## Beyond Nyquist sampling: a cost-based approach |

JOSA A, Vol. 30, Issue 4, pp. 645-655 (2013)

http://dx.doi.org/10.1364/JOSAA.30.000645

Enhanced HTML Acrobat PDF (663 KB)

### Abstract

A sampling-based framework for finding the optimal representation of a finite energy optical field using a finite number of bits is presented. For a given bit budget, we determine the optimum number and spacing of the samples in order to represent the field with as low error as possible. We present the associated performance bounds as trade-off curves between the error and the cost budget. In contrast to common practice, which often treats sampling and quantization separately, we explicitly focus on the interplay between limited spatial resolution and limited amplitude accuracy, such as whether it is better to take more samples with lower amplitude accuracy or fewer samples with higher accuracy. We illustrate that in certain cases sampling at rates different from the Nyquist rate is more efficient.

© 2013 Optical Society of America

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(100.3020) Image processing : Image reconstruction-restoration

(350.5730) Other areas of optics : Resolution

(070.2025) Fourier optics and signal processing : Discrete optical signal processing

(110.3055) Imaging systems : Information theoretical analysis

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: August 1, 2012

Manuscript Accepted: January 28, 2013

Published: March 21, 2013

**Citation**

Ayça Özçelikkale and Haldun M. Ozaktas, "Beyond Nyquist sampling: a cost-based approach," J. Opt. Soc. Am. A **30**, 645-655 (2013)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-4-645

Sort: Year | Journal | Reset

### References

- G. T. D. Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955). [CrossRef]
- D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1961), Vol. I, pp. 109–153.
- W. Lukozs, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966). [CrossRef]
- W. Lukozs, “Optical systems with resolving powers exceeding the classical limit II,” J. Opt. Soc. Am. 57, 932–941 (1967). [CrossRef]
- G. T. Di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969). [CrossRef]
- F. Gori and G. Guattari, “Effects of coherence on the degrees of freedom of an image,” J. Opt. Soc. Am. 61, 36–39 (1971). [CrossRef]
- F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973). [CrossRef]
- A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982). [CrossRef]
- G. Newsam and R. Barakat, “Essential dimension as a well-defined number of degrees of freedom of finite-convolution operators appearing in optics,” J. Opt. Soc. Am. A 2, 2040–2045 (1985). [CrossRef]
- O. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989). [CrossRef]
- F. Gori, Advanced Topics in Shannon Sampling and Interpolation Theory (Springer-Verlag, 1993), pp. 37–83.
- A. Lohmann, R. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996). [CrossRef]
- D. Mendlovic and A. W. Lohmann, “Space-bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14, 558–562 (1997). [CrossRef]
- R. Pierri and F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998). [CrossRef]
- R. Piestun and D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A 17, 892–902 (2000). [CrossRef]
- D. A. B. Miller, “Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt. 39, 1681–1699 (2000). [CrossRef]
- J. Xu and R. Janaswamy, “Electromagnetic degrees of freedom in 2-d scattering environments,” IEEE Trans. Antennas Propag. 54, 3882–3894 (2006). [CrossRef]
- F. S. Oktem and H. M. Ozaktas, “Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space–bandwidth product,” J. Opt. Soc. Am. A 27, 1885–1895 (2010). [CrossRef]
- F. S. Roux, “Complex-valued Fresnel-transform sampling,” Appl. Opt. 34, 3128–3135 (1995). [CrossRef]
- A. Stern, and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004). [CrossRef]
- L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359–367 (2007). [CrossRef]
- J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008). [CrossRef]
- D. MacKay, “Quantal aspects of scientific information,” Trans. IRE Prof. Group Inf. Theory 1, 60–80 (1953). [CrossRef]
- J. T. Winthrop, “Propagation of structural information in optical wave fields,” J. Opt. Soc. Am. 61, 15–30 (1971). [CrossRef]
- T. W. Barret, “Structural information theory,” J. Acoust. Soc. Am. 54, 1092–1098 (1973). [CrossRef]
- H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006). [CrossRef]
- A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008). [CrossRef]
- F. Oktem and H. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009). [CrossRef]
- J. J. Healy and J. T. Sheridan, “Space–bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790 (2011). [CrossRef]
- F. T. Yu, Optics and Information Theory (Wiley, 1976).
- M. J. Bastiaans, “Uncertainty principle and informational entropy for partially coherent light,” J. Opt. Soc. Am. A 3, 1243–1246 (1986). [CrossRef]
- M. S. Hughes, “Analysis of digitized wave-forms using Shannon entropy,” J. Acoust. Soc. Am. 93, 892–906 (1993). [CrossRef]
- D. Blacknell and C. J. Oliver, “Information-content of coherent images,” J. Phys. D 26, 1364–1370 (1993). [CrossRef]
- R. Barakat, “Some entropic aspects of optical diffraction imagery,” Opt. Commun. 156, 235–239 (1998). [CrossRef]
- M. A. Neifeld, “Information, resolution, and space-bandwidth product,” Opt. Lett. 23, 1477–1479 (1998). [CrossRef]
- F. T. Yu, Entropy and Information Optics (Marcel Dekker, 2000).
- H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19, 1563–1571 (2002). [CrossRef]
- A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003). [CrossRef]
- A. Stern and B. Javidi, “Shannon number and information capacity of three-dimensional integral imaging,” J. Opt. Soc. Am. A 21, 1602–1612 (2004). [CrossRef]
- M. A. Porras and R. Medina, “Entropy-based definition of laser beam spot size,” Appl. Opt. 34, 8247–8251 (1995). [CrossRef]
- P. Réfrégier and J. Morio, “Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations,” J. Opt. Soc. Am. A 23, 3036–3044 (2006). [CrossRef]
- M. Migliore, “On electromagnetics and information theory,” IEEE Trans. Antennas Propag. 56, 3188–3200 (2008). [CrossRef]
- E. D. Micheli and G. A. Viano, “Inverse optical imaging viewed as a backward channel communication problem,” J. Opt. Soc. Am. A 26, 1393–1402 (2009). [CrossRef]
- A. Özçelikkale, H. M. Ozaktas, and E. Arıkan, “Signal recovery with cost constrained measurements,” IEEE Trans. Signal Process. 58, 3607–3617 (2010). [CrossRef]
- R. Konsbruck, E. Telatar, and M. Vetterli, “On sampling and coding for distributed acoustic sensing,” IEEE Trans. Inf. Theory 58, 3198–3214 (2012). [CrossRef]
- A. Özçelikkale and H. M. Ozaktas, “Representation of optical fields using finite numbers of bits,” Opt. Lett. 37, 2193–2195 (2012). [CrossRef]
- B. Dulek and S. Gezici, “Average Fisher information maximisation in presence of cost-constrained measurements,” Electron. Lett. 47, 654–656 (2011). [CrossRef]
- B. Dulek and S. Gezici, “Cost minimization of measurement devices under estimation accuracy constraints in the presence of Gaussian noise,” Digit. Signal Process. 22, 828–840 (2012). [CrossRef]
- H. L. Van Trees, Detection, Estimation and Modulation Theory, Part I (Wiley, 2001).
- E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980). [CrossRef]
- F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980). [CrossRef]
- A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian–Schell model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982). [CrossRef]
- A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982). [CrossRef]
- A. T. Friberg and J. Turunen, “Imaging of Gaussian–Schell model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988). [CrossRef]
- P. Jixiong, “Waist location and Rayleigh range for Gaussian Schell-model beams,” J. Opt. 22, 157–159 (1991). [CrossRef]
- G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
- Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian–Schell model beams,” Opt. Lett. 27, 1672–1674 (2002). [CrossRef]
- S. Zhu, Y. Cai, and O. Korotkova, “Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam,” Opt. Express 18, 12587–12598 (2010). [CrossRef]
- M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979). [CrossRef]
- H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
- K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).
- A. Balakrishnan, “A note on the sampling principle for continuous signals,” IEEE Trans. Inf. Theory 3, 143–146 (1957). [CrossRef]
- W. A. Gardner, “A sampling theorem for nonstationary random processes,” IEEE Trans. Inf. Theory 18, 808–809 (1972). [CrossRef]
- F. Garcia, I. Lourtie, and J. Buescu, “L2(R) nonstationary processes and the sampling theorem,” IEEE Signal Process. Lett. 8, 117–119 (2001). [CrossRef]
- W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, scalar sources,” Opt. Acta 28, 245–259 (1981). [CrossRef]
- R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1990).
- T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
- J. Buescu, “Positive integral operators in unbounded domains,” J. Math. Anal. Appl. 296, 244–255 (2004). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.