Abstract

A generalization of the Karhunen–Loève (KL) transform to Hilbert spaces is developed. It allows one to find the best low-dimensional approximation of an ensemble of images with respect to a variety of distance functions other than the traditional mean square error (L2 norm). A simple and intuitive characterization of the family of Hilbert norms in finite-dimensional spaces leads to an algorithm for calculating the Hilbert-KL expansion. KL approximations of ensembles of objects and faces optimized with respect to a norm based on the modulation transfer function of the human visual system are compared with the standard L2 approximations.

© 1999 Optical Society of America

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References

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  1. H. Murase, S. K. Nayar, “Visual learning and recognition of 3-D objects, from appearance,” Int. J. Comput. Vis. 14, 5–24 (1995).
    [CrossRef]
  2. L. Sirovich, M. Kirbi, “Low-dimensional procedure for characterization of human faces,” J. Opt. Soc. Am. A 4, 519–524 (1987).
    [CrossRef] [PubMed]
  3. J. Mannos, D. Sakrison, “The effect of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inf. Theory 20, 525–536 (1974).
    [CrossRef]
  4. M. Miyahara, “Quality assessments for visual service,” IEEE Commun. Mag. (October1988), pp. 51–60.
  5. J. E. Farell, A. E. Fitzhugh, “Discriminability metric based on human contrast sensitivity,” J. Opt. Soc. Am. A 7, 1976–1984 (1990).
    [CrossRef]
  6. R. A. DeVore, B. Jawerth, B. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
    [CrossRef]
  7. N. Chaddha, H. Y. Meng, “Psycho-visual based distortion measure for monochrome image compression,” in Visual Communications and Image Processing ’93, B. G. Haskell, H.-M. Hang, eds., Proc. SPIE2094, 1680–1690 (1993).
    [CrossRef]
  8. M. Kirby, “Minimal dynamical systems from PDE’s usingSobolev eigenfunctions,” Physica D 57, 466–475 (1992).
    [CrossRef]
  9. B. W. Silverman, “Smoothed functional principal components analysis by choice of norm,” Ann. Statistics 24, 1–24 (1996).
    [CrossRef]
  10. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, Orlando, Fla., 1990).
  11. N. N. Vakhania, V. I. Tarieladze, S. A. Chobaryan, Probability Distribution on Banach Spaces (Reidel, Dordrecht, The Netherlands, 1987).
  12. G. Golub, C. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996).
  13. A. Levy, M. Lindenbaum, “Sequential Karhunen–Loève basis extraction,” preprint, 1998, available from A. Levy, Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel.
  14. E. Biglier, K. Yao, “Some properties of singular value decomposition and their application to digital signal processing,” Signal Process. 18, 277–289 (1989).
    [CrossRef]
  15. M. Salman, M. Lindenbaum, “A layered representation for model-based filtering and recognition,” (Technion–Israel Institute of Technology, Haifa 32000, Israel, 1997).
  16. M. Unser, M. Eder, “Nonlinear operator for improving texture segmentation based on features extracted by spatial filtering,” IEEE Trans. Syst. Man Cybern. 20, 804–815 (1990).
    [CrossRef]

1996 (1)

B. W. Silverman, “Smoothed functional principal components analysis by choice of norm,” Ann. Statistics 24, 1–24 (1996).
[CrossRef]

1995 (1)

H. Murase, S. K. Nayar, “Visual learning and recognition of 3-D objects, from appearance,” Int. J. Comput. Vis. 14, 5–24 (1995).
[CrossRef]

1992 (2)

R. A. DeVore, B. Jawerth, B. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
[CrossRef]

M. Kirby, “Minimal dynamical systems from PDE’s usingSobolev eigenfunctions,” Physica D 57, 466–475 (1992).
[CrossRef]

1990 (2)

J. E. Farell, A. E. Fitzhugh, “Discriminability metric based on human contrast sensitivity,” J. Opt. Soc. Am. A 7, 1976–1984 (1990).
[CrossRef]

M. Unser, M. Eder, “Nonlinear operator for improving texture segmentation based on features extracted by spatial filtering,” IEEE Trans. Syst. Man Cybern. 20, 804–815 (1990).
[CrossRef]

1989 (1)

E. Biglier, K. Yao, “Some properties of singular value decomposition and their application to digital signal processing,” Signal Process. 18, 277–289 (1989).
[CrossRef]

1988 (1)

M. Miyahara, “Quality assessments for visual service,” IEEE Commun. Mag. (October1988), pp. 51–60.

1987 (1)

1974 (1)

J. Mannos, D. Sakrison, “The effect of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inf. Theory 20, 525–536 (1974).
[CrossRef]

Biglier, E.

E. Biglier, K. Yao, “Some properties of singular value decomposition and their application to digital signal processing,” Signal Process. 18, 277–289 (1989).
[CrossRef]

Chaddha, N.

N. Chaddha, H. Y. Meng, “Psycho-visual based distortion measure for monochrome image compression,” in Visual Communications and Image Processing ’93, B. G. Haskell, H.-M. Hang, eds., Proc. SPIE2094, 1680–1690 (1993).
[CrossRef]

Chobaryan, S. A.

N. N. Vakhania, V. I. Tarieladze, S. A. Chobaryan, Probability Distribution on Banach Spaces (Reidel, Dordrecht, The Netherlands, 1987).

DeVore, R. A.

R. A. DeVore, B. Jawerth, B. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
[CrossRef]

Eder, M.

M. Unser, M. Eder, “Nonlinear operator for improving texture segmentation based on features extracted by spatial filtering,” IEEE Trans. Syst. Man Cybern. 20, 804–815 (1990).
[CrossRef]

Farell, J. E.

Fitzhugh, A. E.

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, Orlando, Fla., 1990).

Golub, G.

G. Golub, C. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996).

Jawerth, B.

R. A. DeVore, B. Jawerth, B. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
[CrossRef]

Kirbi, M.

Kirby, M.

M. Kirby, “Minimal dynamical systems from PDE’s usingSobolev eigenfunctions,” Physica D 57, 466–475 (1992).
[CrossRef]

Levy, A.

A. Levy, M. Lindenbaum, “Sequential Karhunen–Loève basis extraction,” preprint, 1998, available from A. Levy, Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel.

A. Levy, M. Lindenbaum, “Sequential Karhunen–Loève basis extraction,” preprint, 1998, available from A. Levy, Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel.

Lindenbaum, M.

A. Levy, M. Lindenbaum, “Sequential Karhunen–Loève basis extraction,” preprint, 1998, available from A. Levy, Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel.

M. Salman, M. Lindenbaum, “A layered representation for model-based filtering and recognition,” (Technion–Israel Institute of Technology, Haifa 32000, Israel, 1997).

Lucier, B. J.

R. A. DeVore, B. Jawerth, B. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
[CrossRef]

Mannos, J.

J. Mannos, D. Sakrison, “The effect of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inf. Theory 20, 525–536 (1974).
[CrossRef]

Meng, H. Y.

N. Chaddha, H. Y. Meng, “Psycho-visual based distortion measure for monochrome image compression,” in Visual Communications and Image Processing ’93, B. G. Haskell, H.-M. Hang, eds., Proc. SPIE2094, 1680–1690 (1993).
[CrossRef]

Miyahara, M.

M. Miyahara, “Quality assessments for visual service,” IEEE Commun. Mag. (October1988), pp. 51–60.

Murase, H.

H. Murase, S. K. Nayar, “Visual learning and recognition of 3-D objects, from appearance,” Int. J. Comput. Vis. 14, 5–24 (1995).
[CrossRef]

Nayar, S. K.

H. Murase, S. K. Nayar, “Visual learning and recognition of 3-D objects, from appearance,” Int. J. Comput. Vis. 14, 5–24 (1995).
[CrossRef]

Sakrison, D.

J. Mannos, D. Sakrison, “The effect of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inf. Theory 20, 525–536 (1974).
[CrossRef]

Salman, M.

M. Salman, M. Lindenbaum, “A layered representation for model-based filtering and recognition,” (Technion–Israel Institute of Technology, Haifa 32000, Israel, 1997).

Silverman, B. W.

B. W. Silverman, “Smoothed functional principal components analysis by choice of norm,” Ann. Statistics 24, 1–24 (1996).
[CrossRef]

Sirovich, L.

Tarieladze, V. I.

N. N. Vakhania, V. I. Tarieladze, S. A. Chobaryan, Probability Distribution on Banach Spaces (Reidel, Dordrecht, The Netherlands, 1987).

Unser, M.

M. Unser, M. Eder, “Nonlinear operator for improving texture segmentation based on features extracted by spatial filtering,” IEEE Trans. Syst. Man Cybern. 20, 804–815 (1990).
[CrossRef]

Vakhania, N. N.

N. N. Vakhania, V. I. Tarieladze, S. A. Chobaryan, Probability Distribution on Banach Spaces (Reidel, Dordrecht, The Netherlands, 1987).

Van Loan, C.

G. Golub, C. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996).

Yao, K.

E. Biglier, K. Yao, “Some properties of singular value decomposition and their application to digital signal processing,” Signal Process. 18, 277–289 (1989).
[CrossRef]

Ann. Statistics (1)

B. W. Silverman, “Smoothed functional principal components analysis by choice of norm,” Ann. Statistics 24, 1–24 (1996).
[CrossRef]

IEEE Commun. Mag. (1)

M. Miyahara, “Quality assessments for visual service,” IEEE Commun. Mag. (October1988), pp. 51–60.

IEEE Trans. Inf. Theory (2)

R. A. DeVore, B. Jawerth, B. J. Lucier, “Image compression through wavelet transform coding,” IEEE Trans. Inf. Theory 38, 719–746 (1992).
[CrossRef]

J. Mannos, D. Sakrison, “The effect of a visual fidelity criterion on the encoding of images,” IEEE Trans. Inf. Theory 20, 525–536 (1974).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

M. Unser, M. Eder, “Nonlinear operator for improving texture segmentation based on features extracted by spatial filtering,” IEEE Trans. Syst. Man Cybern. 20, 804–815 (1990).
[CrossRef]

Int. J. Comput. Vis. (1)

H. Murase, S. K. Nayar, “Visual learning and recognition of 3-D objects, from appearance,” Int. J. Comput. Vis. 14, 5–24 (1995).
[CrossRef]

J. Opt. Soc. Am. A (2)

Physica D (1)

M. Kirby, “Minimal dynamical systems from PDE’s usingSobolev eigenfunctions,” Physica D 57, 466–475 (1992).
[CrossRef]

Signal Process. (1)

E. Biglier, K. Yao, “Some properties of singular value decomposition and their application to digital signal processing,” Signal Process. 18, 277–289 (1989).
[CrossRef]

Other (6)

M. Salman, M. Lindenbaum, “A layered representation for model-based filtering and recognition,” (Technion–Israel Institute of Technology, Haifa 32000, Israel, 1997).

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, Orlando, Fla., 1990).

N. N. Vakhania, V. I. Tarieladze, S. A. Chobaryan, Probability Distribution on Banach Spaces (Reidel, Dordrecht, The Netherlands, 1987).

G. Golub, C. Van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996).

A. Levy, M. Lindenbaum, “Sequential Karhunen–Loève basis extraction,” preprint, 1998, available from A. Levy, Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel.

N. Chaddha, H. Y. Meng, “Psycho-visual based distortion measure for monochrome image compression,” in Visual Communications and Image Processing ’93, B. G. Haskell, H.-M. Hang, eds., Proc. SPIE2094, 1680–1690 (1993).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Approximated objects: upper row, SKLT; lower row, FKLT.

Fig. 2
Fig. 2

Approximated objects: upper row, FKLT; lower row, SKLT.

Fig. 3
Fig. 3

Approximated faces: upper row, SKLT; lower row, FKLT.

Fig. 4
Fig. 4

Approximated faces: upper row, FKLT; lower row, SKLT.

Tables (2)

Tables Icon

Table 1 Comparison of SKLT and FKLT on COIL-20 Objects

Tables Icon

Table 2 Comparison of SKLT and FKLT on ORL Database of Faces

Equations (22)

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

aj(ω)=(fω, uj),j=1, i, , I.
j=1I|aj(ω)|2dω.
F(ϕ)=|(fω, ϕ)|2dω.
F(ϕ)=ϕ, ϕ,fω(x)fω(y)¯dω.
(ϕ, Kϕ)=maxψ=1(ψ, Kψ).
(f, g)=αfg¯dx+βfg¯dx,
ut+4uxxxx+α(uxx+12ux2)=0.
w0u-v2+w1ux-vx2++w4uxxxx-vxxxx2,
ϕ(x)c(x, y)ϕ(y)dxdy,
v, u=(v, Bu),
v, u=(ΣUv, ΣUu).
Kϕ=ϕ, K=(ΣUϕ, ΣUK)=λϕ,
(ΣUKUTΣ)(ΣUϕ)=λ(ΣUϕ).
(ΣUA)(ΣUA)Tv=λv.
d(u, v)=[Fg(u)-Fg(v)]2dxdy,
F(fr)=2.6(0.0192+0.114fr)exp[-(0.114fr)1.1].
u, v=(ΣFUFu, ΣFUFv).
|(fω, ϕ)|2dω=(fω(x), ϕ(x))(fω(y), ϕ(y))¯dω=(ϕ(y), (ϕ(x), fω(x))fω(y))dω=ϕ(y), (ϕ(x), fω(x))fω(y)dω)=ϕ(y),ϕ(x), fω(x)fω(y)¯dω.
(Kh, g)=h(x), fω(x)fω(y)¯dω, g(y)=(h(x), fω(x))(fω(y), g(y))dω=h(x), g(y), fω(y)fω(x)¯dω=(h, Kg).
Kg2=g(x), fω(x)fω(y)¯dω, g(x), fω(x)fω(y)¯dω=((g(x), fω(x))fω(y), (g(x), fω(x))fω(y))dω dω=(g(x), fω(x))(g(x), fω(x))¯×(fω(y), fω(y))dω dωg2fω2dω2,
Kϕn(ϕn(x), fω(x))fωdω.
(ϕn(x), fω(x))fωϕn fω2,

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