Practical band-limited extrapolation relying on Slepian series and compressive sampling

Laurent Gosse (1)

1 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"

We consider a rather simple algorithm to address the fascinating field of numerical extrapolation of (analytic) band-limited functions. It relies on two main elements: namely, the lower frequencies are treated by projecting the known part of the signal to be extended onto the space generated by ``Prolate Spheroidal Wave Functions" (PSWF, as originally proposed by Slepian), whereas the higher ones can be handled by the recent so--called ``Compressive Sampling" (CS, proposed by Candès) algorithms which are independent of the largeness of the bandwidth. Slepian functions are recalled and their numerical computation is explained in full detail whereas Compressive Sampling techniques are summarized together with a recent iterative algorithm which has been proved to work efficiently on so--called ``compressible signals" which appear to match rather well the class of smooth bandlimited functions. Numerical results are displayed for both numerical techniques and the accuracy of the process consisting in putting them altogether is studied for several test-signals.

Mots clés: Band-limited extrapolation; Prolate spheroidal wave functions; Slepian series; analytic continuation; finite Fourier transform; compressive sampling; sparse and compressible signals recovery

Domaines: Analyse numérique [math.NA]; Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Laurent Gosse. Practical band-limited extrapolation relying on Slepian series and compressive sampling. Computers & Mathematics with Applications, 2010, 60, pp.1259-1279. ⟨10.1016/j.camwa.2010.06.006⟩. ⟨hal-00414212⟩