## A coherent framework for fingerprint analysis: are fingerprints holograms?

Optics Express, Vol. 15, Issue 14, pp. 8667-8677 (2007)

http://dx.doi.org/10.1364/OE.15.008667

Acrobat PDF (1489 KB)

### Abstract

We propose a coherent mathematical model for human fingerprint images. Fingerprint structure is represented simply as a hologram – namely a phase modulated fringe pattern. The holographic form unifies analysis, classification, matching, compression, and synthesis of fingerprints in a self-consistent formalism. Hologram phase is at the heart of the method; a phase that uniquely decomposes into two parts via the Helmholtz decomposition theorem. Phase also circumvents the infinite frequency singularities that always occur at minutiae. Reliable analysis is possible using a recently discovered two-dimensional demodulator. The parsimony of this model is demonstrated by the reconstruction of a fingerprint image with an extreme compression factor of 239.

© 2007 Optical Society of America

## 1. Introduction

8. K. G. Larkin, D. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns: I. General background to the spiral phase quadrature transform.,” J. Opt. Soc. Am. A **18**, 1862–1870 (2001). http://www.opticsinfobase.org/abstract.cfm?URI=josaa-18-8-1862 [CrossRef]

8. K. G. Larkin, D. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns: I. General background to the spiral phase quadrature transform.,” J. Opt. Soc. Am. A **18**, 1862–1870 (2001). http://www.opticsinfobase.org/abstract.cfm?URI=josaa-18-8-1862 [CrossRef]

## 2. The hologram model

23. A. M. Turing, “The chemical basis of morphogenesis, reprinted from Philosophical Transactions of the Royal Society (Part B), 237, 37-72 (1953)," Bull. Math. Biol. **52**, 153–197 (1990). [CrossRef] [PubMed]

24. A. Witkin and M. Kass, “Reaction-diffusion textures,” Comput. Graphics **25**, 299–308 (1991). [CrossRef]

25. J. P. Crutchfield, ed., *Is Anything Ever New? Considering Emergence, in Complexity: Metaphors, Models, and Reality*, (Addison-Wesley, Redwood City, 1994). http://www.santafe.edu/research/publications/wpabstract/199403011

26. J. Myung and M. Pitt, “Model Selection Methods,” in *Amsterdam Workshop on Model Selection*(Amsterdam, 2004). http://www2.fmg.uva.nl/modelselection/presentation.cfm?presenter=5

27. D. Gabor, “Microscopy by reconstructed wave-fronts,” Pro. R. Soc. London **197**, 454–487 (1949). [CrossRef]

*a*(

*x, y*), the amplitude

*b*(

*x, y*), and the phase

*ψ*(

*x,y*) are suitably smooth real functions. We note that the sign of the phase

*ψ*(

*x,y*) contains a global ambiguity that we also disregard here. A noise term

*n*(

*x,y*) formally completes the model, and may contain finer details, such as pores, as well as noise and other artifacts that do not easily fit the hologram model. It transpires that the AM-FM fingerprint model has been attempted several times before: in 1987 Kass proposed a dominant frequency that is locally distorted by curvilinear co-ordinates [19

19. M. Kass and A. Witkin, “Analyzing oriented patterns,” Computer vision, graphics, and image processing **37**, 362–385 (1987). [CrossRef]

28. J. G. Daugman and C. J. Downing, “Demodulation, predictive coding, and spatial vision,” J. Opt. Soc. Am. A **12**, 641–660 (1995). [CrossRef]

*p*=±1, and its location by (

_{n}*x*). We note that the relation between polarity and the incidence of ridge endings or ridge bifurcations is dependent on the direction of the local phase gradient. The spiral phase allows an abrupt change in the local fringe density, either inserting or deleting a ridge. Figure 2 shows an artificial fingerprint pattern generated using Eq. (1) – Eq. (3). The pattern shows a loop and a delta structure [18

_{n},y_{n}18. R. Penrose, “The topology of ridge systems,” Ann. Hum. Genet.,Lond. **42**, 435–444 (1979). [CrossRef]

22. B. G. Sherlock and D. M. Monro, “A model for interpreting fingerprint topology,” Pattern Recogn. **26**, 1047–1055 (1993). [CrossRef]

29. D. Kosz, “New numerical methods of fingerprint recognition based on mathematical description of arrangement of dermatoglyphics and creation of minutiae,” in *Biometrics in Human Service User Group Newsletter*,Mintie D., ed., (1999). http://www.ct.gov/dss/cwp/view.asp?A=2349&Q=304724

30. W. Bicz, “The idea of description (reconstruction) of fingerprints with mathematical algorithms and history of the development of this idea at Optel,” (Optel, 2003), http://www.optel.pl/article/english/idea.htm, (Accessed 9 May 2006),

## 3. Two-dimensional demodulation

8. K. G. Larkin, D. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns: I. General background to the spiral phase quadrature transform.,” J. Opt. Soc. Am. A **18**, 1862–1870 (2001). http://www.opticsinfobase.org/abstract.cfm?URI=josaa-18-8-1862 [CrossRef]

31. K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns: II. Stationary phase analysis of the spiral phase quadrature transform.,” J. Opt. Soc. Am. A **18**, 1871–1881 (2001). [CrossRef]

*iϕ*(

*u,v*)]:

*u,v*) and requires forward

**F**, and inverse

**F**

^{-1}, Fourier transforms, the method can also be implemented in the spatial domain using convolution. Note that the sought after quadrature term sin[

*ψ*(

*x,y*)] is, almost magically, expressed by the transform. But there is a catch: a directional phase multiplier -

*i*exp[

*iβ*(

*x,y*)] has also appeared.

## 4. Orientation and direction estimation

*x,y*) can be obtained from the orientation phase map by unwrapping. A relatively sophisticated unwrapping technique, using the topological properties of the ridge flow fields [22

22. B. G. Sherlock and D. M. Monro, “A model for interpreting fingerprint topology,” Pattern Recogn. **26**, 1047–1055 (1993). [CrossRef]

36. K. G. Larkin, “Natural demodulation of 2D fringe patterns,” in *Fringe´01 - The Fourth International Workshop on Automatic Processing of Fringe Patterns*, Juptner W. and Osten W., eds., (Elsevier, Bremen, Germany, 2001). http://citeseer.ist.psu.edu/458598.html

33. K. G. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Optics Express **13**, 8097–8121 (2005). [CrossRef] [PubMed]

*ψ*(

*x,y*):

*x, y*) modulo 2π.

## 5. Helmholtz phase decomposition

15. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. **30**, 3627–3323 (1991). [CrossRef] [PubMed]

37. Y. Tong, S. Lombey, A. N. Hirani, and M. Desbrun, “Discrete multiscale vector field decomposition,” ACM Transactions on Graphics **22**, 445–452 (2003). [CrossRef]

*ψ*(

*x,y*)-

*ψ*(

_{s}*x,y*) =

*ψ*(

_{c}*x,y*). With the spiral phases removed it is trivial to unwrap the continuous phase, as shown in Fig. 5(c).

## 6. Fingerprint image synthesis

*f*(

*x, y*) from the four elemental sub-images in the canonical equation:

*a, b*, and

*ψ*are all smooth functions and each can be compressed considerably with little resultant loss in fingerprint detail. Note that the smoothness constraint can be formalized mathematically so that the first and second terms,

_{c}*a*and

*b*, do not do all the work of capturing the variations in

*f*(

*x,y*). In our present implementation we have not optimized each of the four component compression algorithms, and there is plenty of scope for improvement using wavelet or other popular compression techniques. In the example presented the images

*a, b*, and

*ψ*were compressed using a fast Fourier transform followed by arithmetic coding of the quantized Fourier coefficients. The last image

_{c}*ψ*is represented as a (sparse) polarity map (see Fig. 4) and is compressed directly by run length encoding of the ternary (i.e. -1, 0, +1) image.

_{s}38. NIST Image Group´s Fingerprint Research, “Fingerprint Test Data on CD-ROM,” (NIST), http://www.itl.nist.gov/iad/894.03/fing/fing.html.

*a*(

*x, y*),

*b*(

*x, y*),

*ψ*(

_{c}*x,y*), and

*ψ*(

_{s}*x, y*).

*ψ*looks the smoothest and most compressible; however it covers a range of about 100 radians and controls the exact fringe spacing, so it requires more dynamic range to encode adequately. Much better compression might be achieved by 2-D polynomial or spline modeling, but is left for future refinements of the technique.

_{c}*ψ*looks very complicated because of the (seemingly) arbitrary phase wraps of the cyclic phase, but is really quite compressible in terms of the spiral polarity map (Fig. 4).

_{S}39. S. Kasaei, M. Deriche, and B. Boashash, “A novel fingerprint image compression technique using wavelets packets and pyramid lattice vector quantization,” IEEE Transactions On Image Processing **11**, 1365–1378 (2002). [CrossRef]

*ψ*(

_{c}*x,y*), shown in Fig. 5(c), can be interpreted as an absolute ridge index. Trustworthiness of the index is, of course, dependant on identifying the correct topology in section 4. Correct categorization of the deltas is particularly important for reliable indexing, and proper encoding of topology within the continuous phase.

## 7. Discussion

## Acknowledgments

41. K. G. Larkin and P. A. Fletcher, “Extreme compression of fingerprint images: squeezing patterns until the spirals pop out,” in *Fifth International Workshop on Information Optics* (Toledo, Spain, 2006). http://scitation.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=860&Issue=1

## References and links

1. | D. Maltoni, D. Maio, A. K. Jain, and S. Prabhakar, |

2. | N. Ratha and R. Bolle, eds., |

3. | S. Chikkerur, A. N. Cartwright, and V. Govindaraju, “Fingerprint Image Enhancement using STFT Analysis,” in |

4. | A. K. Jain and S. Pankanti, “Automated Fingerprint Identification and Imaging Systems,” in |

5. | U. Grasemann and R. Miikkulainen, “Effective image compression using evolved wavelets,” in |

6. | J. Tharna, K. Nilsson, and J. Bigun, “Orientation scanning to improve lossless compression of fingerprint images.,” in |

7. | C. M. Brislawn, “Fingerprints go digital,” Notices of the AMS |

8. | K. G. Larkin, D. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns: I. General background to the spiral phase quadrature transform.,” J. Opt. Soc. Am. A |

9. | F. Galton, |

10. | J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A. |

11. | A. W. Senior, R. M. Bolle, N. K. Ratha, and S. Pankanti, “Fingerprint Minutiae: A Constructive Definition,” in |

12. | A. Ross, J. Shah, and A. K. Jain, “From Template to Image: reconstructing fingerprints from minutiae points,” IEEE Trans PAMI |

13. | R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Science |

14. | J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. |

15. | D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. |

16. | D. L. Fried and J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. |

17. | D. C. Ghiglia and M. D. Pritt, |

18. | R. Penrose, “The topology of ridge systems,” Ann. Hum. Genet.,Lond. |

19. | M. Kass and A. Witkin, “Analyzing oriented patterns,” Computer vision, graphics, and image processing |

20. | C. F. Shu and R. C. Jain, “Direct Estimation and Error Analysis For Oriented Patterns,” CVGIP-Image Understanding |

21. | D. A. Egolf, I. V. Melnikov, and E. Bodenschatz, “Importance of local pattern properties in spiral defect chaos,” Phys. Rev. Lett. |

22. | B. G. Sherlock and D. M. Monro, “A model for interpreting fingerprint topology,” Pattern Recogn. |

23. | A. M. Turing, “The chemical basis of morphogenesis, reprinted from Philosophical Transactions of the Royal Society (Part B), 237, 37-72 (1953)," Bull. Math. Biol. |

24. | A. Witkin and M. Kass, “Reaction-diffusion textures,” Comput. Graphics |

25. | J. P. Crutchfield, ed., |

26. | J. Myung and M. Pitt, “Model Selection Methods,” in |

27. | D. Gabor, “Microscopy by reconstructed wave-fronts,” Pro. R. Soc. London |

28. | J. G. Daugman and C. J. Downing, “Demodulation, predictive coding, and spatial vision,” J. Opt. Soc. Am. A |

29. | D. Kosz, “New numerical methods of fingerprint recognition based on mathematical description of arrangement of dermatoglyphics and creation of minutiae,” in |

30. | W. Bicz, “The idea of description (reconstruction) of fingerprints with mathematical algorithms and history of the development of this idea at Optel,” (Optel, 2003), http://www.optel.pl/article/english/idea.htm, (Accessed 9 May 2006), |

31. | K. G. Larkin, “Natural demodulation of two-dimensional fringe patterns: II. Stationary phase analysis of the spiral phase quadrature transform.,” J. Opt. Soc. Am. A |

32. | B. Jähne, |

33. | K. G. Larkin, “Uniform estimation of orientation using local and nonlocal 2-D energy operators,” Optics Express |

34. | G. H. Granlund and H. Knutsson, |

35. | V. A. Soifer, V. V. Kotlyar, S. N. Khonina, and A. G. Khramov, “The method of the directional field in the interpretation and recognition of images with structure redundancy,” Image Analysis and Signal Processing: Adv. Math. Theory Appl. |

36. | K. G. Larkin, “Natural demodulation of 2D fringe patterns,” in |

37. | Y. Tong, S. Lombey, A. N. Hirani, and M. Desbrun, “Discrete multiscale vector field decomposition,” ACM Transactions on Graphics |

38. | NIST Image Group´s Fingerprint Research, “Fingerprint Test Data on CD-ROM,” (NIST), http://www.itl.nist.gov/iad/894.03/fing/fing.html. |

39. | S. Kasaei, M. Deriche, and B. Boashash, “A novel fingerprint image compression technique using wavelets packets and pyramid lattice vector quantization,” IEEE Transactions On Image Processing |

40. | P. A. Fletcher and K. G. Larkin, –Direct embedding and detection of RST invariant Watermarks,” in |

41. | K. G. Larkin and P. A. Fletcher, “Extreme compression of fingerprint images: squeezing patterns until the spirals pop out,” in |

**OCIS Codes**

(070.5010) Fourier optics and signal processing : Pattern recognition

(090.2880) Holography : Holographic interferometry

(100.2650) Image processing : Fringe analysis

(100.5070) Image processing : Phase retrieval

(110.2960) Imaging systems : Image analysis

(350.5030) Other areas of optics : Phase

**ToC Category:**

Fourier optics and signal processing

**History**

Original Manuscript: May 24, 2007

Revised Manuscript: June 22, 2007

Manuscript Accepted: June 22, 2007

Published: June 26, 2007

**Virtual Issues**

Vol. 2, Iss. 8 *Virtual Journal for Biomedical Optics*

**Citation**

Kieran G. Larkin and Peter A. Fletcher, "A coherent framework for fingerprint analysis: are fingerprints Holograms?," Opt. Express **15**, 8667-8677 (2007)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-14-8667

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### References

- D. Maltoni, D. Maio, A. K. Jain, and S. Prabhakar, Handbook of fingerprint recognition (Springer, New York, 2003).
- N. Ratha and R. Bolle, eds., Automatic Fingerprint Recognition Systems (Springer, New York, 2003).
- S. Chikkerur, A. N. Cartwright, and V. Govindaraju, "Fingerprint image enhancement using STFT analysis," in ICAPR, S. Singh, M. Singh, C. Apte, and P. Perner, eds., (Springer-Verlag, Bath, UK, 2005).
- A. K. Jain and S. Pankanti, "Automated fingerprint identification and imaging systems," in Advances in Fingerprint Technology, H. C. Lee, and R. E. Gaensslen, eds., (CRC Press, 2001).
- U. Grasemann, and R. Miikkulainen, "Effective image compression using evolved wavelets," in Genetic and Evolutionary Computation Conference (GECCO-05),(ACM, Washington DC, 2005), pp. 1961 - 1968.
- J. Tharna, K. Nilsson, and J. Bigun, "Orientation scanning to improve lossless compression of fingerprint images," in Audio and Video based Person Authentication - AVBPA03, J. Kittler, and M. S. Nixon, eds., (Springer, Heidelberg, 2003), pp. 343-350.
- C. M. Brislawn, "Fingerprints go digital," Not. Am. Math. Soc. 42, 1278-1283 (1995).
- K. G. Larkin, D. Bone, and M. A. Oldfield, "Natural demodulation of two-dimensional fringe patterns: I. General background to the spiral phase quadrature transform.," J. Opt. Soc. Am. A 18, 1862-1870 (2001). http://www.opticsinfobase.org/abstract.cfm?URI=josaa-18-8-1862 [CrossRef]
- F. Galton, Finger Prints (Macmillan, London, 1892). http://galton.org/books/finger-prints/index.htm
- J. F. Nye and M. V. Berry, "Dislocations in wave trains," Proc. R. Soc. Lond. A. 336, 165-190 (1974). [CrossRef]
- A. W. Senior, R. M. Bolle, N. K. Ratha, and S. Pankanti, "Fingerprint Minutiae: A Constructive Definition," in Workshop on biometrics, IEEE ECCV, (Copenhagen, Denmark, 2002).
- A. Ross, J. Shah, and A. K. Jain, "From template to image: reconstructing fingerprints from minutiae points," IEEE Trans PAMI 29, 544-560 (2007). [CrossRef]
- R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry: two-dimensional phase unwrapping," Radio Science 23, 713-720 (1988). [CrossRef]
- J. M. Huntley, "Noise-immune phase unwrapping algorithm," Appl. Opt. 28, 3268-3270 (1989). [CrossRef] [PubMed]
- D. J. Bone, "Fourier fringe analysis: the two-dimensional phase unwrapping problem," Appl. Opt. 30, 3627-3632 (1991). [CrossRef] [PubMed]
- D. L. Fried and J. L. Vaughn, "Branch cuts in the phase function," Appl. Opt. 31, 2865-2882 (1992). [CrossRef] [PubMed]
- D. C. Ghiglia, and M. D. Pritt, Two-dimensional phase unwrapping (John Wiley and Sons, New York, 1998).
- R. Penrose, "The topology of ridge systems," Ann. Hum. Genet. 42, 435-444 (1979). [CrossRef]
- M. Kass and A. Witkin, "Analyzing oriented patterns," Computer vision, graphics, and image processing 37, 362-385 (1987). [CrossRef]
- C. F. Shu and R. C. Jain, "Direct Estimation and Error Analysis for Oriented Patterns," CVGIP-Image Understanding 58, 383-398 (1993). [CrossRef]
- D. A. Egolf, I. V. Melnikov, and E. Bodenschatz, "Importance of local pattern properties in spiral defect chaos," Phys. Rev. Lett. 80, 3228-3231 (1998). [CrossRef]
- B. G. Sherlock and D. M. Monro, "A model for interpreting fingerprint topology," Pattern Recogn. 26, 1047-1055 (1993). [CrossRef]
- A. M. Turing, "The chemical basis of morphogenesis, reprinted from Philosophical Transactions of the Royal Society (Part B), 237, 37-72 (1953)," Bull. Math. Biol. 52, 153-197 (1990). [CrossRef] [PubMed]
- A. Witkin and M. Kass, "Reaction-diffusion textures," Comput. Graphics 25, 299-308 (1991). [CrossRef]
- J. P. Crutchfield, ed., Is Anything Ever New? Considering Emergence, in Complexity: Metaphors, Models, and Reality, (Addison-Wesley, Redwood City, 1994). http://www.santafe.edu/research/publications/wpabstract/199403011
- J. Myung and M. Pitt, "Model Selection Methods," in Amsterdam Workshop on Model Selection(Amsterdam, 2004). http://www2.fmg.uva.nl/modelselection/presentation.cfm?presenter=5
- D. Gabor, "Microscopy by reconstructed wave-fronts," Proc. R. Soc., London 197, 454-487 (1949). [CrossRef]
- J. G. Daugman and C. J. Downing, "Demodulation, predictive coding, and spatial vision," J. Opt. Soc. Am. A 12, 641-660 (1995). [CrossRef]
- D. Kosz, "New numerical methods of fingerprint recognition based on mathematical description of arrangement of dermatoglyphics and creation of minutiae," in Biometrics in Human Service User Group Newsletter, D. Mintie, ed., (1999). http://www.ct.gov/dss/cwp/view.asp?A=2349&Q=304724
- W. Bicz, "The idea of description (reconstruction) of fingerprints with mathematical algorithms and history of the development of this idea at Optel," (Optel, 2003), http://www.optel.pl/article/english/idea.htm, (Accessed 9 May 2006),
- K. G. Larkin, "Natural demodulation of two-dimensional fringe patterns: II. Stationary phase analysis of the spiral phase quadrature transform.," J. Opt. Soc. Am. A 18, 1871-1881 (2001). [CrossRef]
- B. Jähne, Practical handbook on Image processing for Scientific applications (CRC Press, Boca Raton, Florida, 1997).
- K. G. Larkin, "Uniform estimation of orientation using local and nonlocal 2-D energy operators," Opt. Express 13, 8097 - 8121 (2005). [CrossRef] [PubMed]
- G. H. Granlund, and H. Knutsson, Signal processing for computer vision (Kluwer, Dordrecht, Netherlands, 1995).
- V. A. Soifer, V. V. Kotlyar, S. N. Khonina, and A. G. Khramov, "The method of the directional field in the interpretation and recognition of images with structure redundancy," Image Analysis and Signal Processing: Adv. Math. Theory Appl. 6, 710-724 (1996).
- K. G. Larkin, "Natural demodulation of 2D fringe patterns," in Fringe'01 - The Fourth International Workshop on Automatic Processing of Fringe Patterns, W. Juptner, and W. Osten, eds., (Elsevier, Bremen, Germany, 2001). http://citeseer.ist.psu.edu/458598.html
- Y. Tong, S. Lombey, A. N. Hirani, and M. Desbrun, "Discrete multiscale vector field decomposition," ACM Transactions on Graphics 22, 445 - 452 (2003). [CrossRef]
- NIST Image Group's Fingerprint Research, "Fingerprint Test Data on CD-ROM," (NIST), http://www.itl.nist.gov/iad/894.03/fing/fing.html.
- S. Kasaei, M. Deriche, and B. Boashash, "A novel fingerprint image compression technique using wavelets packets and pyramid lattice vector quantization," IEEE Trans. Image Process. 11, 1365-1378 (2002). [CrossRef]
- P. A. Fletcher and K. G. Larkin, "Direct embedding and detection of RST invariant Watermarks," in IH2002, Fifth International Workshop on Information Hiding, F. A. P. Petitcolas, ed., (Springer Verlag, Noordwijkerhout, The Netherlands, 2002), pp. 129-144.
- K. G. Larkin, and P. A. Fletcher, "Extreme compression of fingerprint images: squeezing patterns until the spirals pop out," in Fifth International Workshop on Information Optics (Toledo, Spain, 2006). http://scitation.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=860&Issue=1

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