## Image-based registration for synthetic aperture holography |

Optics Express, Vol. 19, Issue 12, pp. 11716-11731 (2011)

http://dx.doi.org/10.1364/OE.19.011716

Acrobat PDF (3103 KB)

### Abstract

High pixel count apertures for digital holography may be synthesized by scanning smaller aperture detector arrays. Characterization and compensation for registration errors in the detector array position and pitch and for phase instability between the reference and object field is a major challenge in scanned systems. We use a secondary sensor to monitor phase and image-based registration parameter estimators to demonstrate near diffraction-limited resolution from a 63.4 mm aperture synthesized by scanning a 5.28 mm subaperture over 144 transverse positions. We demonstrate 60 *μ*m resolution at 2 m range.

© 2011 OSA

## 1. Introduction

1. C. W. Sherwin, P. Ruina, and R. D. Rawcliffe, “Some early developments in synthetic aperture radar systems,” IRE Trans. Mil. Electron. **6**, 111–115 (1962). [CrossRef]

2. L. G. Brown, “A survey of image registration techniques,” ACM Comput. Surv. **24**, 4 (1992). [CrossRef]

4. J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. **27**, 24, 2179–2181 (2002). [CrossRef]

*et al.*presented a 27.4 × 1.7 mm aperture synthesis by scanning a sensor with an effective area of 1.7 × 1.7 mm at 1.5 m [5

5. R. Binet, J. Colineau, and J.-C. Lehureau, “Short-range synthetic aperture imaging at 633 nm by digital holography,” Appl. Opt. **41**, 4775–4782 (2002). [CrossRef] [PubMed]

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

*et al.*synthesized nine measurements in a regular octagon geometry, improving resolution by a factor of 3 in digital holographic microscopy [8

8. V. Mico, Z. Zalevsky, C. Ferreira, and J. Garca, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express **16**, 19260–19270 (2008). [CrossRef]

*et al.*used the same phase factors for a 25.9 × 23.3 mm aperture synthesis in digital Fresnel holography [9

9. H. Jiang, J. Zhao, J. Di, and C. Qin, “Numerically correcting the joint misplacement of the sub-holograms in spatial synthetic aperture digital Fresnel holography,” Opt. Express **17**, 18836–18842 (2009). [CrossRef]

11. D. J. Brady, *Optical Imaging and Spectroscopy* (Wiley, 2009). [CrossRef]

12. U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. **13**, R85–R101 (2002). [CrossRef]

13. B. Javidi, P. Ferraro, S.-H. Hong, S. De Nicola, A. Finizio, D. Alfieri, and G. Pierattini, “Three-dimensional image fusion by use of multiwavelength digital holography,” Opt. Lett. **30**, 144–146 (2005). [CrossRef] [PubMed]

## 2. Problem formulation

*Object plane*and

*detector plane*are defined with the spatial coordinates of (

*x*,

*y*) and (

*u*,

*v*), respectively. Also,

*z*and

_{d}*z*denote the ranges of the object and the reference point from the detector array along the optical axis. A

_{r}*hologram patch*indicates a measurement of scanning aperture. A

*hologram block*is composed of the hologram patches, and a

*wide aperture (WA) hologram*is a collection of the hologram blocks. A symbol

*A*denotes the set of spatial coordinates of all locations in the area of the synthetic aperture as shown in Fig. 1(b). The

*A*is partitioned into

*I*×

*J*subsets, where

*A*denotes the (

_{i j}*i*,

*j*)-th subset of

*A*: ∪

_{i j}*A*=

_{i j}*A*.

*I*(

^{i j}*u*,

*v;z*,

_{d}*z*) for all

_{r}*i*and

*j*are collected, we can register them together by placing in the positions according to the

*A*as shown in Fig. 1(b). By doing so, we can synthesize the full synthetic aperture intensity. Then a typical holographic filtering process [10] can be performed to extract the scattered field

_{i j}*E*(

_{s}*u*,

*v; z*), which can then be backpropagated by using an adjoint operator to form a coherent image of the object field

_{d}*E*(

_{o}*x*,

*y*).

### 2.1. Error sources

*Piston phase errors*: unknown changes in the constant phase of the interference intensity measurements*I*(^{i j}*u*,*v; z*,_{d}*z*),_{r}*Detector registration errors*: unknown errors in the exact positions of*I*(^{i j}*u*,*v; z*,_{d}*z*) caused by the inaccuracy of the 2D translation stage that scans the FPA,_{r}*Reference field errors*: unknown relative changes in the position of the reference field to the object field, which may be caused by the experimental instability (e.g. vibration and temperature fluctuations),*Reference field discrepancy*: unknown discrepancy in the phase of the reference field caused by the non-ideal generation of the spherical field.

### 2.2. Mathematical modeling of errors

*I*(

^{i j}*u*,

*v; z*,

_{d}*z*) is captured by placing the sensor array at the (i,j)-th position as shown in Fig. 1(b). It consists of the reference and scattered field,

_{r}*R*(

^{i j}*u*,

*v; z*) and

_{r}*D̃*(

^{i j}*u*,

*v; z*,

_{d}*z*) is re-defined by Error parameters vector is defined as

_{r}*R̃*(

^{i j}*u*,

*v; z*) and the inaccurate scattered field

_{r}11. D. J. Brady, *Optical Imaging and Spectroscopy* (Wiley, 2009). [CrossRef]

*R̃*(

^{i j}*u*,

*v; z*) is impacted by the transverse error

_{r}## 3. Computational methods for aperture synthesis

*i*,

*j*∈

*B*where

_{mn}*B*is the set of all indices i and j for the (m,n)-th block. Thus the error parameters vector becomes, WA hologram synthesis (subsection 3.3) makes the error parameters vector become, Finally, the reference field discrepancy

_{mn}### 3.1. Piston phase compensation

*s*(

^{i j}*u*,

*v*) indicates the complex hologram image in the (i,j)-th hologram patch. To figure out the relative field variation to the first hologram image,

*s*(

^{i j}*u*,

*v*) is divided by

*s*

^{11}(

*u*,

*v*) in exponential form. The MATLAB function sign avoids the phase variation from being affected by phase wrapping problem. The MATLAB function angle returns the phase angles, in radians, of complex elements.

### 3.2. Hologram block synthesis (hologram patch based process)

*i*= 1,...,

*I*and

*j*= 1,...,

*J*. The (i,j)-th hologram patch is defined in the (m,n)-th hologram block. where

*R*(

^{mn}*u*,

*v; z*) is the numerically generated spherical field with the hologram block size. The backpropagated field image is evaluated using the sharpness metric [14

_{r}14. S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A **25**, 983–994 (2008). [CrossRef]

*(*

_{SM}*e*) on the guiding feature images,

_{dp}### 3.3. Hologram synthesis (hologram block based process)

*B*) is denoted in the m-th row and in the n-th column of the matrix of the blocks. The detector registration errors in hologram blocks are defined as

_{k}*m*= 1, ...,

*M*and

*n*= 1, ...,

*N*. Then the WA hologram field measurement is expressed by summing the estimated hologram blocks

*D*(

^{mn}*u*,

*v; z*,

_{d}*z*), The transverse errors in the hologram blocks,

_{r}*j*∈

*B*, are also considered. The estimated WA reference field is expressed by,

_{mn}**E**

*(*

_{s}*u*,

*v; z*) is obtained by multiplying the estimated WA hologram field measurement by the estimated WA hologram reference field, Then the estimated WA object field

_{d}**E**

*(*

_{o}*x*,

*y*) is obtained by using the backpropagation method,

*e*

_{db}_{,}

*= (*

_{r}*e*

_{db}_{,}

*,*

_{u}*e*

_{db}_{,}

*,*

_{v}*e*

_{t}_{,}

*,*

_{u}*e*

_{t}_{,}

*), we again use the sharpness metric [14*

_{v}14. S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A **25**, 983–994 (2008). [CrossRef]

### 3.4. Reference field estimation

*P*(

_{k}*u*,

*v*) and

*C*are Chebyshev basis function and coefficient, respectively. Here the Chebyshev polynomials of the first kind are used up to 4th order (15 terms representing piston, tilt, focal shift, primary coma, and primary spherical phase error).

_{k}**E**

*(*

_{s}*u*,

*v; z*). Then the phase estimation is performed by minimizing the sharpness metric on the guiding feature images. The estimated WA object field

_{d}**E**

*(*

_{o}*x*,

*y*) is obtained by using the backpropagation method, The sharpness metric [14

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

## 4. Experiment

*μ*m pinhole and a 0.65 numerical aperture (NA) microscopic objective lens. The high NA microscopic objective lens generates a wide spherical field in the detector plane. Note the center of the reference field was vertically 100 mm lower than the center of the guiding features and axially 2.032 m (within ±2 mm accuracy) away from the surface of FPA.

11. D. J. Brady, *Optical Imaging and Spectroscopy* (Wiley, 2009). [CrossRef]

*λ*is the illumination wavelength, z is the propagation distance,

*δu*is the pixel pitch, and N is the number of pixels. Our experiments use a pixel pitch of 4.4

*μ*m, corresponding to the monochrome CCD PointGrey GRAS-20S4M-C FPA. We use a square patch of 1200×1200 pixels on the array, meaning that a single aperture hologram captures a 5.28×5.28

*mm*aperture and a 12×12 synthetic aperture captures a 63.36×63.36

*mm*aperture. At the object range of 2 meters, the theoretical resolution and FOV are 20

*μ*m and 288×288 mm (where N= 12 hologram patches × 1200 pixels/hologram patch).

15. T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE **3098**, 224–233 (1997). [CrossRef]

*μ*m equivalent to the pixel pitch of FPA. The angular spectrum method was used since the Fresnel approximation method is designed for the small FOV object at the center.

### 4.1. Stereo camera system for piston phase compensation

### 4.2. Reinitialization points scheme for hologram scanning

*μ*m and a bi-directional repeatability of 1.0 – 2.5

*μ*m. However, the guaranteed accuracy is 15

*μ*m and the inaccuracy linearly accumulates along the translation axis.

## 5. Processes and Results

*μm*of the translation stage. To speed up the estimation of the hologram block, each row of the hologram block was considered to have identical detector registration errors. This is a reasonable assumption since the translation stage achieves bi-directional repeatability in the adjacent row measurement. So the detector registration errors were transversely searched for by sweeping the possible errors within the range for about one hour.

*μ*m.

*μ*m in Fig. 5(f). Also, Fig. 5(e) and (f) show the effect of the reference field discrepancy estimation on the zoomed-in images. The estimation helps to resolve the features (group 4, element1) that are marked by a red circle. The estimated phase of the reference field is shown in Fig. 5(g). Note that the theoretical resolution is calculated by multiplying the speckle factor of 3 by the theoretical resolution [11

11. D. J. Brady, *Optical Imaging and Spectroscopy* (Wiley, 2009). [CrossRef]

16. A. Kozmat and C. R. Christensent, “Effects of speckle on resolution,” J. Opt. Soc. Am. **66**, 1257–1260 (1976). [CrossRef]

*μ*m in Fig. 6(b) and (e). The image of the 12×12 hologram patch resolves the features (group 4, element 1) whose resolution corresponds to the theoretical resolution 62.5

*μ*m in Fig. 6(c) and (f). Here we used the speckle-affected resolution [16

16. A. Kozmat and C. R. Christensent, “Effects of speckle on resolution,” J. Opt. Soc. Am. **66**, 1257–1260 (1976). [CrossRef]

## 6. Conclusion

## Acknowledgments

## References and links

1. | C. W. Sherwin, P. Ruina, and R. D. Rawcliffe, “Some early developments in synthetic aperture radar systems,” IRE Trans. Mil. Electron. |

2. | L. G. Brown, “A survey of image registration techniques,” ACM Comput. Surv. |

3. | L. Romero and F. Calderon, |

4. | J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. |

5. | R. Binet, J. Colineau, and J.-C. Lehureau, “Short-range synthetic aperture imaging at 633 nm by digital holography,” Appl. Opt. |

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

7. | J. W. Goodman, |

8. | V. Mico, Z. Zalevsky, C. Ferreira, and J. Garca, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express |

9. | H. Jiang, J. Zhao, J. Di, and C. Qin, “Numerically correcting the joint misplacement of the sub-holograms in spatial synthetic aperture digital Fresnel holography,” Opt. Express |

10. | J. W. Goodman, |

11. | D. J. Brady, |

12. | U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. |

13. | B. Javidi, P. Ferraro, S.-H. Hong, S. De Nicola, A. Finizio, D. Alfieri, and G. Pierattini, “Three-dimensional image fusion by use of multiwavelength digital holography,” Opt. Lett. |

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

15. | T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE |

16. | A. Kozmat and C. R. Christensent, “Effects of speckle on resolution,” J. Opt. Soc. Am. |

**OCIS Codes**

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

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: March 10, 2011

Revised Manuscript: April 11, 2011

Manuscript Accepted: April 13, 2011

Published: June 1, 2011

**Citation**

Sehoon Lim, Kerkil Choi, Joonku Hahn, Daniel L. Marks, and David J. Brady, "Image-based registration for synthetic aperture holography," Opt. Express **19**, 11716-11731 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11716

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### References

- C. W. Sherwin, P. Ruina, and R. D. Rawcliffe, “Some early developments in synthetic aperture radar systems,” IRE Trans. Mil. Electron. 6, 111–115 (1962). [CrossRef]
- L. G. Brown, “A survey of image registration techniques,” ACM Comput. Surv. 24, 4 (1992). [CrossRef]
- L. Romero and F. Calderon, A Tutorial on Parametric Image Registration (I-Tech, 2007).
- J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. 27, 24, 2179–2181 (2002). [CrossRef]
- R. Binet, J. Colineau, and J.-C. Lehureau, “Short-range synthetic aperture imaging at 633 nm by digital holography,” Appl. Opt. 41, 4775–4782 (2002). [CrossRef] [PubMed]
- J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A 20, 609–620 (2003). [CrossRef]
- J. W. Goodman, Speckle Phenomena in Optics - Theory and Applications (Roberts and Company, 2007).
- V. Mico, Z. Zalevsky, C. Ferreira, and J. Garca, “Superresolution digital holographic microscopy for three-dimensional samples,” Opt. Express 16, 19260–19270 (2008). [CrossRef]
- H. Jiang, J. Zhao, J. Di, and C. Qin, “Numerically correcting the joint misplacement of the sub-holograms in spatial synthetic aperture digital Fresnel holography,” Opt. Express 17, 18836–18842 (2009). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics , 3rd ed. (Roberts and Company, 2005).
- D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009). [CrossRef]
- U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002). [CrossRef]
- B. Javidi, P. Ferraro, S.-H. Hong, S. De Nicola, A. Finizio, D. Alfieri, and G. Pierattini, “Three-dimensional image fusion by use of multiwavelength digital holography,” Opt. Lett. 30, 144–146 (2005). [CrossRef] [PubMed]
- S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A 25, 983–994 (2008). [CrossRef]
- T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997). [CrossRef]
- A. Kozmat and C. R. Christensent, “Effects of speckle on resolution,” J. Opt. Soc. Am. 66, 1257–1260 (1976). [CrossRef]

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