Abstract

Wavelet networks (WNs) were introduced in 1992 as a combination of artificial neural radial basis function (RBF) networks and wavelet decomposition. Since then, however, WNs have received only a little attention. We believe that the potential of WNs has been generally underestimated. WNs have the advantage that the wavelet coefficients are directly related to the image data through the wavelet transform. In addition, the parameters of the wavelets in the WNs are subject to optimization, which results in a direct relation between the represented function and the optimized wavelets, leading to considerable data reduction (thus making subsequent algorithms much more efficient) as well as to wavelets that can be used as an optimized filter bank. In our study we analyze some WN properties and highlight their advantages for object representation purposes. We then present a series of results of experiments in which we used WNs for face tracking. We exploit the efficiency that is due to data reduction for face recognition and face-pose estimation by applying the optimized-filter-bank principle of the WNs.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Q. Zhang, A. Benveniste, “Wavelet networks,” IEEE Trans. Neural Netw. 3, 889–898 (1992).
    [CrossRef] [PubMed]
  2. H. Szu, B. Telfer, S. Kadambe, “Neural network adaptive wavelets for signal representation and classification,” Opt. Eng. 31, 1907–1961 (1992).
    [CrossRef]
  3. H. Szu, B. Telfer, J. Garcia, “Wavelet transforms and neural networks for compression and recognition,” Neural Networks 9, 695–708 (1996).
    [CrossRef]
  4. Q. Zhang, “Using wavelet network in nonparametric estimation,” IEEE Trans. Neural Netw. 8, 227–236 (1997).
    [CrossRef] [PubMed]
  5. C. C. Holmes, B. K. Mallick, “Bayesian wavelet networks for nonparametric regression,” IEEE Trans. Neural Netw. 11, 27–35 (2000).
    [CrossRef]
  6. L. Reyneri, “Unification of neural and wavelet networks and fuzzy systems,” IEEE Trans. Neural Netw. 10, 801–814 (1999).
    [CrossRef]
  7. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
    [CrossRef]
  8. J. Daugman, “Complete discrete 2D Gabor transform by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
    [CrossRef]
  9. T. S. Lee, “Image representation using 2D Gabor wavelets,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 959–971 (1996).
    [CrossRef]
  10. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).
  11. J. Daugman, “Uncertainty relation for resolution in space, spatial frequency, and orientation optimized two-dimensional visual cortical filters,” J. Opt. Soc. Am. A 2, 1160–1168 (1985).
    [CrossRef] [PubMed]
  12. R. Feris, V. Krüger, R. Cesar, “Efficient real-time face tracking in wavelet subspace,” in Proceedings of the International Workshop on Recognition, Analysis and Tracking of Faces and Gestures in Real-Time Systems (IEEE Computer Society, Santa Ana, Calif., 2001), pp. 113–118.
  13. V. Krüger, G. Sommer, “Affine real-time face tracking using gabor wavelet networks” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society, Santa Ana, Calif., 2000), pp. 127–130.
  14. A. Pentland, “Looking at people: sensing for ubiquitous and sensable computing,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 107–119 (2000).
    [CrossRef]
  15. V. Krüger, S. Bruns, G. Sommer. Efficient head pose estimation with gabor wavelet networks, in Proceedings of the British Machine Vision Conference (British Machine Vision Association, Malverne, UK, 2000), pp. 72–81.
  16. J. Bruske, G. Sommer, “Dynamic cell structure learns perfectly topology preserving map,” Neural Comput. 7, 845–865 (1995).
    [CrossRef]
  17. H. Ritter, T. Martinez, K. Schulten, Neuronale Netze (Addison-Wesley, Reading, Mass., 1991).
  18. V. Krüger, “Gabor wavelet networks for object representation,” (Center for Automation Research, University of Maryland, College Park, Md., 2001).
  19. P. N. Belhumeur, J. P. Hespanha, D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
    [CrossRef]

2000 (2)

C. C. Holmes, B. K. Mallick, “Bayesian wavelet networks for nonparametric regression,” IEEE Trans. Neural Netw. 11, 27–35 (2000).
[CrossRef]

A. Pentland, “Looking at people: sensing for ubiquitous and sensable computing,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 107–119 (2000).
[CrossRef]

1999 (1)

L. Reyneri, “Unification of neural and wavelet networks and fuzzy systems,” IEEE Trans. Neural Netw. 10, 801–814 (1999).
[CrossRef]

1997 (2)

Q. Zhang, “Using wavelet network in nonparametric estimation,” IEEE Trans. Neural Netw. 8, 227–236 (1997).
[CrossRef] [PubMed]

P. N. Belhumeur, J. P. Hespanha, D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
[CrossRef]

1996 (2)

T. S. Lee, “Image representation using 2D Gabor wavelets,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 959–971 (1996).
[CrossRef]

H. Szu, B. Telfer, J. Garcia, “Wavelet transforms and neural networks for compression and recognition,” Neural Networks 9, 695–708 (1996).
[CrossRef]

1995 (1)

J. Bruske, G. Sommer, “Dynamic cell structure learns perfectly topology preserving map,” Neural Comput. 7, 845–865 (1995).
[CrossRef]

1992 (2)

Q. Zhang, A. Benveniste, “Wavelet networks,” IEEE Trans. Neural Netw. 3, 889–898 (1992).
[CrossRef] [PubMed]

H. Szu, B. Telfer, S. Kadambe, “Neural network adaptive wavelets for signal representation and classification,” Opt. Eng. 31, 1907–1961 (1992).
[CrossRef]

1990 (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

1988 (1)

J. Daugman, “Complete discrete 2D Gabor transform by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
[CrossRef]

1985 (1)

Belhumeur, P. N.

P. N. Belhumeur, J. P. Hespanha, D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
[CrossRef]

Benveniste, A.

Q. Zhang, A. Benveniste, “Wavelet networks,” IEEE Trans. Neural Netw. 3, 889–898 (1992).
[CrossRef] [PubMed]

Bruns, S.

V. Krüger, S. Bruns, G. Sommer. Efficient head pose estimation with gabor wavelet networks, in Proceedings of the British Machine Vision Conference (British Machine Vision Association, Malverne, UK, 2000), pp. 72–81.

Bruske, J.

J. Bruske, G. Sommer, “Dynamic cell structure learns perfectly topology preserving map,” Neural Comput. 7, 845–865 (1995).
[CrossRef]

Cesar, R.

R. Feris, V. Krüger, R. Cesar, “Efficient real-time face tracking in wavelet subspace,” in Proceedings of the International Workshop on Recognition, Analysis and Tracking of Faces and Gestures in Real-Time Systems (IEEE Computer Society, Santa Ana, Calif., 2001), pp. 113–118.

Daubechies, I.

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

Daugman, J.

J. Daugman, “Complete discrete 2D Gabor transform by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
[CrossRef]

J. Daugman, “Uncertainty relation for resolution in space, spatial frequency, and orientation optimized two-dimensional visual cortical filters,” J. Opt. Soc. Am. A 2, 1160–1168 (1985).
[CrossRef] [PubMed]

Feris, R.

R. Feris, V. Krüger, R. Cesar, “Efficient real-time face tracking in wavelet subspace,” in Proceedings of the International Workshop on Recognition, Analysis and Tracking of Faces and Gestures in Real-Time Systems (IEEE Computer Society, Santa Ana, Calif., 2001), pp. 113–118.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Garcia, J.

H. Szu, B. Telfer, J. Garcia, “Wavelet transforms and neural networks for compression and recognition,” Neural Networks 9, 695–708 (1996).
[CrossRef]

Hespanha, J. P.

P. N. Belhumeur, J. P. Hespanha, D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
[CrossRef]

Holmes, C. C.

C. C. Holmes, B. K. Mallick, “Bayesian wavelet networks for nonparametric regression,” IEEE Trans. Neural Netw. 11, 27–35 (2000).
[CrossRef]

Kadambe, S.

H. Szu, B. Telfer, S. Kadambe, “Neural network adaptive wavelets for signal representation and classification,” Opt. Eng. 31, 1907–1961 (1992).
[CrossRef]

Kriegman, D. J.

P. N. Belhumeur, J. P. Hespanha, D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
[CrossRef]

Krüger, V.

V. Krüger, “Gabor wavelet networks for object representation,” (Center for Automation Research, University of Maryland, College Park, Md., 2001).

V. Krüger, S. Bruns, G. Sommer. Efficient head pose estimation with gabor wavelet networks, in Proceedings of the British Machine Vision Conference (British Machine Vision Association, Malverne, UK, 2000), pp. 72–81.

V. Krüger, G. Sommer, “Affine real-time face tracking using gabor wavelet networks” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society, Santa Ana, Calif., 2000), pp. 127–130.

R. Feris, V. Krüger, R. Cesar, “Efficient real-time face tracking in wavelet subspace,” in Proceedings of the International Workshop on Recognition, Analysis and Tracking of Faces and Gestures in Real-Time Systems (IEEE Computer Society, Santa Ana, Calif., 2001), pp. 113–118.

Lee, T. S.

T. S. Lee, “Image representation using 2D Gabor wavelets,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 959–971 (1996).
[CrossRef]

Mallick, B. K.

C. C. Holmes, B. K. Mallick, “Bayesian wavelet networks for nonparametric regression,” IEEE Trans. Neural Netw. 11, 27–35 (2000).
[CrossRef]

Martinez, T.

H. Ritter, T. Martinez, K. Schulten, Neuronale Netze (Addison-Wesley, Reading, Mass., 1991).

Pentland, A.

A. Pentland, “Looking at people: sensing for ubiquitous and sensable computing,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 107–119 (2000).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Reyneri, L.

L. Reyneri, “Unification of neural and wavelet networks and fuzzy systems,” IEEE Trans. Neural Netw. 10, 801–814 (1999).
[CrossRef]

Ritter, H.

H. Ritter, T. Martinez, K. Schulten, Neuronale Netze (Addison-Wesley, Reading, Mass., 1991).

Schulten, K.

H. Ritter, T. Martinez, K. Schulten, Neuronale Netze (Addison-Wesley, Reading, Mass., 1991).

Sommer, G.

J. Bruske, G. Sommer, “Dynamic cell structure learns perfectly topology preserving map,” Neural Comput. 7, 845–865 (1995).
[CrossRef]

V. Krüger, S. Bruns, G. Sommer. Efficient head pose estimation with gabor wavelet networks, in Proceedings of the British Machine Vision Conference (British Machine Vision Association, Malverne, UK, 2000), pp. 72–81.

V. Krüger, G. Sommer, “Affine real-time face tracking using gabor wavelet networks” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society, Santa Ana, Calif., 2000), pp. 127–130.

Szu, H.

H. Szu, B. Telfer, J. Garcia, “Wavelet transforms and neural networks for compression and recognition,” Neural Networks 9, 695–708 (1996).
[CrossRef]

H. Szu, B. Telfer, S. Kadambe, “Neural network adaptive wavelets for signal representation and classification,” Opt. Eng. 31, 1907–1961 (1992).
[CrossRef]

Telfer, B.

H. Szu, B. Telfer, J. Garcia, “Wavelet transforms and neural networks for compression and recognition,” Neural Networks 9, 695–708 (1996).
[CrossRef]

H. Szu, B. Telfer, S. Kadambe, “Neural network adaptive wavelets for signal representation and classification,” Opt. Eng. 31, 1907–1961 (1992).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

Zhang, Q.

Q. Zhang, “Using wavelet network in nonparametric estimation,” IEEE Trans. Neural Netw. 8, 227–236 (1997).
[CrossRef] [PubMed]

Q. Zhang, A. Benveniste, “Wavelet networks,” IEEE Trans. Neural Netw. 3, 889–898 (1992).
[CrossRef] [PubMed]

IEEE Trans. Acoust. Speech Signal Process. (1)

J. Daugman, “Complete discrete 2D Gabor transform by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech Signal Process. 36, 1169–1179 (1988).
[CrossRef]

IEEE Trans. Inf. Theory (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

IEEE Trans. Neural Netw. (4)

Q. Zhang, A. Benveniste, “Wavelet networks,” IEEE Trans. Neural Netw. 3, 889–898 (1992).
[CrossRef] [PubMed]

Q. Zhang, “Using wavelet network in nonparametric estimation,” IEEE Trans. Neural Netw. 8, 227–236 (1997).
[CrossRef] [PubMed]

C. C. Holmes, B. K. Mallick, “Bayesian wavelet networks for nonparametric regression,” IEEE Trans. Neural Netw. 11, 27–35 (2000).
[CrossRef]

L. Reyneri, “Unification of neural and wavelet networks and fuzzy systems,” IEEE Trans. Neural Netw. 10, 801–814 (1999).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

T. S. Lee, “Image representation using 2D Gabor wavelets,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 959–971 (1996).
[CrossRef]

A. Pentland, “Looking at people: sensing for ubiquitous and sensable computing,” IEEE Trans. Pattern Anal. Mach. Intell. 22, 107–119 (2000).
[CrossRef]

P. N. Belhumeur, J. P. Hespanha, D. J. Kriegman, “Eigenfaces vs. fisherfaces: recognition using class specific linear projection,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 711–720 (1997).
[CrossRef]

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

Neural Comput. (1)

J. Bruske, G. Sommer, “Dynamic cell structure learns perfectly topology preserving map,” Neural Comput. 7, 845–865 (1995).
[CrossRef]

Neural Networks (1)

H. Szu, B. Telfer, J. Garcia, “Wavelet transforms and neural networks for compression and recognition,” Neural Networks 9, 695–708 (1996).
[CrossRef]

Opt. Eng. (1)

H. Szu, B. Telfer, S. Kadambe, “Neural network adaptive wavelets for signal representation and classification,” Opt. Eng. 31, 1907–1961 (1992).
[CrossRef]

Other (6)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes. The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1986).

R. Feris, V. Krüger, R. Cesar, “Efficient real-time face tracking in wavelet subspace,” in Proceedings of the International Workshop on Recognition, Analysis and Tracking of Faces and Gestures in Real-Time Systems (IEEE Computer Society, Santa Ana, Calif., 2001), pp. 113–118.

V. Krüger, G. Sommer, “Affine real-time face tracking using gabor wavelet networks” in Proceedings of the International Conference on Pattern Recognition (IEEE Computer Society, Santa Ana, Calif., 2000), pp. 127–130.

V. Krüger, S. Bruns, G. Sommer. Efficient head pose estimation with gabor wavelet networks, in Proceedings of the British Machine Vision Conference (British Machine Vision Association, Malverne, UK, 2000), pp. 72–81.

H. Ritter, T. Martinez, K. Schulten, Neuronale Netze (Addison-Wesley, Reading, Mass., 1991).

V. Krüger, “Gabor wavelet networks for object representation,” (Center for Automation Research, University of Maryland, College Park, Md., 2001).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Images indicate the variability in precision with a varying number of wavelets. The wavelets were chosen in the order in which they were optimized.

Fig. 2
Fig. 2

Examples for two different mother wavelets, the odd Gabor function (two left images) and the nonisotropic difference-of-Gaussian (two right images). The images show the reconstruction with 16 wavelets and their superimposed optimized wavelet positions.

Fig. 3
Fig. 3

Function gL2(R2) mapped by the linear mapping Ψ˜ onto the vector wRN in the wavelet subspace. The mapping of w into L2(R2) is achieved with the linear mapping Ψ. Both mappings constitute an orthogonal projection of a function gL2(R2) into the (image) subspace ΨL2(R2).

Fig. 4
Fig. 4

Images of a wooden toy block on which a WN was trained. The black line segments sketch the positions, sizes, and orientations of all the wavelets of the WN. The third image (from left) shows the residual image R between the original image and the approximation by the wavelets. The far-right image sketches the parameters of the largest optimized anisotropic difference-of-Gaussian wavelets.

Fig. 5
Fig. 5

Sample frames of our wavelet subspace tracking experiment. Note that the tracking method is robust to variations of facial expressions as well as to affine deformations of the face image.

Fig. 6
Fig. 6

These images show what happens when for the same individual the optimal coefficients vector is computed with a correct (left) and with a wrong (right) template WN.

Fig. 7
Fig. 7

Various images of subject01 (top) and their projections into the image subspace (bottom). The applied WN was optimized on the “normal” expression of “subject01.”

Fig. 8
Fig. 8

Various images of subjects other than subject01 (top) and their projections into the image subspace (bottom). The applied WN was optimized on the “normal” expression of subject 01.

Fig. 9
Fig. 9

Distance measurements 1/Ψ of the images of the various subjects in the face database to the gallery WN (Ψ01,v01) of subject 01. Higher values indicate a smaller difference between the two compared wavelet coefficient vectors. One sees that the values on the left (subject 01) indicate a much smaller difference than the values on the right (different subjects).

Fig. 10
Fig. 10

Left, original doll face image I; right, its reconstruction I^52 by formula (3) with an optimized WN Ψ of just N=52 odd Gabor wavelets, distributed over the inner face region.

Fig. 11
Fig. 11

Different orientations of the doll’s head. The head is connected to a robot arm so that the ground truth is known. The white square indicates the detected position, scale, and orientation of the WN.

Equations (24)

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

ψn(x)=ψ(SR(x-t)).
E=minni,wi for allif-i=1Nwiψni22
Ψ=(ψn1 ,, ψnN)T and w=(w1 ,, wN)T
fˆ=i=1Nwiψni=ΨTw.
ψni, ψ˜nj=δi,j.
[Ψ, Ψ˜]=I,
ψ˜ni=j(Ψ)i,j-1ψnj,
gˆ=i=1Nwiψni,w=Ψ˜g.
f-i=1nwiψni>f-i=1nwiψni-wn+1ψnn+1.
ψn1 ,, ψnn=ψn1 ,, ψnn+1,
ψn1 ,, ψnn=ψn1 ,, ψnn+1.
f-i=1nwiψni(ψn1 ,, ψnn+1),
f-i=1nwiψni, ψnn+1=0.
f-i=1nwiψni, ψnn+10.
w=Ψ˜g
i=1Nviψni-i=1Nwiψni2,
v-wΨ  i,j(vi-wi)(vj-wj)ψni, ψnj1/2
=(v-w)t(Ψ)i,j(v-w).
R=f-wiψni.
x[R(x)-wnewψnnew(x)]2=min
xR(x)ψnnew(x)=max
Iˆ(SR(x-t))=i=1Nviψni(SR(x-t)),
v-wΨ,
E=minS,R,tv-wΨ

Metrics