## Fidelity optimization for aberration-tolerant hybrid imaging systems

Optics Express, Vol. 18, Issue 9, pp. 9220-9228 (2010)

http://dx.doi.org/10.1364/OE.18.009220

Acrobat PDF (1130 KB)

### Abstract

Several phase-modulation functions have been reported to decrease the aberration variance of the modulation-transfer-function (MTF) in aberration-tolerant hybrid imaging systems. The choice of this phase-modulation function is crucial for optimization of the overall system performance. To prevent a significant loss in signal-to-noise ratio, it is common to enforce restorability constraints on the MTF, requiring trade of aberration-tolerance and noise-gain. Instead of optimizing specific MTF characteristics, we directly minimize the expected imaging-error of the joint design. This method is used to compare commonly used phase-modulation functions: the antisymmetric generalized cubic polynomial and fourth-degree rotational symmetric phase-modulation. The analysis shows how optimal imaging performance is obtained using moderate phase-modulation, and more importantly, the relative merits of the above functions.

© 2010 OSA

## 1. Introduction

1. E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. **34**(11), 1859–1866 (1995). [CrossRef] [PubMed]

4. S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A **23**(5), 1058–1062 (2006). [CrossRef]

5. K. S. Kubala, H. B. Wach, V. V. Chumachenko, and E. R. Dowski, “Increasing the depth of field in an LWIR system for improved object identification,” Proc. SPIE **5784**, 146–156 (2005). [CrossRef]

*x*and

*y*are normalized pupil coordinates, and the parameters

*α*and

*β*characterize the phase-modulation. Phase-masks such as the separable logarithmic [6

6. S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. **43**(13), 2709–2721 (2004). [CrossRef] [PubMed]

7. Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. **272**(1), 56–66 (2007). [CrossRef]

8. N. Caron and Y. Sheng, “Polynomial phase masks for extending the depth of field of a microscope,” Appl. Opt. **47**(22), E39–E43 (2008). [CrossRef] [PubMed]

*β*≡0. Also non-separable profiles, such as those described by Eq. (1) with

*β*≈-3

*α*have been found to yield systems with good defocus tolerance [9

9. S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE **5108**, 1–12 (2003). [CrossRef]

10. S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE **5559**, 335–345 (2004). [CrossRef]

11. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. **26**(12), 875–877 (2001). [CrossRef] [PubMed]

3. S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. **28**(10), 771–773 (2003). [CrossRef] [PubMed]

4. S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A **23**(5), 1058–1062 (2006). [CrossRef]

12. M. D. Robinson, G. Feng, and D. Stork, “Spherical coded imagers: improving lens speed, depth-of-field, and manufacturing yield through enhanced spherical aberration and compensating image processing,” Proc. SPIE **7429**, 74290M (2009). [CrossRef]

*ρ*is the normalized radius, and the parameters

*γ*and

*δ*characterize the phase-modulation.

13. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. **13**(4), 600–612 (2004). [CrossRef] [PubMed]

## 2. Hybrid imaging fidelity evaluation

*I*, and the noiseless diffraction-limited image,

_{r}*I*: where

_{dl}*H*is the aberrated optical-transfer-function(OTF) including pupil phase-modulation,

_{ab}*H*is the Fourier transform of the image restoration filter (for example a Wiener filter), and

_{W}*H*is the diffraction-limited OTF. The Fourier-transforms of scene and noise are represented by

_{dl}*S*and

*N*respectively,

*H*is a function of

_{W}*H*and

_{ab}*P*[14], knowledge of the variance of the scene spectrum,

_{S}/P_{N}*P*, and noise spectrum,

_{S}*P*is sufficient to predict the imaging error of a given system. Often the spatial frequency spectrum of the noise is modeled as white and Gaussian, in which case the comparison of hybrid imaging systems can be based solely on

_{N}*H*and

_{ab}*P*. While the latter is scene-dependent, a representative measure of

_{S}/P_{N}*ε*can be obtained by incorporation of the typical amplitude spectrum of scenes: this is approximately proportional to

*1/|f|*, where

^{κ}*|f|*is the spatial frequency normalized to the optical cut-off, and the exponent

*κ*is typically between 0.9 and 1.20 [15

15. D. L. Ruderman and W. Bialek, “Statistics of natural images: Scaling in the woods,” Phys. Rev. Lett. **73**(6), 814–817 (1994). [CrossRef] [PubMed]

17. A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. **36**(17), 2759–2770 (1996). [CrossRef] [PubMed]

18. G. Muyo, A. Singh, M. Andersson, D. Huckridge, A. Wood, and A. R. Harvey, “Infrared imaging with a wavefront-coded singlet lens,” Opt. Express **17**(23), 21118–21123 (2009). [CrossRef] [PubMed]

19. M. Demenikov, E. Findlay, and A. R. Harvey, “Miniaturization of zoom lenses with a single moving element,” Opt. Express **17**(8), 6118–6127 (2009). [CrossRef] [PubMed]

*a priori*knowledge of the PSF, near-optimal image recovery is possible using blind-deconvolution techniques [20

20. J. Markham and J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A **16**(10), 2377–2391 (1999). [CrossRef]

21. Y. L. You and M. Kaveh, “Blind image restoration by anisotropic regularization,” IEEE Trans. Image Process. **8**(3), 396–407 (1999). [CrossRef] [PubMed]

22. M. Demenikov and A. R. Harvey, “A technique to remove image artefacts in optical systems with wavefront coding,” Proc. SPIE **7429**, 74290 (2009). [CrossRef]

## 3. Defocus tolerance with third-order phase-modulation

*α*, for a pure-cubic phase function. The top row shows the error magnitudes of simulated images obtained with a hybrid imaging system incorporating a sub-optimal phase-modulation,

*α*= ½

*α*

_{opt}, for defocuses of

*W*

_{20}= 0

*λ*,

*W*

_{20}= 3

*λ*, and

*W*

_{20}= 5

*λ*subsequent to restoration with a Wiener filter. Τhe optimal phase-modulation,

*α*

_{opt}= 2.87

*λ*, is calculated using Eq. (4) for a defocus tolerance, |

*W*

_{20}|

*≤*5

*λ*. Τhe error magnitudes are calculated with respect to the ideal, noiseless, diffraction-limited images. Error magnitudes for systems incorporating the optimal, and twice the optimal modulation depth are shown respectively on following two rows.

*α*, the suppression of the modulation-transfer function (MTF) for larger values of defocus introduces increasing error magnitudes in the recovered images (see for example the image for

*α*= ½

*α*

_{opt},

*W*

_{20}= 5

*λ*). For increasing values of

*α*, the errors become less sensitive to variations in

*W*

_{20}; but for

*α*≥

*α*

_{opt}, the errors averaged across the range of defocus is increased due to the increased suppression of the MTF by the pupil-plane phase-modulation. A large number of such simulations for representative scenes enables identification of the optimal

*α*for these particular scenes, however, it is possible to obtain a more general result by assuming a model for the scene spatial spectrum, as suggested in section 2.

*ε*with a signal-to-noise (SNR) model:

*P*= |10/

_{S}(f)/P_{N}*f*|

^{2}, that is

*κ =*1, and with the standard deviation of the noise equal to 1/280 of the dynamic range. These are typical values for a well-illuminated, high dynamic range detector operating in detector-noise-limited conditions. In Fig. 2a

*ε*is plotted as a function of

*W*

_{20}for several values of

*α*. It can be appreciated from this figure that larger values of

*α*yield larger values of

*ε*for small

*W*

_{20}(due to increased suppression of the MTF). However,

*ε*is then less sensitive to defocus; hence for larger

*W*

_{20,}a larger

*α*yields a smaller

*ε*(by avoiding the zeros in the out-of-focus MTF of a traditional optical system). In applications for which a finite range of defocus up to a certain maximum value

*W*

_{20,max}is encountered, then

*ε*over the range of

*W*

_{20}is a useful figure of merit. The solid-blue trace in Fig. 2b shows the variation of

*λ*≤

*W*

_{20}≤ 5

*λ*as a function of

*α*. For this range of

*W*

_{20}, the optimal value of

*α*is found to be 2.87

*λ*.

*(β*≡

*0*in Eq. (1)), the generalized cubic masks (

*β*≈−3

*α*), and the radial quartic profile given by Eq. (2). In Fig. 2b

*α*for the former two, and in terms of

*γ*for the latter. It can be seen that for these specific parameters minimum values of

*α*and

*β*independently. The contour plot in Fig. 3a shows the variation in imaging error,

*α*and

*β*for a system with a defocus tolerance

*|W*

_{20}

*| ≤*5

*λ*. A global optimum imaging fidelity is obtained near the axis

*β*≡ 0, and a secondary maximum can be noted on the axis

*β*≡ −3

*α*in Fig. 3a.

*α,β*) are plotted in Fig. 4a as a function of the maximum value of defocus to be mitigated,

*W*

_{20}

*. A quasi-linear increase of the global optimum of*

_{,max}*α*≈2

*W*

_{20,max}can be observed. This is in agreement with the analytically derived expression for the optimum value of

*α = (*1

*-f)*3

*W*

_{20}

*given in [23*

_{,max}23. G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. **30**(20), 2715–2717 (2005). [CrossRef] [PubMed]

*f*= 1/3. These calculations also show that

*β*is optimal at approximately 0.3

*λ*, largely independent of

*W*

_{20}

*, although its effect on the performance is essential negligible.*

_{,max}*α*and

*β*for the secondary optimum are shown in Fig. 4b and have an approximately linear dependency with

*W*

_{20,max}. Notice that the dependency between the two parameters closely follows

*β≡-*3

*α*, corresponding to a trefoil wave-front modulation. A very similar phase-modulation has been derived by minimization of the Fisher information in the PSF, while constraining reduction in the Strehl ratio [9

9. S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE **5108**, 1–12 (2003). [CrossRef]

10. S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE **5559**, 335–345 (2004). [CrossRef]

*W*

_{20,max}for both, approximately pure-cubic (

*β*≈0.3

*λ*) and generalized cubic (

*β*≈-3

*α*) phase functions. It can be seen that

*W*

_{20,max}in both cases and that the approximately cubic function provides a slightly lower

18. G. Muyo, A. Singh, M. Andersson, D. Huckridge, A. Wood, and A. R. Harvey, “Infrared imaging with a wavefront-coded singlet lens,” Opt. Express **17**(23), 21118–21123 (2009). [CrossRef] [PubMed]

## 4. Perception of hybrid imaging quality

13. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. **13**(4), 600–612 (2004). [CrossRef] [PubMed]

*W*

_{20}≤ 5λ. Photon-noise, Gaussian sensor-noise, and analog-to-digital conversion at the detector are included in the simulation. The noisy images are then restored using the corresponding Wiener filter, compared to the theoretical diffraction-limited image using the MSSIM metric, and plotted in Fig. 3b. The MSSIM contour plot and the plot in Fig. 3a, obtained using the scene-model are very similar with almost identical optimal values of

*α*and

*β*indicating that the use of Eq. (4) as a metric of image quality yields a valid optimization of perceived image quality.

*W*

_{20}, Fig. 1 and Fig. 5 show false-coloring of the image errors for three defocus positions and two scenes: a spoke and a representative scene. Whereas Fig. 1 depicts image errors for the pure-cubic phase-modulation (

*β*≡0), Fig. 5 depicts image errors for the generalized cubic with

*β =*−3

*α*, and quartic phase functions. Image errors are depicted for

*W*

_{20}= 0, 3

*λ*, and 5

*λ*and for the amplitudes of phase-modulation equal to half the optimal values, the exact optimal values and twice the optimal values, where the optimal amplitudes are

*α*

_{opt,gen}= 1.29

*λ*for the generalized cubic and

*γ*

_{opt}= 2.8

*λ*,

*δ*

_{opt}= −0.7 for the quartic mask.

*W*

_{20}, nevertheless the average values of the errors are larger. Note that only images for positive defocus are shown. The PSF of anti-symmetric phase-modulation varies as an even function of defocus; images for negative defocus are therefore identical to images for positive defocus. The same is not true for the quartic phase-modulation for which simulations were conducted for both positive and negative defocus. In this case however, the images shown are representative because the optimized

*δ*-parameter in Eq. (2) shifts the region of low imaging error to coincide optimally with the defocus tolerance range −5λ ≤

*W*

_{20}≤ 5λ.

## 5. Discussion

*W*

_{20}>

*λ*/2, leading to irretrievable contrast loss. The introduction of pupil-plane phase-modulation can significantly reduce the MTF variation, and thus increase the defocus-tolerance. We have shown however that the phase-modulation parameters should be chosen with care; noise amplification and regularization by the image restoration also reduce the expected image fidelity. The method described here allows identification of the phase-modulation parameters that maximize the expected imaging fidelity for a given defocus tolerance. The numerical phase-modulation parameters presented here can be used as initial values for optimization of specific and rigorously modeled hybrid designs.

24. G. Muyo and A. R. Harvey, “The effect of detector sampling in wavefront-coded imaging systems,” J. Opt. A, Pure Appl. Opt. **11**(5), 054002 (2009). [CrossRef]

*W*

_{20}and this will introduce image errors that increase with defocus mismatch. In the case of the quartic phase-modulation, these errors are similar to those introduced by regularization and are well described by the MSSIM metric, but for antisymmetric phase profiles, such as the cubic and generalized cubic functions, artifacts are introduced that have the form of image replications [22

22. M. Demenikov and A. R. Harvey, “A technique to remove image artefacts in optical systems with wavefront coding,” Proc. SPIE **7429**, 74290 (2009). [CrossRef]

## 6. Conclusion

*α*and

*β*of the generalized cubic mask. Both configurations yield a lower minimum expected imaging error than what can be achieved with the quartic phase-modulation. The imaging fidelity was predicted using a general statistical signal-model, and the results are shown to be compatible with structural similarity evaluations on a set of test-scenes simulated with realistic levels of noise.

## Acknowledgments

## References and links

1. | E. R. Dowski Jr and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. |

2. | S. Mezouari and A. R. Harvey, “Combined amplitude and phase filters for increased tolerance to spherical aberration,” J. Mod. Opt. |

3. | S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. |

4. | S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A |

5. | K. S. Kubala, H. B. Wach, V. V. Chumachenko, and E. R. Dowski, “Increasing the depth of field in an LWIR system for improved object identification,” Proc. SPIE |

6. | S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. |

7. | Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. |

8. | N. Caron and Y. Sheng, “Polynomial phase masks for extending the depth of field of a microscope,” Appl. Opt. |

9. | S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE |

10. | S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE |

11. | W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. |

12. | M. D. Robinson, G. Feng, and D. Stork, “Spherical coded imagers: improving lens speed, depth-of-field, and manufacturing yield through enhanced spherical aberration and compensating image processing,” Proc. SPIE |

13. | Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. |

14. | P. A. Jansson, |

15. | D. L. Ruderman and W. Bialek, “Statistics of natural images: Scaling in the woods,” Phys. Rev. Lett. |

16. | E. P. Simoncelli, “Statistical Modeling of Photographic Images,” in |

17. | A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. |

18. | G. Muyo, A. Singh, M. Andersson, D. Huckridge, A. Wood, and A. R. Harvey, “Infrared imaging with a wavefront-coded singlet lens,” Opt. Express |

19. | M. Demenikov, E. Findlay, and A. R. Harvey, “Miniaturization of zoom lenses with a single moving element,” Opt. Express |

20. | J. Markham and J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A |

21. | Y. L. You and M. Kaveh, “Blind image restoration by anisotropic regularization,” IEEE Trans. Image Process. |

22. | M. Demenikov and A. R. Harvey, “A technique to remove image artefacts in optical systems with wavefront coding,” Proc. SPIE |

23. | G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. |

24. | G. Muyo and A. R. Harvey, “The effect of detector sampling in wavefront-coded imaging systems,” J. Opt. A, Pure Appl. Opt. |

**OCIS Codes**

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

(100.3020) Image processing : Image reconstruction-restoration

(110.0110) Imaging systems : Imaging systems

(110.1758) Imaging systems : Computational imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: March 11, 2010

Revised Manuscript: April 9, 2010

Manuscript Accepted: April 9, 2010

Published: April 16, 2010

**Citation**

Tom Vettenburg, Nicholas Bustin, and Andrew R. Harvey, "Fidelity optimization for aberration-tolerant hybrid imaging systems," Opt. Express **18**, 9220-9228 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-9220

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### References

- E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859–1866 (1995). [CrossRef] [PubMed]
- S. Mezouari and A. R. Harvey, “Combined amplitude and phase filters for increased tolerance to spherical aberration,” J. Mod. Opt. 50(11), 2213–2220 (2003).
- S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28(10), 771–773 (2003). [CrossRef] [PubMed]
- S. Mezouari, G. Muyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A 23(5), 1058–1062 (2006). [CrossRef]
- K. S. Kubala, H. B. Wach, V. V. Chumachenko, and E. R. Dowski, “Increasing the depth of field in an LWIR system for improved object identification,” Proc. SPIE 5784, 146–156 (2005). [CrossRef]
- S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43(13), 2709–2721 (2004). [CrossRef] [PubMed]
- Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272(1), 56–66 (2007). [CrossRef]
- N. Caron and Y. Sheng, “Polynomial phase masks for extending the depth of field of a microscope,” Appl. Opt. 47(22), E39–E43 (2008). [CrossRef] [PubMed]
- S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003). [CrossRef]
- S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization for extended-focus, aberration-corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004). [CrossRef]
- W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26(12), 875–877 (2001). [CrossRef] [PubMed]
- M. D. Robinson, G. Feng, and D. Stork, “Spherical coded imagers: improving lens speed, depth-of-field, and manufacturing yield through enhanced spherical aberration and compensating image processing,” Proc. SPIE 7429, 74290M (2009). [CrossRef]
- Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004). [CrossRef] [PubMed]
- P. A. Jansson, Deconvolution of images and spectra (2nd ed.) (Ac. Press, Inc., Orlando, FL, USA, 1996).
- D. L. Ruderman and W. Bialek, “Statistics of natural images: Scaling in the woods,” Phys. Rev. Lett. 73(6), 814–817 (1994). [CrossRef] [PubMed]
- E. P. Simoncelli, “Statistical Modeling of Photographic Images,” in Handbook of Image and Video Processing(Ac. Press, 2005), pp. 431–441.
- A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36(17), 2759–2770 (1996). [CrossRef] [PubMed]
- G. Muyo, A. Singh, M. Andersson, D. Huckridge, A. Wood, and A. R. Harvey, “Infrared imaging with a wavefront-coded singlet lens,” Opt. Express 17(23), 21118–21123 (2009). [CrossRef] [PubMed]
- M. Demenikov, E. Findlay, and A. R. Harvey, “Miniaturization of zoom lenses with a single moving element,” Opt. Express 17(8), 6118–6127 (2009). [CrossRef] [PubMed]
- J. Markham and J. A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A 16(10), 2377–2391 (1999). [CrossRef]
- Y. L. You and M. Kaveh, “Blind image restoration by anisotropic regularization,” IEEE Trans. Image Process. 8(3), 396–407 (1999). [CrossRef] [PubMed]
- M. Demenikov and A. R. Harvey, “A technique to remove image artefacts in optical systems with wavefront coding,” Proc. SPIE 7429, 74290 (2009). [CrossRef]
- G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. 30(20), 2715–2717 (2005). [CrossRef] [PubMed]
- G. Muyo and A. R. Harvey, “The effect of detector sampling in wavefront-coded imaging systems,” J. Opt. A, Pure Appl. Opt. 11(5), 054002 (2009). [CrossRef]

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