EPFL | Biomedical Imaging Group | Approximation Techniques

Quantitative Fourier Analysis of Approximation Techniques: Part I—Interpolators and Projectors

T. Blu, M. Unser

IEEE Transactions on Signal Processing, vol. 47, no. 10, pp. 2783–2795, October 1999.

We present a general Fourier-based method that provides an accurate prediction of the approximation error as a function of the sampling step T. Our formalism applies to an extended class of convolution-based signal approximation techniques, which includes interpolation, generalized sampling with prefiltering, and the projectors encountered in wavelet theory. We claim that we can predict the L²-approximation error, by integrating the spectrum of the function to approximate—not necessarily bandlimited—against a frequency kernel E(ω) that characterizes the approximation operator. This prediction is easier, yet more precise than was previously available. Our approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function. Our error prediction is exact for stationary processes, as well as for bandlimited signals. We apply this method to the comparison of standard interpolation and approximation techniques.

Our method has interesting implications for approximation theory. In particular, we use our results to obtain some new asymptotic expansions of the error as T tends to 0, and also to derive improved upper bounds of the kind found in the Strang-Fix theory. We finally show how we can design quasi-interpolators that are near-optimal in the least-squares sense.

Please consult also the companion paper by T. Blu, M. Unser, "Quantitative Fourier Analysis of Approximation Techniques: Part II—Wavelets," IEEE Transactions on Signal Processing, vol. 47, no. 10, pp. 2796-2806, October 1999.

@ARTICLE(http://bigwww.epfl.ch/publications/blu9901.html,
AUTHOR="Blu, T. and Unser, M.",
TITLE="Quantitative {F}ourier Analysis of Approximation Techniques:
	{P}art {I}---{I}nterpolators and Projectors",
JOURNAL="{IEEE} Transactions on Signal Processing",
YEAR="1999",
volume="47",
number="10",
pages="2783--2795",
month="October",
note="")

© 1999 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.