## Multi-transmitter aperture synthesis with Zernike based aberration correction |

Optics Express, Vol. 20, Issue 24, pp. 26448-26457 (2012)

http://dx.doi.org/10.1364/OE.20.026448

Acrobat PDF (1215 KB)

### Abstract

Multi-transmitter aperture synthesis increases the effective aperture in coherent imaging by shifting the backscattered speckle field across a physical aperture or set of apertures. Through proper arrangement of the transmitter locations, it is possible to obtain speckle fields with overlapping regions, which allows fast computation of optical aberrations from wavefront differences. In this paper, we present a method where Zernike polynomials are used to model the aberrations and high-order aberrations are estimated without the need to do phase unwrapping of the difference fronts.

© 2012 OSA

## 1. Introduction

1. N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. **46**(23), 5933–5943 (2007). [CrossRef] [PubMed]

2. D. Rabb, D. Jameson, A. Stokes, and J. Stafford, “Distributed aperture synthesis,” Opt. Express **18**(10), 10334–10342 (2010). [CrossRef] [PubMed]

3. B. K. Gunturk, N. J. Miller, and E. A. Watson, “Camera phasing in multi-aperture cohering imaging,” Opt. Express **20**(11), 11796–11805 (2012). [CrossRef] [PubMed]

4. D. J. Rabb, D. F. Jameson, J. W. Stafford, and A. J. Stokes, “Multi-transmitter aperture synthesis,” Opt. Express **18**(24), 24937–24945 (2010). [CrossRef] [PubMed]

5. R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. **64**(9), 1200–1210 (1974). [CrossRef]

8. S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A **25**(4), 983–994 (2008). [CrossRef]

9. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**(3), 207–211 (1976). [CrossRef]

4. D. J. Rabb, D. F. Jameson, J. W. Stafford, and A. J. Stokes, “Multi-transmitter aperture synthesis,” Opt. Express **18**(24), 24937–24945 (2010). [CrossRef] [PubMed]

10. D. Rabb, J. W. Stafford, and D. F. Jameson, “Non-iterative aberration correction of a multiple transmitter system,” Opt. Express **19**(25), 25048–25056 (2011). [CrossRef]

10. D. Rabb, J. W. Stafford, and D. F. Jameson, “Non-iterative aberration correction of a multiple transmitter system,” Opt. Express **19**(25), 25048–25056 (2011). [CrossRef]

11. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. **14**(1), 142–150 (1975). [PubMed]

12. G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. **35**(31), 6162–6172 (1996). [CrossRef] [PubMed]

13. S. Okuda, T. Nomura, K. Kamiya, and H. Miyashiro, “High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials,” Appl. Opt. **39**(28), 5179–5186 (2000). [CrossRef]

14. F. Dai, F. Tang, X. Wang, P. Feng, and O. Sasaki, “Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms,” Opt. Express **20**(2), 1530–1544 (2012). [CrossRef] [PubMed]

10. D. Rabb, J. W. Stafford, and D. F. Jameson, “Non-iterative aberration correction of a multiple transmitter system,” Opt. Express **19**(25), 25048–25056 (2011). [CrossRef]

## 2. Proposed method

4. D. J. Rabb, D. F. Jameson, J. W. Stafford, and A. J. Stokes, “Multi-transmitter aperture synthesis,” Opt. Express **18**(24), 24937–24945 (2010). [CrossRef] [PubMed]

**19**(25), 25048–25056 (2011). [CrossRef]

*U*(

_{b}*x*,

*y*) is shifted by the same amount (in the reverse direction). On the other hand, the phase error is static because the sensor is fixed. Suppose that we measure two pupil-plane fields,

*U*

_{1}(

*x*,

*y*) and

*U*

_{2}(

*x*,

*y*), each with a different transmitter locations: where

*P*(

*x*,

*y*) is the pupil function,

*W*(

_{e}*x*,

*y*) is the wavefront error, and (

*x*

_{1},

*y*

_{1}) and (

*x*

_{2},

*y*

_{2}) are the shift amounts in the backscattered fields due to transmitter locations. Let

*W*(

_{b}*x*,

*y*) be the wavefront of

*U*(

_{b}*x*,

*y*), then the detected wavefronts are The object wavefront is not well defined and may contain several branch points, so it is desirable to calculate the aberration wavefront independent of the target wavefront. It is possible to numerically remove the dependence on the backscattered wavefront by first registering the wavefronts

*W*(

_{i}*x*,

*y*) according to the shift amounts. The registration can be achieved based on calibrated measurement of the shifts due to transmitter locations, as well as by registering the pupil plane speckle intensity returning from the target [10

**19**(25), 25048–25056 (2011). [CrossRef]

*W*(

*x*,

*y*) may have phase wraps although not explicitly written in (3), and we will explain how to handle phase wrapping issue shortly.

*W*(

_{e}*x*,

*y*) = ∑

_{k}*a*(

_{k}Z_{k}*x*,

*y*), where

*Z*(

_{k}*x*,

*y*) is a Zernike polynomial and

*a*is the unknown coefficient corresponding to

_{k}*Z*(

_{k}*x*,

*y*). Then, the wavefront difference becomes where we defined Δ

*Z*(

_{k}*x*,

*y*) ≡

*Z*(

_{k}*x*+

*x*

_{1},

*y*+

*y*

_{1}) −

*Z*(

_{k}*x*+

*x*

_{2},

*y*+

*y*

_{2}).

*a*by forming a linear set of equations for all positions (

_{k}*x*,

*y*). There are, however, two important issues. First, the difference front Δ

*Z*(

_{k}*x*,

*y*) does not form an orthogonal basis, resulting in a set of equations that are not linearly independent. The problem could be alleviated by using multiple measurements corresponding to transmitter separations along different directions giving additional overlapping regions, which could result in an over- or well-determined system necessary for the proposed approach to successfully calculate the desired aberration coefficients. Second, the difference front Δ

*W*(

*x*,

*y*) needs to be phase unwrapped. Looking at Fig.2(a), we see that there could be phase jumps of 2

*π*from one region to another. That is, the equation (4) is not valid for all pixels in the overlap region. On the other hand, the overlap region could be divided into sub-regions, where there is no phase wrap as illustrated in Fig.2(b); and within each sub-region, we could write (4) with a constant but unknown phase offset. In sub-region

*m*, the wavefront difference at the

*i*th position in that sub-region is where

*α*

^{(m)}is the unknown phase offset,

*a*is the coefficient of the

_{k}*k*th Zernike polynomial. Let

*N*be the number of pixels in sub-region

_{m}*m*and

*M*be the number of sub-regions, then we form the following linear set of equations:

*α*

^{(1)}, ...,

*α*

^{(M)},

*a*

_{1}, ...,

*a*can be estimated in a number of ways; in this paper they are estimated using the QR decomposition based pseudo-inverse operation (the “backslash” operator in MATLAB). The Zernike coefficients

_{K}*a*

_{1}, ...,

*a*are then used to correct for the aberrations in each aperture.

_{K}## 3. Experimental results

*L*, which is flood illuminated by a coherent laser source of wavelength

*λ*. The complex-valued field reflected off the object is modeled with amplitude equal to the square root of the object’s intensity reflectance and phase a uniformly distributed random variable over −

*π*to

*π*. The complex-valued field in the receiver plane, subject to the paraxial approximation, is given by the Fresnel diffraction integral [15]. The Fresnel diffraction integral is numerically evaluated using the angular spectrum propagation method [16]. The object plane and receiver pupil planes in the simulation consisted of

*N*= 2048 × 2048 computational grids with identical 182

*μ*m sample spacings in both planes. The optical wavelength,

*λ*, is 1.55

*μ*m, and the range,

*L*, from the receive pupil plane to the object is 100 meters. The numerical propagation consists of 10 partial propagations of 10 meters each to avoid the wraparound effects. The optical field in the receiver pupil plane is collected using a 48mm diameter aperture with three transmitters in the configuration shown in Fig.3(b). The focused optical fields are then aberrated using randomly weighted Zernike polynomials.

*Canny*edge detector [17]. The edge lines are morphologically dilated (with a 3 × 3 kernel) to have a wider coverage of discontinuities. Region segments outside the edges form the sub-regions. The extracted sub-regions are shown in Fig. 4(d) to 4(f). As seen in these sample results, the segmentation procedure may result in over-segmentation, however, this is not an issue. The opposite (under-segmentation), on the other hand, would be an issue as multiple regions with different phase offsets would be forced to have the same phase offset, which would degrade the estimation of the Zernike coefficients. With over-segmentation, the only concern is the introduction of additional phase offset parameters to be estimated, and therefore an increase in the computational cost. One possible approach to reduce computational cost is to discard small sub-regions, which should not affect the performance as the majority of pixels are included. In our experiments we discard sub-regions that are less than 20 pixels, and place the remaining sub-regions into the matrix equation (6).

## 4. Conclusions

## References and links

1. | N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. |

2. | D. Rabb, D. Jameson, A. Stokes, and J. Stafford, “Distributed aperture synthesis,” Opt. Express |

3. | B. K. Gunturk, N. J. Miller, and E. A. Watson, “Camera phasing in multi-aperture cohering imaging,” Opt. Express |

4. | D. J. Rabb, D. F. Jameson, J. W. Stafford, and A. J. Stokes, “Multi-transmitter aperture synthesis,” Opt. Express |

5. | R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. |

6. | R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image sharpness criterion,” in Statistical Optics , Proc. SPIE |

7. | J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A |

8. | S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A |

9. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. |

10. | D. Rabb, J. W. Stafford, and D. F. Jameson, “Non-iterative aberration correction of a multiple transmitter system,” Opt. Express |

11. | M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. |

12. | G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt. |

13. | S. Okuda, T. Nomura, K. Kamiya, and H. Miyashiro, “High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials,” Appl. Opt. |

14. | F. Dai, F. Tang, X. Wang, P. Feng, and O. Sasaki, “Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms,” Opt. Express |

15. | J. W. Goodman, |

16. | J. D. Schmidt, |

17. | R. C. Gonzalez and R. E. Woods, |

18. | J. W. Goodman, |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(090.1995) Holography : Digital holography

**ToC Category:**

Image Processing

**History**

Original Manuscript: August 6, 2012

Revised Manuscript: October 21, 2012

Manuscript Accepted: November 2, 2012

Published: November 8, 2012

**Citation**

Bahadir K. Gunturk, David J. Rabb, and Douglas F. Jameson, "Multi-transmitter aperture synthesis with Zernike based aberration correction," Opt. Express **20**, 26448-26457 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26448

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### References

- N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt.46(23), 5933–5943 (2007). [CrossRef] [PubMed]
- D. Rabb, D. Jameson, A. Stokes, and J. Stafford, “Distributed aperture synthesis,” Opt. Express18(10), 10334–10342 (2010). [CrossRef] [PubMed]
- B. K. Gunturk, N. J. Miller, and E. A. Watson, “Camera phasing in multi-aperture cohering imaging,” Opt. Express20(11), 11796–11805 (2012). [CrossRef] [PubMed]
- D. J. Rabb, D. F. Jameson, J. W. Stafford, and A. J. Stokes, “Multi-transmitter aperture synthesis,” Opt. Express18(24), 24937–24945 (2010). [CrossRef] [PubMed]
- R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am.64(9), 1200–1210 (1974). [CrossRef]
- R. G. Paxman and J. C. Marron, “Aberration correction of speckled imagery with an image sharpness criterion,” in Statistical Optics, Proc. SPIE 976, 37–47 (1988).
- J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A20(4), 609–620 (2003). [CrossRef]
- S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A25(4), 983–994 (2008). [CrossRef]
- R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am.66(3), 207–211 (1976). [CrossRef]
- D. Rabb, J. W. Stafford, and D. F. Jameson, “Non-iterative aberration correction of a multiple transmitter system,” Opt. Express19(25), 25048–25056 (2011). [CrossRef]
- M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt.14(1), 142–150 (1975). [PubMed]
- G. Harbers, P. J. Kunst, and G. W. R. Leibbrandt, “Analysis of lateral shearing interferograms by use of Zernike polynomials,” Appl. Opt.35(31), 6162–6172 (1996). [CrossRef] [PubMed]
- S. Okuda, T. Nomura, K. Kamiya, and H. Miyashiro, “High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials,” Appl. Opt.39(28), 5179–5186 (2000). [CrossRef]
- F. Dai, F. Tang, X. Wang, P. Feng, and O. Sasaki, “Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms,” Opt. Express20(2), 1530–1544 (2012). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2004).
- J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (SPIE, 2010).
- R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice Hall, 2007).
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2010).

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