Abstract

We introduce wavelet packet correlation filter classifiers. Correlation filters are traditionally designed in the image domain by minimization of some criterion function of the image training set. Instead, we perform classification in wavelet spaces that have training set representations that provide better solutions to the optimization problem in the filter design. We propose a pruning algorithm to find these wavelet spaces by using a correlation energy cost function, and we describe a match score fusion algorithm for applying the filters trained across the packet tree. The proposed classification algorithm is suitable for any object-recognition task. We present results by implementing a biometric recognition system that uses the NIST 24 fingerprint database, and show that applying correlation filters in the wavelet domain results in considerable improvement of the standard correlation filter algorithm.

© 2005 Optical Society of America

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References

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  1. M. Vetterli, J. Kovačević, Wavelets and Subband Coding (Prentice-Hall, Englewood Cliffs, N.J., 1995).
  2. S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1999).
  3. C. Daniell, A. Mahalanobis, R. Goodman, “Object recognition in subband transform-compressed images by use of correlation filters,” Appl. Opt. 42, 6474–6487 (2003).
    [CrossRef] [PubMed]
  4. K. Venkataramani, “Reduced complexity correlation filters for fingerprint verification,” M.S. thesis (Carnegie Mellon University, Pittsburgh, Pa., 2002).
  5. B. V. K. Vijaya Kumar, M. Savvides, K. Venkataramani, C. Xie, “Spatial frequency domain image processing for biometric recognition,” in Proceedings of IEEE Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 1, pp. 53–56.
    [CrossRef]
  6. M. Savvides, B. V. K. Vijaya Kumar, P. Khosla, “Face verification using correlation filters,” in Proceedings of IEEE Automatic Identification Advanced Technologies (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 56–61.
  7. B. V. K. Vijaya Kumar, J. Thornton, “Distortion-tolerant iris recognition using advanced correlation filters,” in Proceedings of IEEE Multi-Modal User Authentication Workshop (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 173–180.
  8. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  9. B. V. K. Vijaya Kumar, A. Mahalanobis, “Recent advances in composite correlation filter designs,” Asian J. Phys. 8, 407–420 (1999).
  10. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  11. B. V. K. Vijaya Kumar, “Minimum-variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
    [CrossRef]
  12. P. Réfrégier, “Filter design for optical pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
    [CrossRef] [PubMed]
  13. P. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  14. B. V. K. Vijaya Kumar, D. Carlson, A. Mahalanobis, “Optimal trade-off synthetic discriminant function (OTSDF) filters for arbitrary devices,” Opt. Lett. 19, 1556–1558 (1994).
    [CrossRef]
  15. I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
    [CrossRef]
  16. R. Coifman, Y. Meyer, V. Wickerhauser, “Wavelet analysis and signal processing,” in Wavelets and Their Applications, M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, eds. (Jones and Bartlett, Boston, Mass., 1992), pp. 153–178.
  17. C. Watson, NIST Special Database 24—Live-Scan Digital Video Fingerprint Database (National Institutes of Health, Gaithersburg, Md., 1998), available at http://www.nist.gov/srd/nistsd24.htm .
  18. E. Candès, D. Donoho, “Curvelets—a surprisingly effective non-adaptive representation for objects with edges,” in Curve and Surface Fitting, A. Cohen, C. Rabut, L. L. Schumaker, eds. (Vanderbilt U. Press, Nashville, Tenn., 1999), pp. 123–143.
  19. E. Candès, D. Donoho, “Ridgelets and their derivatives: representation of images with edges,” in Curve and Surface Fitting, A. Cohen, C. Rabut, L. L. Schumaker, eds. (Vanderbilt U. Press, Nashville, Tenn., 1999), available at http://www.acm.caltech.edu/~emmanuel/papers/StMalo.pdf .
  20. A. Mahalanobis, B. V. K. Vijaya Kumar, S. Sims, J. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
    [CrossRef] [PubMed]

2003 (1)

1999 (1)

B. V. K. Vijaya Kumar, A. Mahalanobis, “Recent advances in composite correlation filter designs,” Asian J. Phys. 8, 407–420 (1999).

1994 (2)

1992 (1)

1991 (1)

1990 (1)

1987 (1)

1986 (1)

Candès, E.

E. Candès, D. Donoho, “Curvelets—a surprisingly effective non-adaptive representation for objects with edges,” in Curve and Surface Fitting, A. Cohen, C. Rabut, L. L. Schumaker, eds. (Vanderbilt U. Press, Nashville, Tenn., 1999), pp. 123–143.

Carlson, D.

Casasent, D.

Coifman, R.

R. Coifman, Y. Meyer, V. Wickerhauser, “Wavelet analysis and signal processing,” in Wavelets and Their Applications, M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, eds. (Jones and Bartlett, Boston, Mass., 1992), pp. 153–178.

Daniell, C.

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

Donoho, D.

E. Candès, D. Donoho, “Curvelets—a surprisingly effective non-adaptive representation for objects with edges,” in Curve and Surface Fitting, A. Cohen, C. Rabut, L. L. Schumaker, eds. (Vanderbilt U. Press, Nashville, Tenn., 1999), pp. 123–143.

Epperson, J.

Goodman, R.

Khosla, P.

M. Savvides, B. V. K. Vijaya Kumar, P. Khosla, “Face verification using correlation filters,” in Proceedings of IEEE Automatic Identification Advanced Technologies (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 56–61.

Kovacevic, J.

M. Vetterli, J. Kovačević, Wavelets and Subband Coding (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Mahalanobis, A.

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1999).

Meyer, Y.

R. Coifman, Y. Meyer, V. Wickerhauser, “Wavelet analysis and signal processing,” in Wavelets and Their Applications, M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, eds. (Jones and Bartlett, Boston, Mass., 1992), pp. 153–178.

Réfrégier, P.

Savvides, M.

M. Savvides, B. V. K. Vijaya Kumar, P. Khosla, “Face verification using correlation filters,” in Proceedings of IEEE Automatic Identification Advanced Technologies (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 56–61.

B. V. K. Vijaya Kumar, M. Savvides, K. Venkataramani, C. Xie, “Spatial frequency domain image processing for biometric recognition,” in Proceedings of IEEE Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 1, pp. 53–56.
[CrossRef]

Sims, S.

Thornton, J.

B. V. K. Vijaya Kumar, J. Thornton, “Distortion-tolerant iris recognition using advanced correlation filters,” in Proceedings of IEEE Multi-Modal User Authentication Workshop (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 173–180.

Venkataramani, K.

B. V. K. Vijaya Kumar, M. Savvides, K. Venkataramani, C. Xie, “Spatial frequency domain image processing for biometric recognition,” in Proceedings of IEEE Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 1, pp. 53–56.
[CrossRef]

K. Venkataramani, “Reduced complexity correlation filters for fingerprint verification,” M.S. thesis (Carnegie Mellon University, Pittsburgh, Pa., 2002).

Vetterli, M.

M. Vetterli, J. Kovačević, Wavelets and Subband Coding (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Vijaya Kumar, B. V. K.

B. V. K. Vijaya Kumar, A. Mahalanobis, “Recent advances in composite correlation filter designs,” Asian J. Phys. 8, 407–420 (1999).

B. V. K. Vijaya Kumar, D. Carlson, A. Mahalanobis, “Optimal trade-off synthetic discriminant function (OTSDF) filters for arbitrary devices,” Opt. Lett. 19, 1556–1558 (1994).
[CrossRef]

A. Mahalanobis, B. V. K. Vijaya Kumar, S. Sims, J. Epperson, “Unconstrained correlation filters,” Appl. Opt. 33, 3751–3759 (1994).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, “Minimum-variance synthetic discriminant functions,” J. Opt. Soc. Am. A 3, 1579–1584 (1986).
[CrossRef]

B. V. K. Vijaya Kumar, M. Savvides, K. Venkataramani, C. Xie, “Spatial frequency domain image processing for biometric recognition,” in Proceedings of IEEE Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 1, pp. 53–56.
[CrossRef]

M. Savvides, B. V. K. Vijaya Kumar, P. Khosla, “Face verification using correlation filters,” in Proceedings of IEEE Automatic Identification Advanced Technologies (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 56–61.

B. V. K. Vijaya Kumar, J. Thornton, “Distortion-tolerant iris recognition using advanced correlation filters,” in Proceedings of IEEE Multi-Modal User Authentication Workshop (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 173–180.

Wickerhauser, V.

R. Coifman, Y. Meyer, V. Wickerhauser, “Wavelet analysis and signal processing,” in Wavelets and Their Applications, M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, eds. (Jones and Bartlett, Boston, Mass., 1992), pp. 153–178.

Xie, C.

B. V. K. Vijaya Kumar, M. Savvides, K. Venkataramani, C. Xie, “Spatial frequency domain image processing for biometric recognition,” in Proceedings of IEEE Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 1, pp. 53–56.
[CrossRef]

Appl. Opt. (4)

Asian J. Phys. (1)

B. V. K. Vijaya Kumar, A. Mahalanobis, “Recent advances in composite correlation filter designs,” Asian J. Phys. 8, 407–420 (1999).

J. Opt. Soc. Am. A (1)

Opt. Lett. (3)

Other (11)

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

R. Coifman, Y. Meyer, V. Wickerhauser, “Wavelet analysis and signal processing,” in Wavelets and Their Applications, M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, eds. (Jones and Bartlett, Boston, Mass., 1992), pp. 153–178.

C. Watson, NIST Special Database 24—Live-Scan Digital Video Fingerprint Database (National Institutes of Health, Gaithersburg, Md., 1998), available at http://www.nist.gov/srd/nistsd24.htm .

E. Candès, D. Donoho, “Curvelets—a surprisingly effective non-adaptive representation for objects with edges,” in Curve and Surface Fitting, A. Cohen, C. Rabut, L. L. Schumaker, eds. (Vanderbilt U. Press, Nashville, Tenn., 1999), pp. 123–143.

E. Candès, D. Donoho, “Ridgelets and their derivatives: representation of images with edges,” in Curve and Surface Fitting, A. Cohen, C. Rabut, L. L. Schumaker, eds. (Vanderbilt U. Press, Nashville, Tenn., 1999), available at http://www.acm.caltech.edu/~emmanuel/papers/StMalo.pdf .

M. Vetterli, J. Kovačević, Wavelets and Subband Coding (Prentice-Hall, Englewood Cliffs, N.J., 1995).

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1999).

K. Venkataramani, “Reduced complexity correlation filters for fingerprint verification,” M.S. thesis (Carnegie Mellon University, Pittsburgh, Pa., 2002).

B. V. K. Vijaya Kumar, M. Savvides, K. Venkataramani, C. Xie, “Spatial frequency domain image processing for biometric recognition,” in Proceedings of IEEE Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2002), Vol. 1, pp. 53–56.
[CrossRef]

M. Savvides, B. V. K. Vijaya Kumar, P. Khosla, “Face verification using correlation filters,” in Proceedings of IEEE Automatic Identification Advanced Technologies (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 56–61.

B. V. K. Vijaya Kumar, J. Thornton, “Distortion-tolerant iris recognition using advanced correlation filters,” in Proceedings of IEEE Multi-Modal User Authentication Workshop (Institute of Electrical and Electronics Engineers, New York, 2003), pp. 173–180.

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Figures (10)

Fig. 1
Fig. 1

Application of a correlation filter in the frequency domain.

Fig. 2
Fig. 2

MACE-constrained minimization problem: (a) correlation filter to be designed; (b) transforms of the training images; (c) correlation planes that result when the filter is applied to each training image; (d) energy of each correlation plane. The filter in (a) is designed to give specified correlation peaks at the origins of (c) while minimizing the combined energies at (d). Simply stated, the filter is designed to give the sharpest-possible correlation peaks for the within-class training set.

Fig. 3
Fig. 3

(a) Two-channel analysis filter bank structure used for wavelet analysis. (b) Multichannel filter bank algorithm that uses two-channel filter bank blocks iteratively on the low-pass channel (DWT with three levels).

Fig. 4
Fig. 4

(a) Single-level splitting of an image space V0 into four spaces V1, W11, W12, and W13 with quadtree implementation. (b) Quadtree implementation of single-level wavelet packet decomposition.

Fig. 5
Fig. 5

Samples of image decomposition into subspaces with the best wavelet packet for two different fingerprint classes.

Fig. 6
Fig. 6

PCE values for a range of possible horizontal shifts for a sample authentic image. A solid curve is used for the wavelet packet correlation filters with prefiltering, and a dashed curve for those without prefiltering. Shift invariance improves separation between the scores of authentic and impostor images.

Fig. 7
Fig. 7

Samples of fingerprint images from two different classes: a typical class (top row), and a class with particularly difficult distortions (bottom row).

Fig. 8
Fig. 8

Best decompositions for wavelet-domain correlation filters in each class, with classes 1–10 in the top row, and classes 11–20 in the bottom row.

Fig. 9
Fig. 9

Verification ERR by class: SCF, standard correlation filters; WDCF, wavelet-domain correlation filters. SCF average EER = 7.21%; WDCF average EER = 1.18%.

Fig. 10
Fig. 10

Identification error rate (IER) by class: SCF average IER = 18.41%; WDCF average IER = 1.68%.

Tables (2)

Tables Icon

Table 1 Training Algorithm for Each Class

Tables Icon

Table 2 Algorithm for Applying Wavelet Packet Correlation Filters

Equations (19)

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

y ^ = y k             if f ( x , y k ) = min y i f ( x , y i ) ,
a = { yes if f ( x , y k ) T no otherwise ,
v i = ( DFT ) - 1 C i DFT x ,
PCE ( v i ) = max ( v i ) - mean ( v i ) stdev ( v i ) ,
f ( v i ) = 1 PCE ( v i ) .
ACE = h + Dh ,
h = D - 1 X ( X + D - 1 X ) - 1 u ,
ONV = h + Ph .
E ( h ) = α ( ONV ) + β ( ACE ) ,
h = ( α I + β D ) - 1 X ( X + ( α I + β D ) - 1 X ) - 1 u ,
x ( i ) [ n ] = k Z x g ( i + 1 ) [ k ] g [ n - 2 k ] + k Z x h ( i + 1 ) [ k ] h [ n - 2 k ] ,
x g ( i + 1 ) [ n ] = k Z x ( i ) [ k ] g [ k - 2 n ] ,
x h ( i + 1 ) [ n ] = k Z x ( i ) [ k ] h [ k - 2 n ] .
x l = W P l x ,
E ˜ = ( D - 1 X ( X + D - 1 X ) - 1 u ) + D ( D - 1 X ( X + D - 1 X ) - 1 u ) = u T ( X + D - 1 X ) - 1 u .
F = 1 / E ˜ .
F ( V 0 ) > F ( V 1 ) + F ( W 11 ) + F ( W 12 ) + F ( W 13 ) ,
v i , l = ( DFT ) - 1 C i , l DFT W P i , l x ,
f ( v i ) = 1 l PCE ( v i , l ) .

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