## Using an aliasing operator and a single discrete Fourier transform to down-sample the Fresnel transform |

Optics Express, Vol. 20, Issue 8, pp. 8815-8823 (2012)

http://dx.doi.org/10.1364/OE.20.008815

Acrobat PDF (1507 KB)

### Abstract

In Digital Holography there are applications where computing a few samples of a wavefield is sufficient to retrieve an image of the region of interest. In such cases, the sampling rate achieved by the direct and the spectral methods of the discrete Fresnel transform could be excessive. A few algorithmic methods have been proposed to numerically compute samples of propagated wavefields while allowing down-sampling control. Nevertheless, all of them require the computation of at least two 2D discrete Fourier transforms which increases the computational load. Here, we propose the use of an aliasing operator and a single discrete Fourier transform to achieve an efficient method to down-sample the wavefields obtained by the Fresnel transform.

© 2012 OSA

## 1. Introduction

*d*, and the recovery wavelength

*λ*[1

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

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

*d*and

*λ*[1

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

9. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. **29**(8), 854–856 (2004). [CrossRef] [PubMed]

4. X. Deng, B. Bihari, J. Gan, F. Zhao, and R. T. Chen, “Fast algorithm for chirp transforms with zooming-in ability and its applications,” J. Opt. Soc. Am. A **17**(4), 762–771 (2000). [CrossRef] [PubMed]

5. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. **29**(14), 1668–1670 (2004). [CrossRef] [PubMed]

7. J. C. Li, P. Tankam, Z. J. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett. **34**(5), 572–574 (2009). [CrossRef] [PubMed]

8. L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett. **31**(7), 897–899 (2006). [CrossRef] [PubMed]

9. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. **29**(8), 854–856 (2004). [CrossRef] [PubMed]

8. L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett. **31**(7), 897–899 (2006). [CrossRef] [PubMed]

10. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. **48**(9), 095801 (2009). [CrossRef]

**3098**, 224–233 (1997). [CrossRef]

11. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. **52**(10), 1123–1130 (1962). [CrossRef]

8. L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett. **31**(7), 897–899 (2006). [CrossRef] [PubMed]

9. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. **29**(8), 854–856 (2004). [CrossRef] [PubMed]

10. D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. **48**(9), 095801 (2009). [CrossRef]

## 2. The discrete Fresnel transform

*h*(

_{r}*m*,

*n*). The size and sampling rate of the interference pattern are given by the characteristics of the digital camera. We consider that the digital camera (CCD or CMOS) is an array of

*N*×

_{x}*N*pixels with sampling periods of Δ

_{y}*in the*

_{x}*x*-axis and Δ

*in the*

_{y}*y*-axis, thus the physical size of the interference pattern is (

*N*Δ

_{x}*) × (*

_{x}*N*Δ

_{y}*) =*

_{y}*L*×

_{x}*L*. Given

_{y}*h*(

_{r}*m*,

*n*) for 0

*≤m≤N*-1 and 0

_{x}*≤n≤N*-1, the DFreT can be used to compute samples of the wavefield in the

_{y}*ξ*-

*η*plane propagated at a distance

*d*from the interference pattern in the

*x*-

*y*plane. The computed array of samples is a window of

*N*×

_{ξ}*N*elements with sampling periods of Δ

_{η}*in the*

_{ξ}*ξ*-axis, and Δ

*in the*

_{η}*η*-axis, thus the physical size of the sampled wavefield is (

*N*Δ

_{ξ}*) × (*

_{ξ}*N*Δ

_{η}*)*

_{η}*= L*×

_{ξ}*L*. Here, the computed array of samples is referred to as Γ(

_{η}*k*,

*l*), and defined for 0

*≤k≤N*-1 and 0

_{ξ}*≤l≤N*-1.

_{η}_{2D}is a 2D discrete Fourier transform, and

*N*/2,

_{x}*N*/2,

_{y}*N*/2 and

_{ξ}*N*/2 in Eq. (2) and Eq. (3), and the complex exponential exp[

_{η}*iπ*(

*m*-

*N*/2 +

_{x}*n*-

*N*/2)] in Eq. (2) are introduced to align the symmetry center of the complex exponentials to the center of processed arrays.

_{y}*N*=

_{ξ}*N*and

_{x}*N*. In this case, the sampling periods Δ

_{η}= N_{y}*and Δ*

_{ξ}*, and in consequence the sampling rates (1/Δ*

_{η}*and 1/Δ*

_{ξ}*), are given by*

_{η}**3098**, 224–233 (1997). [CrossRef]

## 3. Down-sampling the Fresnel transform

*N*and

_{ξ}*N*from Eq. (4). Thus, the number of elements in the new array should be given by

_{η}*and 1/Δ*

_{ξ}*should be defined. It is important to highlight that since*

_{η}*N*and

_{ξ}*N*should be integers, the sampling rates can only vary in discrete steps defined by 1/Δ

_{η}*=*

_{ξ}*a*Δ

*/*

_{x}*λd*and 1/Δ

*=*

_{η}*b*Δ

*/*

_{y}*λd*for positive integer values of

*a*and

*b*. A direct alternative of down-sampling (

*N*<

_{ξ}*N*and/or

_{x}*N*<

_{η}*N*) is truncating the input array (interference pattern); in this case, the wavefield is consequently modified. Truncation could be a good alternative when rough objects are recorded since their information is spread out all over the recording medium (far field propagation), but a bad one for smooth objects since their information is spread out only in a portion of the recording medium (near field propagation). In fact, information is lost in both cases when truncating. In the former case, truncation eliminates high frequency information, while in the latter case, truncation additionally eliminates spatial information. The latter effect will be shown later.

_{y}### 3.1. Aliasing operator

*f*(

*m*) can be thought of as DFT coefficients of a periodic sequence

*g*(

*m*) obtained through summing periodic replicas of

*f*(

*m*). These samples are equally spaced in frequency in a range defined by 0≤

*ω*<2

*π*. This relationship is expressed as:

*N*defines the number of required samples of the FT and, in consequence, the length of the DFT (also of

_{s}*g*(

*m*)). The different replicas of

*f*(

*m*) in Eq. (7) are given by the shift index

*r*. Thus, aliasing (superposition of shifted replicas of a signal) can be introduced in the input domain to modify the sampling in the Fourier domain.

*N*does not need to be equal to the length of the aperiodic signal

_{s}*f*(

*m*), here defined as

*N*. If

_{i}*N*<

_{s}*N*, there are some shifted replicas of

_{i}*f*(

*m*) that contribute to the summation in Eq. (7). An example of how

*g*(

*m*) is generated from different replicas of

*f*(

*m*) is shown in Fig. 1 .

*f*(

*m*) is finite with length

*N*, just a few of its replicas are required to obtain

_{i}*g*(

*m*). In fact, due to the different contributions of the shifted replicas, all the elements in

*f*(

*m*) are used to generate

*g*(

*m*). Thus, considering the finite length of

*f*(

*m*),

*g*(

*m*) can be computed aswhere ⌊ ⌋ is the floor operator. The maximum value of the upper bound of the summation in Eq. (8) is obtained when the fraction

*m*/

*N*

_{s}does not surpass the fractional part of (

*N*-1)/

_{i}*N*. Hence, the maximum value of the upper bound occurs when

_{s}*m*>[(

*N*-1) mod

_{i}*N*], the upper bound takes its minimum value. Thus, the computation of

_{s}*g*(

*m*) can be carried out by

*N*and

_{i}*N*can take any integer value that fulfill

_{s}*N*<

_{s}*N*. Thus, our proposal consists in using the aliasing operator in Eq. (10) prior to computing the DFT, which will result in equally-spaced samples of the FT.

_{i}*N*/2 is included in the complex exponential of the FT. For this case, the FT can be computed as

_{i}*g*(

*m*) should be computed as specified by Eq. (10) from the original input sequence

*f*(

*m*).

### 3.2. Using the aliasing operator to down-sample the Fresnel transform

*N*and

_{ξ}*N*obtained with Eq. (5) correspond to the number of equally-spaced samples of the Fresnel transform in each column and each row, respectively, that have to be computed to achieve the required sampling rate. Thus, down-sampling is required when

_{η}*N*<

_{ξ}*N*and/or

_{x}*N*<

_{η}*N*, where

_{y}*N*is the length of the columns and

_{x}*N*of the rows in the digital camera. In order to use the aliasing operator, the array on which this should be applied has to be identified. This array is

_{y}*f*(

*m*,

*n*) as expressed in Eq. (2). Thus, the aliasing operator should be applied over

*f*(

*m*,

*n*) to generate the new array

*g*(

*m*,

*n*) to be transformed by the 2D DFT. Since

*f*(

*m*,

*n*) has columns of length

*N*and rows of length

_{x}*N*, the aliasing operator should be applied over the columns if

_{y}*N*<

_{ξ}*N*and over the rows if

_{x}*N*<

_{η}*N*to generate

_{y}*g*(

*m*,

*n*). Then, the DFreT should be implemented aswhere

*p*(

*k*,

*l*) is defined in Eq. (3), and the term exp[

*iπ*(

*kN*/

_{x}*N*+

_{ξ}*lN*/

_{y}*N*)] is introduced as indicated by Eq. (12) to correctly recover the phase information since the symmetry shifts

_{η}*N*/2 and

_{x}*N*/2 are used in the input domain of the DFreT (Eq. (2)).

_{y}*f*(

*m*,

*n*) cannot be recovered from Γ(

*k*,

*l*)). Thus, in order to obtain recoveries with different sampling rates, different aliasing operations over

*f*(

*m*,

*n*) should be carried out.

## 4. Experimental results

*λ*and inclination angle

_{rec}*θ*that fulfill the Nyquist criterion. The registered objects are smooth transmission-slides. A synthetic interference pattern (

*H*1) with

*N*×

_{x}*N*= 512 × 512 and Δ

_{y}*= Δ*

_{x}*= 6.4 μm is generated using a reference wave with*

_{y}*λ*= 532 nm and

_{rec}*θ*= −0.02078 rad, and a spoke target as object. A second interference pattern (

*H*2) is recorded on a CCD sensor of

*N*×

_{x}*N*= 768 × 1024 and Δ

_{y}*= Δ*

_{x}*= 4.7 μm using a reference wave with*

_{y}*λ*= 632 nm and

_{rec}*θ*= 0.0286 rad, and a USAF test chart as object. For the interference pattern

*H*1, the object was placed at a distance

*d*= 17 cm, and for

*H*2, it was placed at a distance

*d*= 22.7cm. The required DFTs are computed with the FFTW [14

14. M. Frigo and S. G. Johnson, “The FFTW web page,” (2007), http://www.fftw.org/*.*

*H*1 and

*H*2 are shown. It can be observed that since the recorded objects are smooth transmission-slides, the scattered light from the objects does not spread out all over the recording sensor as stated previously.

*= Δ*

_{ξ}*= 27.6 μm in Fig. 4(a), and Δ*

_{η}*= 39.75 μm and Δ*

_{ξ}*= 29.81 μm in Fig. 4(b). In all the following recoveries the complete wavefields are shown together with sub-images of the region of interest.*

_{η}## 5. Conclusions

## References and links

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

2. | U. Schnars and W. Jueptner, |

3. | B. M. Hennelly, D. P. Kelly, D. S. Monaghan, and N. Pandey, “Zoom Algorithms for Digital Holography,” in |

4. | X. Deng, B. Bihari, J. Gan, F. Zhao, and R. T. Chen, “Fast algorithm for chirp transforms with zooming-in ability and its applications,” J. Opt. Soc. Am. A |

5. | F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. |

6. | W. T. Rhodes, “Light tubes, Wigner diagrams and optical signal propagation simulation,” in |

7. | J. C. Li, P. Tankam, Z. J. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett. |

8. | L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett. |

9. | P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett. |

10. | D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. |

11. | E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. |

12. | S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital Holographic Particle Tracking Velocimetry for 3-D Transient Flow around an Obstacle in a Narrow Channel,” Opt. Rev. |

13. | A. V. Oppenheim, R. W. Schafer, and J. R. Buck, |

14. | M. Frigo and S. G. Johnson, “The FFTW web page,” (2007), http://www.fftw.org/ |

**OCIS Codes**

(090.1760) Holography : Computer holography

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: January 13, 2012

Revised Manuscript: March 2, 2012

Manuscript Accepted: March 6, 2012

Published: April 2, 2012

**Citation**

Modesto Medina-Melendrez, Albertina Castro, and Miguel Arias-Estrada, "Using an aliasing operator and a single discrete Fourier transform to down-sample the Fresnel transform," Opt. Express **20**, 8815-8823 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8815

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### References

- T. M. Kreis, M. Adams, and W. P. P. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE3098, 224–233 (1997). [CrossRef]
- U. Schnars and W. Jueptner, Digital Holography (Springer, 2005).
- B. M. Hennelly, D. P. Kelly, D. S. Monaghan, and N. Pandey, “Zoom Algorithms for Digital Holography,” in Information Optics and Photonics: Algorithms, Systems, and Applications, T. Fournel and B. Javidi, eds. (Springer, 2010), pp. 187–204.
- X. Deng, B. Bihari, J. Gan, F. Zhao, and R. T. Chen, “Fast algorithm for chirp transforms with zooming-in ability and its applications,” J. Opt. Soc. Am. A17(4), 762–771 (2000). [CrossRef] [PubMed]
- F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett.29(14), 1668–1670 (2004). [CrossRef] [PubMed]
- W. T. Rhodes, “Light tubes, Wigner diagrams and optical signal propagation simulation,” in Optical Information Processing: A Tribute to Adolf Lohmann, H. J. Caulfield, ed. (SPIE Press, 2002), pp. 343–356.
- J. C. Li, P. Tankam, Z. J. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett.34(5), 572–574 (2009). [CrossRef] [PubMed]
- L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett.31(7), 897–899 (2006). [CrossRef] [PubMed]
- P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, “Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms,” Opt. Lett.29(8), 854–856 (2004). [CrossRef] [PubMed]
- D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng.48(9), 095801 (2009). [CrossRef]
- E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am.52(10), 1123–1130 (1962). [CrossRef]
- S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital Holographic Particle Tracking Velocimetry for 3-D Transient Flow around an Obstacle in a Narrow Channel,” Opt. Rev.11, 162–164 (2004).
- A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing (Prentice Hall, 1999).
- M. Frigo and S. G. Johnson, “The FFTW web page,” (2007), http://www.fftw.org/ .

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