1. We use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the L2-error as h tends to zero. We also propose a new improved L2-error bound for the least squares case. In the process, we provide all the relevant bound constants for polynomial splines. Some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order. We present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations

Published in:

Signal Processing, IEEE Transactions on  (Volume:45 ,  Issue: 7 )