## Camera phasing in multi-aperture coherent imaging |

Optics Express, Vol. 20, Issue 11, pp. 11796-11805 (2012)

http://dx.doi.org/10.1364/OE.20.011796

Acrobat PDF (1271 KB)

### Abstract

The resolution of a diffraction-limited imaging system is inversely proportional to the aperture size. Instead of using a single large aperture, multiple small apertures are used to synthesize a large aperture. Such a multi-aperture system is modular, typically more reliable and less costly. On the other hand, a multi-aperture system requires phasing sub-apertures to within a fraction of a wavelength. So far in the literature, only the piston, tip, and tilt type of inter-aperture errors have been addressed. In this paper, we present an approach to correct for rotational and translational errors as well.

© 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]

### 1.1. Optical field measurement via spatial heterodyne detection

2. E. N. Leith and J. Upatnieks, “Wavefront reconstruction and communication theory,” J. Opt. Soc. Am. **52**(10), 1123–1128 (1962). [CrossRef]

3. J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Hoft, “Atmospheric turbulence correction using digital holographic detection: experimental results,” Opt. Express **17**(14), 11638–11651 (2009). [CrossRef] [PubMed]

*a*(

_{k}*x, y*) is the complex object field captured at the

*k*th sub-aperture and

*r*(

*x, y*) =

*e*

^{−j2π(u0x+v0y)}is the tilted reference beam, also known as the local oscillator. The complex-valued field is obtained by taking the Fourier transform of

*I*(

_{k}*x, y*). Defining

*â*(

_{k}*u, v*) and

*r̂*(

*u, v*) as the Fourier transforms of the pupil plane field

*a*(

_{k}*x, y*) and the local oscillator

*r*(

*x, y*), the Fourier transform of

*I*(

_{k}*x, y*) becomes

*R*(·) is the autocorrelation function, and the offset (

*u*

_{0},

*v*

_{0}) is due to the tilted local oscillator in the spatial heterodyne mixing. The

*â*(

_{k}*u*,

*v*) term is spatially separated from its conjugate term and the autocorrelation terms by virtue of the spatial offset. Therefore, the desired pupil plane field,

*â*(

_{k}*u, v*), is obtained by cropping the region of

**(

*I*(

_{k}*x, y*)) around (

*u*

_{0}

*, v*

_{0}). In addition to the spatial heterodyne technique, the object field

*a*(

_{k}*x, y*) can be measured using other interferometric methods such as phase shifting interferometry or estimated from the object intensity using a phase retrieval algorithm [4

4. D. Malacara, *Optical Shop Testing* (Wiley, 2007). [CrossRef]

5. J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express **14**(2), 498–508 (2006). [CrossRef] [PubMed]

### 1.2. Aperture synthesis

*k*sub-aperture pupils. Each measured pupil plane field

*a*(

_{k}*x, y*) is placed on a blank aperture field at their corresponding spatial locations to form a composite pupil field. Mathematically, the composite pupil field can be written as

*x*) is the center location of the

_{k}, y_{k}*k*th sub-aperture, and

*K*is the number of sub-apertures. (The notation

*a*

_{comp}_{(1:}

_{K}_{)}(

*x, y*) indicates that all sub-apertures from 1 to

*K*are used to form the composite as opposed to

*a*

_{comp}_{(1}

_{,k}_{)}(

*x, y*) where the first and

*k*th sub-apertures are used to form the composite.) The composite pupil field is then digitally propagated to the image plane, where the magnitude squared of the field becomes the intensity image. In Fig. 1, we show realizations

*b*(

*u,v*) = |

*â*

_{comp}_{(1:}

_{K}_{)}(

*u, v*)|

^{2}obtained with single-aperture (

*K*= 1) and multi-aperture (

*K*= 3) configurations. First, notice that a single realization is of poor quality due to speckle noise. When multiple realizations are averaged, the image quality improves since the statistically independent realizations of speckle noise tend to cancel out. It is known that the signal-to-noise ratio in case of speckle noise is inversely proportional with the square root of the number of realizations averaged [6]. Second, notice that in case of the three-aperture configuration, the resolution improves along the horizontal axis but not the vertical axis, which is obviously due to the arrangement of the sub-apertures [7

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

*a*(

_{k}*x, y*) [8

8. 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]

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

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

## 2. Phase correction of sub-apertures

### 2.1. Intra-aperture correction

*a*(

_{k}*x, y*), and can be corrected by determining the Zernike polynomial weights that optimize a measure

*S*(·) applied on the constructed image

*b*(

*u,v*). Depending on whether the measure

*S*(·) outputs a low or a high value for sharp images, the optimization is either a minimization or a maximization problem. In this paper, we use the convention of minimizing the measure

*S*(·). As the correction is applied on the phase of each pupil field, the optimal Zernike coefficients

*ŵ*for the

_{k,p}*k*th sub-aperture are where

*Z*

_{1}(

*x, y*),...,

*Z*(

_{P}*x, y*) are the Zernike polynomials used. When there are multiple realizations, input to the measure

*S*(·) is the average of all realizations.

### 2.2. Piston/Tip/Tilt correction

*k*th fields is where the Zernike polynomials

*Z*include

_{p}*w*are the coefficients of these polynomials. The optimal values of these coefficients are determined by

_{k,p}### 2.3. Rotation correction

*θ*along the optical axis. In other words, the measured pupil plane field at a sub-aperture is

*a*(

_{k}*x*′

*,y*′) instead of

*a*(

_{k}*x, y*), where (

*x*′

*,y*′) = (

*xcosθ*−

*ysinθ, xsinθ*+

*ycosθ*). Such a rotational error cannot be fixed with pupil plane phasing using Zernike polynomials; and it would also degrade the accuracy of piston/tip/tilt correction.

*a*(

_{k}*x, y*), the rotated field

*a*(

_{k}*x*′

*,y*′) will have the polar version

*a*(

_{k}*x, y*) corresponds to a linear phase shift in

*k*th sub-aperture is achieved by determining the optimal coefficient in which case the composite is formed as where

*𝒫*operator transforms from Cartesian to polar coordinates, and

*𝒫*

^{−1}is the inverse transformation.

*θ*) axis. However, one should be careful that the domain of the polynomial is now rectangular unlike the previous cases, where it is circular. (Therefore, strictly speaking, the term Zernike can be avoided; and the polynomial can simply be referred to as a linear phase polynomial.)

13. B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. **5**(8), 1266–1271 (1996). [CrossRef] [PubMed]

### 2.4. Shift correction

*k*th sub-aperture is offset by amount (

*δx*,

_{k}*δy*); that is, the measurement of the camera is

_{k}*I*(

_{k}*x*+

*δx*,

_{k}*y*+

*δy*). This spatial shift (translational error) will correspond to a phase shift in the image plane:

_{k}*u*

_{0},

*v*

_{0}) is then

*â*(

_{k}*u*,

*v*)

*e*

^{−j2π(δxku+δykv)}

*e*

^{−j2π(δxku0+δykv0)}. That is, the phase distortion in

*â*(

_{k}*u, v*) consists of a constant term and two linear terms, one in each direction. This can be handled by the first three Zernike polynomials,

### 2.5. Proposed phasing algorithm

*S*(·), which is applied on

*b*(

*u,v*).

## 3. Experimental evaluation

*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 [14]. Analytic evaluation of this Fresnel diffraction integral is difficult for all but a few very simple object geometries. Therefore, the Fresnel diffraction integral was numerically evaluated using the angular spectrum propagation method [15]. However, the angular spectrum method of wave propagation is limited by discrete Fourier transform wraparound effects that occur when the wavefront spreads in the transverse dimensions and reflects at the computational grid boundaries. This wraparound effect is the most onerous limitation for wavefront propagations from diffuse objects which have inherently large divergence. In order to mitigate this wraparound effect, the wave propagation is performed in multiple partial propagations. After each partial propagation, the wavefront energy reflected at the computational grid boundary is absorbed by an annular attenuating function. The central, on-axis region of the propagating wavefront is unattenuated. The partial propagation distance is limited such that spreading wavefront does not reflect into the central, on-axis region.

*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 consisted of 10 partial propagations of 10 meters each to avoid the wraparound effects described above.

*f minunc*function of MATLAB, minimizing the well-known power measure ∑

*|*

_{u,v}*b*(

*u,v*)|

^{0.5}[10

10. J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A **20**(4), 609–620 (2003). [CrossRef]

## 4. Conclusions

## References and links

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

2. | E. N. Leith and J. Upatnieks, “Wavefront reconstruction and communication theory,” J. Opt. Soc. Am. |

3. | J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Hoft, “Atmospheric turbulence correction using digital holographic detection: experimental results,” Opt. Express |

4. | D. Malacara, |

5. | J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express |

6. | J. W. Goodman, |

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

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

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

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

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

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

13. | B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. |

14. | J. W. Goodman, |

15. | J. D. Schmidt, |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(090.1995) Holography : Digital holography

**ToC Category:**

Image Processing

**History**

Original Manuscript: April 6, 2012

Manuscript Accepted: April 27, 2012

Published: May 9, 2012

**Virtual Issues**

Vol. 7, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Bahadir K. Gunturk, Nicholas J. Miller, and Edward A. Watson, "Camera phasing in multi-aperture coherent imaging," Opt. Express **20**, 11796-11805 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-11796

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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]
- E. N. Leith and J. Upatnieks, “Wavefront reconstruction and communication theory,” J. Opt. Soc. Am.52(10), 1123–1128 (1962). [CrossRef]
- J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Hoft, “Atmospheric turbulence correction using digital holographic detection: experimental results,” Opt. Express17(14), 11638–11651 (2009). [CrossRef] [PubMed]
- D. Malacara, Optical Shop Testing (Wiley, 2007). [CrossRef]
- J. R. Fienup, “Lensless coherent imaging by phase retrieval with an illumination pattern constraint,” Opt. Express14(2), 498–508 (2006). [CrossRef] [PubMed]
- J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2006).
- D. Rabb, D. Jameson, A. Stokes, and J. Stafford, “Distributed aperture synthesis,” Opt. Express18(10), 10334–10342 (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,” Proc. SPIE976, 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]
- B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process.5(8), 1266–1271 (1996). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (Roberts and Company, 2004).
- J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (SPIE, 2010).

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