Abstract

An all-digital ring-wedge detector system is presented that simulates the analog multielement array commonly used in coherent optoelectronic processors. The system is applicable with either hard-copy or digital imagery. Using neural-network software, we demonstrate high accuracy for the recognition of fingerprints, including both orientation and wide-scale size-independent sortings by using ring-only and wedge-only input neurons, respectively. Also, the system is applied on windowed subregions of fingerprint imagery, providing a feature set that summarizes localized information about spatial-frequency content and edge-angle correlations. Examples are presented in which this localized spatial-frequency information is used to produce local ridge-orientation maps and to detect regions of poor print quality. In summary, both direct-image data and spatial-transform data are found to be important.

© 1999 Optical Society of America

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1997 (1)

A. K. Jain, L. Hong, S. Pankanti, R. Bolle, “An identity-authentication system using fingerprints,” Proc. IEEE 85, 1365–1388 (1997).
[CrossRef]

1996 (2)

C. G. Looney, “Advances in feedforward neural networks: demystifying knowledge acquiring black boxes,” IEEE Trans. Knowl. Data Eng. 8, 211–226 (1996).
[CrossRef]

P. R. Vizcaya, L. A. Gerhardt, “A nonlinear orientation model for global description of fingerprints,” Pattern Recogn. 29, 1221–1231 (1996).
[CrossRef]

1995 (2)

Z. Chen, Y. Sun, Y. Zhang, G. Mu, “Hybrid optical/digital access control using fingerprint identification,” Opt. Eng. 34, 834–839 (1995).
[CrossRef]

N. K. Ratha, S. Chen, A. K. Jain, “Adaptive flow orientation-based feature extraction in fingerprint images,” Pattern Recogn. 28, 1657–1672 (1995).
[CrossRef]

1994 (3)

J. Ohta, J. Sharpe, K. Johnson, “An optoelectronic smart detector array for the classification of fingerprints,” Opt. Commun. 111, 451–458 (1994).
[CrossRef]

B. G. Sherlock, D. M. Monro, K. Millard, “Fingerprint enhancement by directional Fourier filtering,” Proc. IEE 141, 87–94 (1994).

N. George, S.-G. Wang, “Neural networks applied to diffraction-pattern sampling,” Appl. Opt. 33, 3127–3134 (1994).
[CrossRef] [PubMed]

1993 (2)

L. Coetzee, E. C. Botha, “Fingerprint recognition in low quality images,” Pattern Recogn. 26, 1441–1460 (1993).
[CrossRef]

P. Baldi, Y. Chauvin, “Neural networks for fingerprint recognition,” Neural Computa. 5, 402–418 (1993).
[CrossRef]

1992 (4)

M. Kamijo, H. Mieno, K. Kojima, “Classification of fingerprint images using a neural network,” Syst. Comput. Jpn. 23, 89–101 (1992).
[CrossRef]

F. T. Gamble, L. M. Frye, D. R. Grieser, “Real-time fingerprint verification system,” Appl. Opt. 31, 652–655 (1992).
[CrossRef] [PubMed]

V. S. Srinivasan, N. N. Murthy, “Detection of singular points in fingerprint images,” Pattern Recogn. 25, 139–153 (1992).
[CrossRef]

C. Charalambous, “Conjugate gradient algorithm for efficient training of artificial neural networks,” Proc. IEEE 139, 301–310 (1992).

1990 (1)

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

1989 (2)

L. O’Gorman, J. V. Nickerson, “An approach to fingerprint filter design,” Pattern Recogn. 22, 29–38 (1989).
[CrossRef]

T. Kanaoka, M. Watanabe, Y. Hamamoto, S. Tomita, “On a criterion for fingerprint image quality using the autocorrelation,” Trans. Inst. Electron. Inform. Commun. Eng. 72, 698–701 (1989).

1988 (1)

D. Clark, D. P. Casasent, “Practical optical Fourier analysis for high speed inspection,” Opt. Eng. 27, 365–371 (1988).
[CrossRef]

1987 (1)

1984 (2)

M. S. Brown, “A multifaceted holographic field lens for diffraction pattern sampling,” Opt. Acta 31, 507–513 (1984).
[CrossRef]

M. Kawagoe, A. Tojo, “Fingerprint pattern classification,” Pattern Recogn. 17, 295–303 (1984).
[CrossRef]

1983 (1)

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging MI-2, 31–39 (1983).
[CrossRef]

1980 (1)

1975 (1)

D. H. McMahon, G. L. Johnson, S. L. Teeter, C. G. Whitney, “A hybrid optical computer processing technique for fingerprint identification,” IEEE Trans. Comput. C-24, 358–369 (1975).
[CrossRef]

Asai, K.

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

Baldi, P.

P. Baldi, Y. Chauvin, “Neural networks for fingerprint recognition,” Neural Computa. 5, 402–418 (1993).
[CrossRef]

Beale, M.

H. Demuth, M. Beale, MatLab Neural-Network Application Toolbox, V. 2.0B (Math Works, Natick, Mass., 1994).

Beyer, J.

J. Beyer, C. Lake, R. Lougheed, “Ridge flow determination in fingerprint images,” in 22nd AIPR Workshop: Interdisciplinary Computer Vision: Applications and Changing Needs, J. M. Selander, ed., Proc. SPIE2103, 32–43 (1993).
[CrossRef]

Bjorn, V.

A. Shmurun, V. Bjorn, S. Tam, M. Holler, “Extraction of fingerprint orientation maps using a radial basis function recognition accelerator,” in 1994 IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1994), Vol. 2, pp. 1186–1190.

Blue, J. L.

J. L. Blue, P. J. Grother, “Training feed-forward neural networks using conjugate gradients,” in Machine Vision Applications in Character Recognition and Industrial Inspection, D. P. D’Amato, ed., Proc. SPIE1661, 179–190 (1992).
[CrossRef]

Bock, B.

M. O. Freeman, A. Fedor, B. Bock, K. Duell, “Optical wavelet processor for producing spatially localized ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
[CrossRef]

Bolle, R.

A. K. Jain, L. Hong, S. Pankanti, R. Bolle, “An identity-authentication system using fingerprints,” Proc. IEEE 85, 1365–1388 (1997).
[CrossRef]

Botha, E. C.

L. Coetzee, E. C. Botha, “Fingerprint recognition in low quality images,” Pattern Recogn. 26, 1441–1460 (1993).
[CrossRef]

Bradley, J. N.

T. Hopper, C. M. Brislawn, J. N. Bradley, “WSQ gray-scale fingerprint image compression specification,” (Criminal Justice Information Services, Washington, D.C., 1993).

Brislawn, C. M.

T. Hopper, C. M. Brislawn, J. N. Bradley, “WSQ gray-scale fingerprint image compression specification,” (Criminal Justice Information Services, Washington, D.C., 1993).

Brown, M. S.

M. S. Brown, “A multifaceted holographic field lens for diffraction pattern sampling,” Opt. Acta 31, 507–513 (1984).
[CrossRef]

Byars, P.

A. D. Kulkarni, P. Byars, “Artificial neural network models for image understanding,” in Image Processing Algorithms and Techniques II, M. R. Civanlar, S. K. Mitra, R. J. Moorhead, eds., Proc. SPIE1452, 512–522 (1991).
[CrossRef]

A. D. Kulkarni, P. Byars, “Artificial neural network models for texture classification via the Radon transform,” in Intelligent Robots and Computer Vision X, D. P. Casasent, ed., Proc. SPIE1608, 518–525 (1991).

Candela, G. T.

G. T. Candela, P. J. Grother, C. I. Watson, R. A. Wilkinson, C. L. Wilson, PCASYS (pattern-level classification automation system) public domain software, V. 23-1.1 (National Institute of Standards and Technology, Gaithersburg, Md., 1995).

Casasent, D. P.

D. Clark, D. P. Casasent, “Practical optical Fourier analysis for high speed inspection,” Opt. Eng. 27, 365–371 (1988).
[CrossRef]

Charalambous, C.

C. Charalambous, “Conjugate gradient algorithm for efficient training of artificial neural networks,” Proc. IEEE 139, 301–310 (1992).

Chauvin, Y.

P. Baldi, Y. Chauvin, “Neural networks for fingerprint recognition,” Neural Computa. 5, 402–418 (1993).
[CrossRef]

Chen, S.

N. K. Ratha, S. Chen, A. K. Jain, “Adaptive flow orientation-based feature extraction in fingerprint images,” Pattern Recogn. 28, 1657–1672 (1995).
[CrossRef]

Chen, Z.

Z. Chen, Y. Sun, Y. Zhang, G. Mu, “Hybrid optical/digital access control using fingerprint identification,” Opt. Eng. 34, 834–839 (1995).
[CrossRef]

Clark, D.

D. Clark, D. P. Casasent, “Practical optical Fourier analysis for high speed inspection,” Opt. Eng. 27, 365–371 (1988).
[CrossRef]

Coetzee, L.

L. Coetzee, E. C. Botha, “Fingerprint recognition in low quality images,” Pattern Recogn. 26, 1441–1460 (1993).
[CrossRef]

Demuth, H.

H. Demuth, M. Beale, MatLab Neural-Network Application Toolbox, V. 2.0B (Math Works, Natick, Mass., 1994).

Duell, K.

M. O. Freeman, A. Fedor, B. Bock, K. Duell, “Optical wavelet processor for producing spatially localized ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
[CrossRef]

Fedor, A.

M. O. Freeman, A. Fedor, B. Bock, K. Duell, “Optical wavelet processor for producing spatially localized ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1995).

Freeman, M. O.

M. O. Freeman, A. Fedor, B. Bock, K. Duell, “Optical wavelet processor for producing spatially localized ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
[CrossRef]

Frye, L. M.

Gamble, F. T.

Gaunt, R. G.

M. R. Lynch, R. G. Gaunt, “Applications of linear weight neural networks to fingerprint recognition,” in Fourth International Conference on Artificial Neural Networks (Institution of Electrical Engineers, Cambridge, UK, 1995), pp. 139–142.
[CrossRef]

George, N.

N. George, S.-G. Wang, “Neural networks applied to diffraction-pattern sampling,” Appl. Opt. 33, 3127–3134 (1994).
[CrossRef] [PubMed]

N. George, S.-G. Wang, D. L. Venable, “Pattern recognition using the ring-wedge photodetector and neural-network software,” in Optical Pattern Recognition II, H. J. Caufield, ed., Proc. SPIE1134, 96–106 (1989).
[CrossRef]

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. patent3,689,772 (5September1972).

S.-G. Wang, N. George, “Facial recognition using image and transform representations,” presented at the 1991 Optical Society of America Annual Meeting, San Jose, Calif., 3–8.

Gerhardt, L. A.

P. R. Vizcaya, L. A. Gerhardt, “A nonlinear orientation model for global description of fingerprints,” Pattern Recogn. 29, 1221–1231 (1996).
[CrossRef]

Grieser, D. R.

Grother, P. J.

G. T. Candela, P. J. Grother, C. I. Watson, R. A. Wilkinson, C. L. Wilson, PCASYS (pattern-level classification automation system) public domain software, V. 23-1.1 (National Institute of Standards and Technology, Gaithersburg, Md., 1995).

J. L. Blue, P. J. Grother, “Training feed-forward neural networks using conjugate gradients,” in Machine Vision Applications in Character Recognition and Industrial Inspection, D. P. D’Amato, ed., Proc. SPIE1661, 179–190 (1992).
[CrossRef]

Hamamoto, Y.

T. Kanaoka, M. Watanabe, Y. Hamamoto, S. Tomita, “On a criterion for fingerprint image quality using the autocorrelation,” Trans. Inst. Electron. Inform. Commun. Eng. 72, 698–701 (1989).

Hestenes, M. R.

M. R. Hestenes, Conjugate Direction Methods in Optimization (Springer-Verlag, New York, 1980).
[CrossRef]

Holler, M.

A. Shmurun, V. Bjorn, S. Tam, M. Holler, “Extraction of fingerprint orientation maps using a radial basis function recognition accelerator,” in 1994 IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1994), Vol. 2, pp. 1186–1190.

Hong, L.

A. K. Jain, L. Hong, S. Pankanti, R. Bolle, “An identity-authentication system using fingerprints,” Proc. IEEE 85, 1365–1388 (1997).
[CrossRef]

Hopper, T.

T. Hopper, C. M. Brislawn, J. N. Bradley, “WSQ gray-scale fingerprint image compression specification,” (Criminal Justice Information Services, Washington, D.C., 1993).

Hoshino, Y.

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

Ikebata, S.

K. Sasakawa, F. Isogai, S. Ikebata, “Personal verification system with high tolerance of poor quality fingerprints,” in Machine Vision Systems Integration in Industry, B. G. Batchelor, F. W. Waltz, eds., Proc. SPIE1386, 265–272 (1990).
[CrossRef]

Ishii, M.

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

Isogai, F.

K. Sasakawa, F. Isogai, S. Ikebata, “Personal verification system with high tolerance of poor quality fingerprints,” in Machine Vision Systems Integration in Industry, B. G. Batchelor, F. W. Waltz, eds., Proc. SPIE1386, 265–272 (1990).
[CrossRef]

Jain, A. K.

A. K. Jain, L. Hong, S. Pankanti, R. Bolle, “An identity-authentication system using fingerprints,” Proc. IEEE 85, 1365–1388 (1997).
[CrossRef]

N. K. Ratha, S. Chen, A. K. Jain, “Adaptive flow orientation-based feature extraction in fingerprint images,” Pattern Recogn. 28, 1657–1672 (1995).
[CrossRef]

Johnson, G. L.

D. H. McMahon, G. L. Johnson, S. L. Teeter, C. G. Whitney, “A hybrid optical computer processing technique for fingerprint identification,” IEEE Trans. Comput. C-24, 358–369 (1975).
[CrossRef]

Johnson, K.

J. Ohta, J. Sharpe, K. Johnson, “An optoelectronic smart detector array for the classification of fingerprints,” Opt. Commun. 111, 451–458 (1994).
[CrossRef]

Kamijo, M.

M. Kamijo, H. Mieno, K. Kojima, “Classification of fingerprint images using a neural network,” Syst. Comput. Jpn. 23, 89–101 (1992).
[CrossRef]

Kanaoka, T.

T. Kanaoka, M. Watanabe, Y. Hamamoto, S. Tomita, “On a criterion for fingerprint image quality using the autocorrelation,” Trans. Inst. Electron. Inform. Commun. Eng. 72, 698–701 (1989).

Kawagoe, M.

M. Kawagoe, A. Tojo, “Fingerprint pattern classification,” Pattern Recogn. 17, 295–303 (1984).
[CrossRef]

Kaymaz, E.

E. Kaymaz, S. Mitra, “Analysis and matching of degraded and noisy fingerprints,” in Applications of Digital Image Processing XV, A. G. Tescher, ed., Proc. SPIE1771, 498–509 (1992).
[CrossRef]

Kenyon, R. V.

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging MI-2, 31–39 (1983).
[CrossRef]

Kiji, K.

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

Kojima, K.

M. Kamijo, H. Mieno, K. Kojima, “Classification of fingerprint images using a neural network,” Syst. Comput. Jpn. 23, 89–101 (1992).
[CrossRef]

Krile, T. F.

T. F. Krile, J. F. Walkup, “Enhancement of fingerprints using digital and optical techniques,” in Image Analysis Applications, R. Kasturi, M. M. Trived, eds. (Marcel Dekker, Inc., New York, 1990), pp. 343–371.

Kulkarni, A. D.

A. D. Kulkarni, P. Byars, “Artificial neural network models for image understanding,” in Image Processing Algorithms and Techniques II, M. R. Civanlar, S. K. Mitra, R. J. Moorhead, eds., Proc. SPIE1452, 512–522 (1991).
[CrossRef]

A. D. Kulkarni, P. Byars, “Artificial neural network models for texture classification via the Radon transform,” in Intelligent Robots and Computer Vision X, D. P. Casasent, ed., Proc. SPIE1608, 518–525 (1991).

Lake, C.

J. Beyer, C. Lake, R. Lougheed, “Ridge flow determination in fingerprint images,” in 22nd AIPR Workshop: Interdisciplinary Computer Vision: Applications and Changing Needs, J. M. Selander, ed., Proc. SPIE2103, 32–43 (1993).
[CrossRef]

Lin, Q.

Q. Lin, R. S. Nutter, “A new architecture for automatic fingerprint matching using neural networks as a feature finder and matcher,” in Applications and Science of Artificial Neural Networks, S. K. Rogers, D. W. Ruck, eds., Proc. SPIE2492, 612–623 (1995).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964).

Looney, C. G.

C. G. Looney, “Advances in feedforward neural networks: demystifying knowledge acquiring black boxes,” IEEE Trans. Knowl. Data Eng. 8, 211–226 (1996).
[CrossRef]

Lougheed, R.

J. Beyer, C. Lake, R. Lougheed, “Ridge flow determination in fingerprint images,” in 22nd AIPR Workshop: Interdisciplinary Computer Vision: Applications and Changing Needs, J. M. Selander, ed., Proc. SPIE2103, 32–43 (1993).
[CrossRef]

Lynch, M. R.

M. R. Lynch, R. G. Gaunt, “Applications of linear weight neural networks to fingerprint recognition,” in Fourth International Conference on Artificial Neural Networks (Institution of Electrical Engineers, Cambridge, UK, 1995), pp. 139–142.
[CrossRef]

Masters, T.

T. Masters, Practical Neural Network Recipes in C++ (Academic, New York, 1993).

Matsumoto, H.

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

McClelland, J. L.

D. E. Rumelhart, J. L. McClellandthe PDP Research Group, Parallel Distributed Processing (MIT Press, Cambridge, Mass., 1988), Vols. 1 and 2.

McMahon, D. H.

D. H. McMahon, G. L. Johnson, S. L. Teeter, C. G. Whitney, “A hybrid optical computer processing technique for fingerprint identification,” IEEE Trans. Comput. C-24, 358–369 (1975).
[CrossRef]

Mieno, H.

M. Kamijo, H. Mieno, K. Kojima, “Classification of fingerprint images using a neural network,” Syst. Comput. Jpn. 23, 89–101 (1992).
[CrossRef]

Millard, K.

B. G. Sherlock, D. M. Monro, K. Millard, “Fingerprint enhancement by directional Fourier filtering,” Proc. IEE 141, 87–94 (1994).

Mitra, S.

E. Kaymaz, S. Mitra, “Analysis and matching of degraded and noisy fingerprints,” in Applications of Digital Image Processing XV, A. G. Tescher, ed., Proc. SPIE1771, 498–509 (1992).
[CrossRef]

Monro, D. M.

B. G. Sherlock, D. M. Monro, K. Millard, “Fingerprint enhancement by directional Fourier filtering,” Proc. IEE 141, 87–94 (1994).

B. G. Sherlock, D. M. Monro, “Optimized wavelets for fingerprint compression,” in 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Proceedings (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 1447–1450.
[CrossRef]

Mu, G.

Z. Chen, Y. Sun, Y. Zhang, G. Mu, “Hybrid optical/digital access control using fingerprint identification,” Opt. Eng. 34, 834–839 (1995).
[CrossRef]

Murthy, N. N.

V. S. Srinivasan, N. N. Murthy, “Detection of singular points in fingerprint images,” Pattern Recogn. 25, 139–153 (1992).
[CrossRef]

Narita, S.

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

Nickerson, J. V.

L. O’Gorman, J. V. Nickerson, “An approach to fingerprint filter design,” Pattern Recogn. 22, 29–38 (1989).
[CrossRef]

Nishiyama, A.

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

Nohmi, H.

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

Nutter, R. S.

Q. Lin, R. S. Nutter, “A new architecture for automatic fingerprint matching using neural networks as a feature finder and matcher,” in Applications and Science of Artificial Neural Networks, S. K. Rogers, D. W. Ruck, eds., Proc. SPIE2492, 612–623 (1995).
[CrossRef]

O’Gorman, L.

L. O’Gorman, J. V. Nickerson, “An approach to fingerprint filter design,” Pattern Recogn. 22, 29–38 (1989).
[CrossRef]

O’Toole, R. K.

Ohta, J.

J. Ohta, J. Sharpe, K. Johnson, “An optoelectronic smart detector array for the classification of fingerprints,” Opt. Commun. 111, 451–458 (1994).
[CrossRef]

Pankanti, S.

A. K. Jain, L. Hong, S. Pankanti, R. Bolle, “An identity-authentication system using fingerprints,” Proc. IEEE 85, 1365–1388 (1997).
[CrossRef]

Parker, J. A.

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging MI-2, 31–39 (1983).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1995).

Ratha, N. K.

N. K. Ratha, S. Chen, A. K. Jain, “Adaptive flow orientation-based feature extraction in fingerprint images,” Pattern Recogn. 28, 1657–1672 (1995).
[CrossRef]

Ripley, B. D.

B. D. Ripley, Pattern Recognition and Neural Networks (Cambridge U. Press, Cambridge, 1996).

Rumelhart, D. E.

D. E. Rumelhart, J. L. McClellandthe PDP Research Group, Parallel Distributed Processing (MIT Press, Cambridge, Mass., 1988), Vols. 1 and 2.

Sasakawa, K.

K. Sasakawa, F. Isogai, S. Ikebata, “Personal verification system with high tolerance of poor quality fingerprints,” in Machine Vision Systems Integration in Industry, B. G. Batchelor, F. W. Waltz, eds., Proc. SPIE1386, 265–272 (1990).
[CrossRef]

Seara, R.

F. A. P. Soares, R. Seara, O. J. Tobias, “Neural network applied to direction map extraction in fingerprint images,” in Applications and Science of Artificial Neural Networks, S. K. Rogers, D. W. Ruck, eds., Proc. SPIE2492, 884–891 (1995).
[CrossRef]

Senuma, N.

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

Sharpe, J.

J. Ohta, J. Sharpe, K. Johnson, “An optoelectronic smart detector array for the classification of fingerprints,” Opt. Commun. 111, 451–458 (1994).
[CrossRef]

Sherlock, B. G.

B. G. Sherlock, D. M. Monro, K. Millard, “Fingerprint enhancement by directional Fourier filtering,” Proc. IEE 141, 87–94 (1994).

B. G. Sherlock, D. M. Monro, “Optimized wavelets for fingerprint compression,” in 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Proceedings (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 1447–1450.
[CrossRef]

Shmurun, A.

A. Shmurun, V. Bjorn, S. Tam, M. Holler, “Extraction of fingerprint orientation maps using a radial basis function recognition accelerator,” in 1994 IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1994), Vol. 2, pp. 1186–1190.

Shoha, K.

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

Simpson, M. J.

Soares, F. A. P.

F. A. P. Soares, R. Seara, O. J. Tobias, “Neural network applied to direction map extraction in fingerprint images,” in Applications and Science of Artificial Neural Networks, S. K. Rogers, D. W. Ruck, eds., Proc. SPIE2492, 884–891 (1995).
[CrossRef]

Spindel, A.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. patent3,689,772 (5September1972).

Srinivasan, V. S.

V. S. Srinivasan, N. N. Murthy, “Detection of singular points in fingerprint images,” Pattern Recogn. 25, 139–153 (1992).
[CrossRef]

Stark, H.

Sun, Y.

Z. Chen, Y. Sun, Y. Zhang, G. Mu, “Hybrid optical/digital access control using fingerprint identification,” Opt. Eng. 34, 834–839 (1995).
[CrossRef]

Tam, S.

A. Shmurun, V. Bjorn, S. Tam, M. Holler, “Extraction of fingerprint orientation maps using a radial basis function recognition accelerator,” in 1994 IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1994), Vol. 2, pp. 1186–1190.

Teeter, S. L.

D. H. McMahon, G. L. Johnson, S. L. Teeter, C. G. Whitney, “A hybrid optical computer processing technique for fingerprint identification,” IEEE Trans. Comput. C-24, 358–369 (1975).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1995).

Thomasson, J. T.

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. patent3,689,772 (5September1972).

Thomopoulos, S. C. A.

S. C. A. Thomopoulos, Ver-i-Fus fingerprint access control systems (Intelnet, State College, Pa., 1998).

Tobias, O. J.

F. A. P. Soares, R. Seara, O. J. Tobias, “Neural network applied to direction map extraction in fingerprint images,” in Applications and Science of Artificial Neural Networks, S. K. Rogers, D. W. Ruck, eds., Proc. SPIE2492, 884–891 (1995).
[CrossRef]

Tojo, A.

M. Kawagoe, A. Tojo, “Fingerprint pattern classification,” Pattern Recogn. 17, 295–303 (1984).
[CrossRef]

Tomita, S.

T. Kanaoka, M. Watanabe, Y. Hamamoto, S. Tomita, “On a criterion for fingerprint image quality using the autocorrelation,” Trans. Inst. Electron. Inform. Commun. Eng. 72, 698–701 (1989).

Trenkle, J. M.

J. M. Trenkle, “Region of interest detection for fingerprint classification,” in 22nd AIPR Workshop: Interdisciplinary Computer Vision: Applications and Changing Needs, J. M. Selander, ed., Proc. SPIE2103, 48–59 (1994).
[CrossRef]

Troxel, D. E.

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging MI-2, 31–39 (1983).
[CrossRef]

Venable, D. L.

N. George, S.-G. Wang, D. L. Venable, “Pattern recognition using the ring-wedge photodetector and neural-network software,” in Optical Pattern Recognition II, H. J. Caufield, ed., Proc. SPIE1134, 96–106 (1989).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1995).

Vizcaya, P. R.

P. R. Vizcaya, L. A. Gerhardt, “A nonlinear orientation model for global description of fingerprints,” Pattern Recogn. 29, 1221–1231 (1996).
[CrossRef]

Walkup, J. F.

T. F. Krile, J. F. Walkup, “Enhancement of fingerprints using digital and optical techniques,” in Image Analysis Applications, R. Kasturi, M. M. Trived, eds. (Marcel Dekker, Inc., New York, 1990), pp. 343–371.

Wang, S.-G.

N. George, S.-G. Wang, “Neural networks applied to diffraction-pattern sampling,” Appl. Opt. 33, 3127–3134 (1994).
[CrossRef] [PubMed]

N. George, S.-G. Wang, D. L. Venable, “Pattern recognition using the ring-wedge photodetector and neural-network software,” in Optical Pattern Recognition II, H. J. Caufield, ed., Proc. SPIE1134, 96–106 (1989).
[CrossRef]

S.-G. Wang, N. George, “Facial recognition using image and transform representations,” presented at the 1991 Optical Society of America Annual Meeting, San Jose, Calif., 3–8.

Watanabe, M.

T. Kanaoka, M. Watanabe, Y. Hamamoto, S. Tomita, “On a criterion for fingerprint image quality using the autocorrelation,” Trans. Inst. Electron. Inform. Commun. Eng. 72, 698–701 (1989).

Watson, C. I.

C. I. Watson, C. L. Wilson, NIST Special Database 4, fingerprint database (National Institute of Standards and Technology, Gaithersburg, Md., 1992).

G. T. Candela, P. J. Grother, C. I. Watson, R. A. Wilkinson, C. L. Wilson, PCASYS (pattern-level classification automation system) public domain software, V. 23-1.1 (National Institute of Standards and Technology, Gaithersburg, Md., 1995).

Welstead, S. T.

S. T. Welstead, Neural Network and Fuzzy Logic Application in C/C++ (Wiley, New York, 1994).

Whitney, C. G.

D. H. McMahon, G. L. Johnson, S. L. Teeter, C. G. Whitney, “A hybrid optical computer processing technique for fingerprint identification,” IEEE Trans. Comput. C-24, 358–369 (1975).
[CrossRef]

Wilkinson, R. A.

G. T. Candela, P. J. Grother, C. I. Watson, R. A. Wilkinson, C. L. Wilson, PCASYS (pattern-level classification automation system) public domain software, V. 23-1.1 (National Institute of Standards and Technology, Gaithersburg, Md., 1995).

Wilson, C. L.

G. T. Candela, P. J. Grother, C. I. Watson, R. A. Wilkinson, C. L. Wilson, PCASYS (pattern-level classification automation system) public domain software, V. 23-1.1 (National Institute of Standards and Technology, Gaithersburg, Md., 1995).

C. I. Watson, C. L. Wilson, NIST Special Database 4, fingerprint database (National Institute of Standards and Technology, Gaithersburg, Md., 1992).

Zhang, Y.

Z. Chen, Y. Sun, Y. Zhang, G. Mu, “Hybrid optical/digital access control using fingerprint identification,” Opt. Eng. 34, 834–839 (1995).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Comput. (1)

D. H. McMahon, G. L. Johnson, S. L. Teeter, C. G. Whitney, “A hybrid optical computer processing technique for fingerprint identification,” IEEE Trans. Comput. C-24, 358–369 (1975).
[CrossRef]

IEEE Trans. Knowl. Data Eng. (1)

C. G. Looney, “Advances in feedforward neural networks: demystifying knowledge acquiring black boxes,” IEEE Trans. Knowl. Data Eng. 8, 211–226 (1996).
[CrossRef]

IEEE Trans. Med. Imaging (1)

J. A. Parker, R. V. Kenyon, D. E. Troxel, “Comparison of interpolating methods for image resampling,” IEEE Trans. Med. Imaging MI-2, 31–39 (1983).
[CrossRef]

Neural Computa. (1)

P. Baldi, Y. Chauvin, “Neural networks for fingerprint recognition,” Neural Computa. 5, 402–418 (1993).
[CrossRef]

Nippon Electr. Co. Res. Dev. (1)

H. Matsumoto, S. Narita, M. Ishii, A. Nishiyama, N. Senuma, H. Nohmi, K. Kiji, Y. Hoshino, K. Asai, K. Shoha, “Exclusive use equipment,” Nippon Electr. Co. Res. Dev. 96, 143–159 (1990).

Opt. Acta (1)

M. S. Brown, “A multifaceted holographic field lens for diffraction pattern sampling,” Opt. Acta 31, 507–513 (1984).
[CrossRef]

Opt. Commun. (1)

J. Ohta, J. Sharpe, K. Johnson, “An optoelectronic smart detector array for the classification of fingerprints,” Opt. Commun. 111, 451–458 (1994).
[CrossRef]

Opt. Eng. (2)

D. Clark, D. P. Casasent, “Practical optical Fourier analysis for high speed inspection,” Opt. Eng. 27, 365–371 (1988).
[CrossRef]

Z. Chen, Y. Sun, Y. Zhang, G. Mu, “Hybrid optical/digital access control using fingerprint identification,” Opt. Eng. 34, 834–839 (1995).
[CrossRef]

Pattern Recogn. (6)

L. Coetzee, E. C. Botha, “Fingerprint recognition in low quality images,” Pattern Recogn. 26, 1441–1460 (1993).
[CrossRef]

L. O’Gorman, J. V. Nickerson, “An approach to fingerprint filter design,” Pattern Recogn. 22, 29–38 (1989).
[CrossRef]

P. R. Vizcaya, L. A. Gerhardt, “A nonlinear orientation model for global description of fingerprints,” Pattern Recogn. 29, 1221–1231 (1996).
[CrossRef]

M. Kawagoe, A. Tojo, “Fingerprint pattern classification,” Pattern Recogn. 17, 295–303 (1984).
[CrossRef]

N. K. Ratha, S. Chen, A. K. Jain, “Adaptive flow orientation-based feature extraction in fingerprint images,” Pattern Recogn. 28, 1657–1672 (1995).
[CrossRef]

V. S. Srinivasan, N. N. Murthy, “Detection of singular points in fingerprint images,” Pattern Recogn. 25, 139–153 (1992).
[CrossRef]

Proc. IEE (1)

B. G. Sherlock, D. M. Monro, K. Millard, “Fingerprint enhancement by directional Fourier filtering,” Proc. IEE 141, 87–94 (1994).

Proc. IEEE (2)

A. K. Jain, L. Hong, S. Pankanti, R. Bolle, “An identity-authentication system using fingerprints,” Proc. IEEE 85, 1365–1388 (1997).
[CrossRef]

C. Charalambous, “Conjugate gradient algorithm for efficient training of artificial neural networks,” Proc. IEEE 139, 301–310 (1992).

Syst. Comput. Jpn. (1)

M. Kamijo, H. Mieno, K. Kojima, “Classification of fingerprint images using a neural network,” Syst. Comput. Jpn. 23, 89–101 (1992).
[CrossRef]

Trans. Inst. Electron. Inform. Commun. Eng. (1)

T. Kanaoka, M. Watanabe, Y. Hamamoto, S. Tomita, “On a criterion for fingerprint image quality using the autocorrelation,” Trans. Inst. Electron. Inform. Commun. Eng. 72, 698–701 (1989).

Other (32)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, New York, 1995).

S. C. A. Thomopoulos, Ver-i-Fus fingerprint access control systems (Intelnet, State College, Pa., 1998).

FIDS fingerprint identification system (Jasper Consulting, Bemidji, Minn., 1998).

E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal Press, London, 1964).

J. L. Blue, P. J. Grother, “Training feed-forward neural networks using conjugate gradients,” in Machine Vision Applications in Character Recognition and Industrial Inspection, D. P. D’Amato, ed., Proc. SPIE1661, 179–190 (1992).
[CrossRef]

M. R. Hestenes, Conjugate Direction Methods in Optimization (Springer-Verlag, New York, 1980).
[CrossRef]

T. Masters, Practical Neural Network Recipes in C++ (Academic, New York, 1993).

S. T. Welstead, Neural Network and Fuzzy Logic Application in C/C++ (Wiley, New York, 1994).

C. I. Watson, C. L. Wilson, NIST Special Database 4, fingerprint database (National Institute of Standards and Technology, Gaithersburg, Md., 1992).

B. D. Ripley, Pattern Recognition and Neural Networks (Cambridge U. Press, Cambridge, 1996).

D. E. Rumelhart, J. L. McClellandthe PDP Research Group, Parallel Distributed Processing (MIT Press, Cambridge, Mass., 1988), Vols. 1 and 2.

NeuralWorks Professional II/PLUS neural-network software, V. 5.30 (Neural Ware, Inc., Sewickley, Pa., 1988).

H. Demuth, M. Beale, MatLab Neural-Network Application Toolbox, V. 2.0B (Math Works, Natick, Mass., 1994).

Q. Lin, R. S. Nutter, “A new architecture for automatic fingerprint matching using neural networks as a feature finder and matcher,” in Applications and Science of Artificial Neural Networks, S. K. Rogers, D. W. Ruck, eds., Proc. SPIE2492, 612–623 (1995).
[CrossRef]

T. Hopper, C. M. Brislawn, J. N. Bradley, “WSQ gray-scale fingerprint image compression specification,” (Criminal Justice Information Services, Washington, D.C., 1993).

B. G. Sherlock, D. M. Monro, “Optimized wavelets for fingerprint compression,” in 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Proceedings (Institute of Electrical and Electronics Engineers, New York, 1996), pp. 1447–1450.
[CrossRef]

T. F. Krile, J. F. Walkup, “Enhancement of fingerprints using digital and optical techniques,” in Image Analysis Applications, R. Kasturi, M. M. Trived, eds. (Marcel Dekker, Inc., New York, 1990), pp. 343–371.

K. Sasakawa, F. Isogai, S. Ikebata, “Personal verification system with high tolerance of poor quality fingerprints,” in Machine Vision Systems Integration in Industry, B. G. Batchelor, F. W. Waltz, eds., Proc. SPIE1386, 265–272 (1990).
[CrossRef]

E. Kaymaz, S. Mitra, “Analysis and matching of degraded and noisy fingerprints,” in Applications of Digital Image Processing XV, A. G. Tescher, ed., Proc. SPIE1771, 498–509 (1992).
[CrossRef]

N. George, J. T. Thomasson, A. Spindel, “Photodetector light pattern detector,” U.S. patent3,689,772 (5September1972).

N. George, S.-G. Wang, D. L. Venable, “Pattern recognition using the ring-wedge photodetector and neural-network software,” in Optical Pattern Recognition II, H. J. Caufield, ed., Proc. SPIE1134, 96–106 (1989).
[CrossRef]

Federal Bureau of Investigation, The Science of Fingerprints (U.S. Government Printing Office, Washington, D.C., 1984).

G. T. Candela, P. J. Grother, C. I. Watson, R. A. Wilkinson, C. L. Wilson, PCASYS (pattern-level classification automation system) public domain software, V. 23-1.1 (National Institute of Standards and Technology, Gaithersburg, Md., 1995).

J. Beyer, C. Lake, R. Lougheed, “Ridge flow determination in fingerprint images,” in 22nd AIPR Workshop: Interdisciplinary Computer Vision: Applications and Changing Needs, J. M. Selander, ed., Proc. SPIE2103, 32–43 (1993).
[CrossRef]

A. Shmurun, V. Bjorn, S. Tam, M. Holler, “Extraction of fingerprint orientation maps using a radial basis function recognition accelerator,” in 1994 IEEE International Conference on Neural Networks (Institute of Electrical and Electronics Engineers, New York, 1994), Vol. 2, pp. 1186–1190.

F. A. P. Soares, R. Seara, O. J. Tobias, “Neural network applied to direction map extraction in fingerprint images,” in Applications and Science of Artificial Neural Networks, S. K. Rogers, D. W. Ruck, eds., Proc. SPIE2492, 884–891 (1995).
[CrossRef]

M. R. Lynch, R. G. Gaunt, “Applications of linear weight neural networks to fingerprint recognition,” in Fourth International Conference on Artificial Neural Networks (Institution of Electrical Engineers, Cambridge, UK, 1995), pp. 139–142.
[CrossRef]

J. M. Trenkle, “Region of interest detection for fingerprint classification,” in 22nd AIPR Workshop: Interdisciplinary Computer Vision: Applications and Changing Needs, J. M. Selander, ed., Proc. SPIE2103, 48–59 (1994).
[CrossRef]

A. D. Kulkarni, P. Byars, “Artificial neural network models for image understanding,” in Image Processing Algorithms and Techniques II, M. R. Civanlar, S. K. Mitra, R. J. Moorhead, eds., Proc. SPIE1452, 512–522 (1991).
[CrossRef]

A. D. Kulkarni, P. Byars, “Artificial neural network models for texture classification via the Radon transform,” in Intelligent Robots and Computer Vision X, D. P. Casasent, ed., Proc. SPIE1608, 518–525 (1991).

S.-G. Wang, N. George, “Facial recognition using image and transform representations,” presented at the 1991 Optical Society of America Annual Meeting, San Jose, Calif., 3–8.

M. O. Freeman, A. Fedor, B. Bock, K. Duell, “Optical wavelet processor for producing spatially localized ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) Coherent optoelectronic hybrid processor incorporating the analog ring-wedge detector array and neural-network software: O, input object; L, optical transform lens; R/W, ring-wedge photodetector; NS, neural-network software; C, digital computer; A, amplifier and interface. (b) All-digital Fourier domain feature extractor incorporating the ring-wedge data format and neural-network software: O, input object; L, imaging lens; S, sampling system; FFT, fast Fourier transform; R/W, digital ring-wedge detector; NS, neural-network software.

Fig. 2
Fig. 2

Sampling geometry used by the all-digital ring-wedge detector, incorporating 32 ring and 32 wedge sampling regions chosen to simulate the analog multielement array (figure reproduced from Ref. 15 with permission).

Fig. 3
Fig. 3

Example fingerprint images with corresponding discrete Fourier spectra characterized by bands of frequencies whose angular extents follow from the ridge orientation within each print. (a) Image of a whorl fingerprint characterized by a circular ridge structure. (b) Fourier power spectrum corresponding to (a) and containing a radial band of frequencies. Noise in the image results in significant spectral energy outside this characteristic band. (c) Image of a loop fingerprint. (d) Fourier power spectrum corresponding to (c) and containing radial bands of frequencies with limited angular extent about the direction perpendicular to the principal direction of the ridges in the print.

Fig. 4
Fig. 4

All-digital ring-wedge data for fingerprints in Fig. 3. Left, logarithm of total power versus ring number. Right, total power versus wedge number.

Fig. 5
Fig. 5

Thumbprints used in fingerprint-sorting studies: F1 to F8, left to right and top to bottom.

Fig. 6
Fig. 6

Variations in the locations and orientations of thumbprint images taken from person F1. Print locations vary by approximately 20 pixels, and orientations vary by approximately 10°.

Fig. 7
Fig. 7

Local ridge-orientation determination with a sliding-window ring-wedge transform. (a) Gray-scale fingerprint image for localized frequency analysis. (b) Orientation map. The orientations of ridges in the print are determined from a 32 × 32 pixel window sliding across and down in 8-pixel increments. Ridge orientations are given as the orientation of the mean wedge number.

Fig. 8
Fig. 8

Local quality estimation with a sliding-window ring-wedge transform. (a) Gray-scale fingerprint image for localized frequency analysis. (b) Local quality map. Local quality is determined from a 32 × 32 pixel window sliding across and down in 8-pixel increments. Below a prescribed threshold, quality estimates are given as the correlation coefficient between local ring patterns and the average local ring pattern across the print. Dark regions correspond to regions of lower quality.

Tables (9)

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Table 1 Fingerprint-Sorting Accuracy for Both Ring and Wedge Data from Gray-Scale Imagery for a Data Set of 160 Separate Thumbprints (20 from Each Person) in the Testing Set with 2 Errorsa

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Table 2 Fingerprint-Sorting Accuracy for Only Ring Data from Gray-Scale Imagery for a Data Set of 160 Separate Thumbprints in the Testing Set with 10 Errors

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Table 3 Fingerprint-Sorting Accuracy for Only Wedge Data from Gray-Scale Imagery for a Data Set of 160 Separate Thumbprints in the Testing Set with 7 Errors

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Table 4 Fingerprint-Sorting Accuracy for Binarized Print Imagery Data for a Data Set of 160 Separate Thumbprints in the Testing Set with Zero Errors

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Table 5 Fingerprint-Sorting Accuracy for Only Ring Data from Binarized Imagery for a Data Set of 160 Separate Thumbprints in the Testing Set with 1 Error

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Table 6 Fingerprint-Sorting Accuracy for Only Wedge Data from Binarized Imagery for a Data Set of 160 Separate Thumbprints in the Testing Set with 1 Error

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Table 7 Fingerprint-Sorting Accuracy for Binarized Print Imagery for a Data Set of 160 Separate Thumbprints in the Testing Set as Well as 160 Print Images from Unknown Persons with 0 Errors, 0 False Accepts, and 4 False Rejects

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Table 8 Fingerprint-Verification Results for 8 Independent Trials, Each Using Binarized Imagery from Live Scanned Fingerprints with an Overall False-Accept Rate of 0/160 and an Overall False-Reject Rate of 5/160

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Table 9 Summary of Results for Fingerprint Experiments for the All-Digital Ring-Wedge Detector

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

fx, y=gx, yg*x, y,
Ffx, fy=-  dfxdfyfx, yexp-i2πfxx-i2πfyy,
Ffx, fy=Gfx, fyGfx, fy,
mj=Rjdfxdfy|Ffx, fy|,
|Ffx, fy|=u=-v=- |FuΔfx, vΔfy|×Ifx-uΔfx, fy-vΔfy,
|Ffx, fy|=u=-v=- |FuΔfx, vΔfy|sinc2Xfx-uΔfxsinc2Yfy-vΔfy,
|Ffx, fy|=u=-v=- |FuΔfx, vΔfy|rect1Δfxfx-uΔfx×rect1Δfyfy-vΔfy.
mjRjdfxdfyu=-v=-|FuΔfx, vΔfy|×Ifx-uΔfx, fy-vΔfy.
mju=-v=- |FuΔfx, vΔfy|×RjdfxdfyIfx-uΔfx, fy-vΔfy.
Mju, v=RjdfxdfyIfx-uΔfx, fy-vΔfy,
mju=-v=- |FuΔfx, vΔfy|Mju, v.
FuΔfx, vΔfyF˜u, v=1NMn=0N-1m=0M-1 fnΔx, mΔy×exp-2πunN+vmM,
mju=-N/2N/2-1v=-M/2M/2-1 |F˜u, v|Mju, v,
M˜ju, v=M˜ju-kN, v-IM=Mju, v,
F˜u, v=F˜u-kN, v-IM,
mju=0N-1v=0M-1 |F˜u, v|M˜ju, v.

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