A derandomization approach to recovering bandlimited signals across a wide range of random sampling rates
Reconstructing bandlimited functions from random sampling is an important problem in signal processing. Strohmer and Vershynin obtained good results for this problem by using a randomized version of the Kaczmarz algorithm (RK) and assigning to every equation a probability weight proportional to the average distance of the sample from its two nearest neighbors. However, their results are valid only for moderate to high sampling rates; in practice, it may not always be possible to obtain many samples. Experiments show that the number of projections required by RK and other Kaczmarz variants rises seemingly exponentially when the equations/variables ratio (EVR) falls below 5. CGMN, which is a CG acceleration of Kaczmarz, provides very good results for low values of EVR and it is much better than CGNR and CGNE. A derandomization method, based on an extension of the bit-reversal permutation, is combined with the weights and shown to improve the performance of CGMN and the regular (cyclic) Kaczmarz, which even outperforms RK. A byproduct of our results is the finding that signals composed mainly of high-frequency components are easier to recover.
KeywordsBandlimited functions Bit-reversal CGMN Derandomization Extended bit-reversal Low sampling rates Randomized Kaczmarz RK Signal processing
Unable to display preview. Download preview PDF.
The author would like to thank the anonymous reviewers for their helpful comments. Section 4 was added in response to the issues raised by one of the reviewers.
- 2.Blum, M., Floyd, R.W., Pratt, V., Rivest, R.L., Tarjan, R.E.: Linear time bounds for median computations Proc.4th Annual ACM Symp.on Theory of Computing, STOC ’72, pp. 119–124. ACM, New York (1972)Google Scholar
- 4.Cenker, C., Feichtinger, H., Mayer, M., Steier, H., Strohmer, T.: New variants of the POCS method using affine subspaces of finite codimension, with applications to irregular sampling. In: Maragos, P. (ed.) Visual Communications and Image Processing ’92, pp. 299–310. SPIE (1992)Google Scholar
- 5.Censor, Y., Elfving, T., Herman, G.T.: Averaging strings of sequential iterations for convex feasibility problems. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, volume 8 of Studies in Computational Mathematics, pp. 101–113. Elsevier, Amsterdam (2001)Google Scholar
- 11.Feichtinger, H., Gröchenig, K.: Theory and practice of irregular sampling. In: Frazier, M. (ed.) Wavelets: Mathematics and Applications, pp. 305–363. CRC Press, Boca Raton (1994)Google Scholar
- 17.Gordon, D., Gordon, R., Turkel, E.: Compact high order schemes with gradient-direction derivatives for absorbing boundary conditions. J. Comput. Phys. 297(9), 295–315 (Sept. 2015)Google Scholar
- 18.Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction From Projections, 2nd edn. Springer (2009)Google Scholar
- 22.Liu, J., Wright, S.J.: An accelerated randomized Kaczmarz algorithm. Math. Comput. 85(297), 153–178 (Jan. 2016)Google Scholar
- 23.Margolis, E., Eldar, Y.C.: Nonuniform sampling of periodic bandlimited signals. IEEE Trans. Signal Process. 56(7), 2728–2745 (Jul. 2008)Google Scholar
- 24.Mayer, M.: POCS-Methoden. PhD thesis, University of Vienna, Austria. http://univie.ac.at/nuhag-php/bibtex/open_files/ma00_mayerPOCS.pdf (2000)