## Motion detection using extended fractional Fourier transform and digital speckle photography

Optics Express, Vol. 18, Issue 11, pp. 11396-11405 (2010)

http://dx.doi.org/10.1364/OE.18.011396

Acrobat PDF (945 KB)

### Abstract

Digital speckle photography is a useful tool for measuring the motion of optically rough surfaces from the speckle shift that takes place at the recording plane. A simple correlation based digital speckle photographic system has been proposed that implements two simultaneous optical extended fractional Fourier transforms (EFRTs) of different orders using only a single lens and detector to simultaneously detect both the magnitude and direction of translation and tilt by capturing only two frames: one before and another after the object motion. The dynamic range and sensitivity of the measurement can be varied readily by altering the position of the mirror/s used in the optical setup. Theoretical analysis and experiment results are presented.

© 2010 OSA

## 1. Introduction

1. H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. **5**(4), 271–276 (1972). [CrossRef]

9. D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A **23**(11), 2861–2870 (2006). [CrossRef]

1. H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. **5**(4), 271–276 (1972). [CrossRef]

10. E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. U.S.A. **23**(3), 158–164 (1937). [CrossRef] [PubMed]

11. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transformation,” J. Opt. Soc. Am. A **10**(10), 2181–2186 (1993). [CrossRef]

7. D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform,” Appl. Opt. **44**(14), 2720–2727 (2005). [CrossRef] [PubMed]

8. R. E. Patten, B. M. Hennelly, D. P. Kelly, F. T. O’Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. **31**(1), 32–34 (2006). [CrossRef] [PubMed]

9. D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A **23**(11), 2861–2870 (2006). [CrossRef]

12. J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A **14**(12), 3316–3322 (1997). [CrossRef]

## 2. Theoretical analysis

### 2.1 Extended fractional Fourier transform

12. J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A **14**(12), 3316–3322 (1997). [CrossRef]

*a*,

*b*,

*θ*are the extended fractional orders,

*l*is the input distance i.e. the distance of the object from the lens,

*f*is the focal length of the lens. If the input field is translated in the X-direction by

*ξ*and tilted by a small angle

8. R. E. Patten, B. M. Hennelly, D. P. Kelly, F. T. O’Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. **31**(1), 32–34 (2006). [CrossRef] [PubMed]

*s*and

*q*has been shifted by an amount

*Q*.

### 2.2 Simultaneous motion detection

*l*from the lens. As

*l*, there are two EFRTs with extended fractional orders (

_{1}and P

_{2}) are placed at orthogonal orientation.

*Q*’s, which are locations of CC peaks for different EFRTs, can be written asandwhere

*s*and

_{j}*t'*(

_{j}*j*= 1,2) are known. The values of

*s*and

_{j}*t'*can be obtained experimentally through calibration. Thus with only two recordings; one before and another after the object motion we can measure the translation and tilt simultaneously.

_{j}## 3. Experimental work

### 3.1. Experimental setup

### 3.2. Controlling the measurement sensitivity and range

*s*) is independent of the input distance

*l*, and changes linearly as a function of the ratio of the output distance to the focal length (

*s*, as a function of the ratio

*f*) remains unchanged. It can be seen from Fig. 2(a) that

*s*decreases from a positive to a negative value as

*t'*) as a function of the ratio

*f*) remains unchanged with the input distance (

*l*) fixed at

*t'*, also decreases lineally from a positive to a negative value as

*s*and

*t'*. The shift of the CC peak (in pixels) from the center (location of the auto-correlation peak) for a unit magnitude of translation (mm) or tilt (arcmin) is a measure of the sensitivity. A certain translation or tilt was prescribed while the output distance,

*f*) of the lens used in the setup was 250 mm and the input distance,

*l*, was fixed at 500 mm. Figure 3(a) shows the variation of sensitivity for translation measurement (

*s*) while Fig. 3(b) shows the variation of sensitivity for tilt measurement (

*t'*) with the ratio of the output distance to the focal length (

_{1}or M

_{2}in Fig. 1 is shifted along the optic axis by an amount

*D*, due to reflection geometry the output path in the EFRT configuration changes by 2

*D*. One can see from Eqs. (2)–(4) that the extended fractional orders will vary even though the focal length and the input distance remain unchanged. Thus one can readily change the measurement range and sensitivity simply by shifting either or both of the mirrors M

_{1}and M

_{2}.

### 3.3. Experimental results

*s*) and 4.17 pixels/arcmin (

_{1}*t'*), respectively whereas for arm 2 the corresponding values are 56 pixels/mm (

_{1}*s*) and −4 pixels/arcmin (

_{2}*t'*), respectively.

_{2}#### 3.3.1 Pure translation measurement

#### 3.3.2 Pure tilt measurement

#### 3.3.3 Simultaneous translation and tilt measurement

## 4. General discussions

13. M. Sjodahl and L. R. Benckert, “Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. **32**, 2278–2284 (1993). [CrossRef] [PubMed]

_{1}) while arm 2 is blocked. This is followed by recording of arm 2 intensity (I

_{2}) while arm 1 is blocked. Then after the motion of the object, the intensity of arm 1 is recorded (I

_{3}) while arm 2 is blocked. Subsequently the intensity of arm 2 is recorded (I

_{4}) while arm 1 is blocked. Now by cross-correlating the intensities (I

_{1}and I

_{3}) and (I

_{2}and I

_{4}) we can measure the CC peak shifts for a particular EFRT, and then solving simultaneous equations we can measure translation and tilt without any ambiguity. The CC peak values obtained using this method show a much higher values compared to the case when two fields are superposed at the CCD plane.

*s*and

*t*' by changing the focal length (

*f*) alone as can be evident from Eqs. (16) and (17). However, it is difficult to remount the lenses every time with different focal lengths. Use of spatial light modulator (SLM) as Fresnel lens can be useful in those circumstances.

## 5. Conclusions

## Acknowledgement

## References and links

1. | H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. |

2. | P. K. Rastogi, “Techniques of displacement and deformation measurements in speckle metrology,” in |

3. | M. Sjödahl, “Electronic speckle photography: measurement of in-plane strain fields through the use of defocused laser speckle,” Appl. Opt. |

4. | K. J. Gåsvik, |

5. | T. Fricke-Begemann, “Three-dimensional deformation field measurement with digital speckle correlation,” Appl. Opt. |

6. | J. M. Diazdelacruz, “Multiwindowed defocused electronic speckle photographic system for tilt measurement,” Appl. Opt. |

7. | D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform,” Appl. Opt. |

8. | R. E. Patten, B. M. Hennelly, D. P. Kelly, F. T. O’Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. |

9. | D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A |

10. | E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. U.S.A. |

11. | A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transformation,” J. Opt. Soc. Am. A |

12. | J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A |

13. | M. Sjodahl and L. R. Benckert, “Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(120.3940) Instrumentation, measurement, and metrology : Metrology

(330.4150) Vision, color, and visual optics : Motion detection

(070.2575) Fourier optics and signal processing : Fractional Fourier transforms

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: March 12, 2010

Revised Manuscript: May 2, 2010

Manuscript Accepted: May 4, 2010

Published: May 14, 2010

**Citation**

Basanta Bhaduri, C. J. Tay, C. Quan, and Colin J. R. Sheppard, "Motion detection using extended fractional Fourier transform and digital speckle photography," Opt. Express **18**, 11396-11405 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11396

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### References

- H. Tiziani, “A study of the use of laser speckle to measure small tilts of optically rough surfaces accurately,” Opt. Commun. 5(4), 271–276 (1972). [CrossRef]
- P. K. Rastogi, “Techniques of displacement and deformation measurements in speckle metrology,” in Speckle Metrology, R. S. Sirohi, ed. (Marcel Dekker, 1993).
- M. Sjödahl, “Electronic speckle photography: measurement of in-plane strain fields through the use of defocused laser speckle,” Appl. Opt. 34(25), 5799–5808 (1995). [CrossRef] [PubMed]
- K. J. Gåsvik, Optical Metrology, 3rd ed., (John Wiley & Sons Ltd, Chichester, 2002).
- T. Fricke-Begemann, “Three-dimensional deformation field measurement with digital speckle correlation,” Appl. Opt. 42(34), 6783–6796 (2003). [CrossRef] [PubMed]
- J. M. Diazdelacruz, “Multiwindowed defocused electronic speckle photographic system for tilt measurement,” Appl. Opt. 44(12), 2250–2257 (2005). [CrossRef] [PubMed]
- D. P. Kelly, B. M. Hennelly, and J. T. Sheridan, “Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform,” Appl. Opt. 44(14), 2720–2727 (2005). [CrossRef] [PubMed]
- R. E. Patten, B. M. Hennelly, D. P. Kelly, F. T. O’Neill, Y. Liu, and J. T. Sheridan, “Speckle photography: mixed domain fractional Fourier motion detection,” Opt. Lett. 31(1), 32–34 (2006). [CrossRef] [PubMed]
- D. P. Kelly, J. E. Ward, B. M. Hennelly, U. Gopinathan, F. T. O’Neill, and J. T. Sheridan, “Paraxial speckle-based metrology systems with an aperture,” J. Opt. Soc. Am. A 23(11), 2861–2870 (2006). [CrossRef]
- E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. U.S.A. 23(3), 158–164 (1937). [CrossRef] [PubMed]
- A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transformation,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993). [CrossRef]
- J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14(12), 3316–3322 (1997). [CrossRef]
- M. Sjodahl and L. R. Benckert, “Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. 32, 2278–2284 (1993). [CrossRef] [PubMed]

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