Abstract

Soft-partition-weighted-sum (Soft-PWS) filters are a class of spatially adaptive moving-window filters for signal and image restoration. Their performance is shown to be promising. However, optimization of the Soft-PWS filters has received only limited attention. Earlier work focused on a stochastic-gradient method that is computationally prohibitive in many applications. We describe a novel radial basis function interpretation of the Soft-PWS filters and present an efficient optimization procedure. We apply the filters to the problem of noise reduction. The experimental results show that the Soft-PWS filter outperforms the standard partition-weighted-sum filter and the Wiener filter.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. R. Castleman, Digital Image Processing (Prentice-Hall, 1995).
  2. K. E. Barner, A. M. Sarhan, and R. C. Hardie, " Partition-based weighted sum filters for image restoration," IEEE Trans. Image Process. 8, 740- 745 ( 1999).
    [CrossRef]
  3. Y. Linde, A. Buzo, and R. M. Gray, " An algorithm for vector quantization," IEEE Trans. Commun. Theory COMM-28, 84- 95 ( 1980).
    [CrossRef]
  4. A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, The Kluwer International Series in Engineering and Computer Science (Kluwer, 1992).
    [CrossRef]
  5. K. E. Barner, G. R. Arce, and J. -H. Lin, " On the performance of stack filters and vector detection in image restoration," Circuits Syst. Signal Process. 11, 153- 169 ( 1992).
    [CrossRef]
  6. K. E. Barner, R. C. Hardie, and G. R. Arce, " On the permutation and quantization partitioning of RN and the filtering problem," presented at the 28th Annual Conference on Information Sciences and Systems, Princeton, N.J., March 1994.
  7. D. G. Sheppard, K. Panchapakesan, A. Bilgin, B. R. Hunt, and M. W. Marcellin, " Lapped nonlinear interpolative vector quantization and image super-resolution," IEEE Trans. Image Process. 9, 295- 298 ( 2000).
    [CrossRef]
  8. E. Abreu, M. Lightstone, S. Mitra, and K. Arakawa, " A new efficient approach for the removal of impulsive noise from highly corrupted images," IEEE Trans. Image Process. 5, 1012- 1025 ( 1996).
    [CrossRef] [PubMed]
  9. T. Chen and H. R. Wu, " Application of partition-based median type filters for suppressing noise in images," IEEE Trans. Image Processing 10, 829- 836 ( 2001).
    [CrossRef]
  10. R. Nakagaki and A. K. Katsaggelos, " A VQ-based blind image restoration algorithm," IEEE Trans. Image Process. 12, 1044- 1053 ( 2003).
    [CrossRef]
  11. Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image deconvolution," presented at the 2005 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2005), Philadelphia, Pa., 18-23 March 2005.
  12. M. Shao and K. E. Barner, " Optimization of partition based weighted sum filters," presented at the IEEE EURASIP Nonlinear Signal and Image Processing (NSIP) Workshop, Baltimore, Md., 3-6 June 2001.
  13. M. Shao and K. E. Barner, " Optimization of partition-based weighted sum filters and their application to image denoising," IEEE Trans. Image Process. (to be published).
    [PubMed]
  14. R. Fletcher, Practical Methods of Optimization (Wiley, 1987).
  15. K. E. Barner and G. R. Arce, " Permutation filters: a class of non-linear filters based on set permutations," IEEE Trans. Signal Process. 42, 782- 798 ( 1994).
    [CrossRef]
  16. Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image restoration," IEEE Signal Process. Lett. 12, 613- 616 ( 2005).
    [CrossRef]
  17. H. Peng, T. Ozaki, V. Haggan-Ozaki, and Y. Toyoda, " A parameter optimization method for radial basis function type models," IEEE Trans. Neural Net. 14, 432- 438 ( 2003).
    [CrossRef]
  18. L. Ungar, T. Johnson, and R. de Veaux, " Radial basis function neural networks for process control," presented at the Rutger's Conference on Computer Integrated Manufacturing in the Process Industries, New Brunswick, N.J., 25-26 April 1994.
  19. T. H. Sidebotham, The A to Z of Mathematics, A Basic Guide (Wiley, 2002).

2005 (1)

Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image restoration," IEEE Signal Process. Lett. 12, 613- 616 ( 2005).
[CrossRef]

2003 (2)

H. Peng, T. Ozaki, V. Haggan-Ozaki, and Y. Toyoda, " A parameter optimization method for radial basis function type models," IEEE Trans. Neural Net. 14, 432- 438 ( 2003).
[CrossRef]

R. Nakagaki and A. K. Katsaggelos, " A VQ-based blind image restoration algorithm," IEEE Trans. Image Process. 12, 1044- 1053 ( 2003).
[CrossRef]

2001 (1)

T. Chen and H. R. Wu, " Application of partition-based median type filters for suppressing noise in images," IEEE Trans. Image Processing 10, 829- 836 ( 2001).
[CrossRef]

2000 (1)

D. G. Sheppard, K. Panchapakesan, A. Bilgin, B. R. Hunt, and M. W. Marcellin, " Lapped nonlinear interpolative vector quantization and image super-resolution," IEEE Trans. Image Process. 9, 295- 298 ( 2000).
[CrossRef]

1999 (1)

K. E. Barner, A. M. Sarhan, and R. C. Hardie, " Partition-based weighted sum filters for image restoration," IEEE Trans. Image Process. 8, 740- 745 ( 1999).
[CrossRef]

1996 (1)

E. Abreu, M. Lightstone, S. Mitra, and K. Arakawa, " A new efficient approach for the removal of impulsive noise from highly corrupted images," IEEE Trans. Image Process. 5, 1012- 1025 ( 1996).
[CrossRef] [PubMed]

1994 (2)

K. E. Barner and G. R. Arce, " Permutation filters: a class of non-linear filters based on set permutations," IEEE Trans. Signal Process. 42, 782- 798 ( 1994).
[CrossRef]

L. Ungar, T. Johnson, and R. de Veaux, " Radial basis function neural networks for process control," presented at the Rutger's Conference on Computer Integrated Manufacturing in the Process Industries, New Brunswick, N.J., 25-26 April 1994.

1992 (1)

K. E. Barner, G. R. Arce, and J. -H. Lin, " On the performance of stack filters and vector detection in image restoration," Circuits Syst. Signal Process. 11, 153- 169 ( 1992).
[CrossRef]

1980 (1)

Y. Linde, A. Buzo, and R. M. Gray, " An algorithm for vector quantization," IEEE Trans. Commun. Theory COMM-28, 84- 95 ( 1980).
[CrossRef]

Abreu, E.

E. Abreu, M. Lightstone, S. Mitra, and K. Arakawa, " A new efficient approach for the removal of impulsive noise from highly corrupted images," IEEE Trans. Image Process. 5, 1012- 1025 ( 1996).
[CrossRef] [PubMed]

Arakawa, K.

E. Abreu, M. Lightstone, S. Mitra, and K. Arakawa, " A new efficient approach for the removal of impulsive noise from highly corrupted images," IEEE Trans. Image Process. 5, 1012- 1025 ( 1996).
[CrossRef] [PubMed]

Arce, G. R.

K. E. Barner and G. R. Arce, " Permutation filters: a class of non-linear filters based on set permutations," IEEE Trans. Signal Process. 42, 782- 798 ( 1994).
[CrossRef]

K. E. Barner, G. R. Arce, and J. -H. Lin, " On the performance of stack filters and vector detection in image restoration," Circuits Syst. Signal Process. 11, 153- 169 ( 1992).
[CrossRef]

K. E. Barner, R. C. Hardie, and G. R. Arce, " On the permutation and quantization partitioning of RN and the filtering problem," presented at the 28th Annual Conference on Information Sciences and Systems, Princeton, N.J., March 1994.

Barner, K. E.

Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image restoration," IEEE Signal Process. Lett. 12, 613- 616 ( 2005).
[CrossRef]

K. E. Barner, A. M. Sarhan, and R. C. Hardie, " Partition-based weighted sum filters for image restoration," IEEE Trans. Image Process. 8, 740- 745 ( 1999).
[CrossRef]

K. E. Barner and G. R. Arce, " Permutation filters: a class of non-linear filters based on set permutations," IEEE Trans. Signal Process. 42, 782- 798 ( 1994).
[CrossRef]

K. E. Barner, G. R. Arce, and J. -H. Lin, " On the performance of stack filters and vector detection in image restoration," Circuits Syst. Signal Process. 11, 153- 169 ( 1992).
[CrossRef]

K. E. Barner, R. C. Hardie, and G. R. Arce, " On the permutation and quantization partitioning of RN and the filtering problem," presented at the 28th Annual Conference on Information Sciences and Systems, Princeton, N.J., March 1994.

Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image deconvolution," presented at the 2005 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2005), Philadelphia, Pa., 18-23 March 2005.

M. Shao and K. E. Barner, " Optimization of partition based weighted sum filters," presented at the IEEE EURASIP Nonlinear Signal and Image Processing (NSIP) Workshop, Baltimore, Md., 3-6 June 2001.

M. Shao and K. E. Barner, " Optimization of partition-based weighted sum filters and their application to image denoising," IEEE Trans. Image Process. (to be published).
[PubMed]

Bilgin, A.

D. G. Sheppard, K. Panchapakesan, A. Bilgin, B. R. Hunt, and M. W. Marcellin, " Lapped nonlinear interpolative vector quantization and image super-resolution," IEEE Trans. Image Process. 9, 295- 298 ( 2000).
[CrossRef]

Buzo, A.

Y. Linde, A. Buzo, and R. M. Gray, " An algorithm for vector quantization," IEEE Trans. Commun. Theory COMM-28, 84- 95 ( 1980).
[CrossRef]

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, 1995).

Chen, T.

T. Chen and H. R. Wu, " Application of partition-based median type filters for suppressing noise in images," IEEE Trans. Image Processing 10, 829- 836 ( 2001).
[CrossRef]

de Veaux, R.

L. Ungar, T. Johnson, and R. de Veaux, " Radial basis function neural networks for process control," presented at the Rutger's Conference on Computer Integrated Manufacturing in the Process Industries, New Brunswick, N.J., 25-26 April 1994.

Fletcher, R.

R. Fletcher, Practical Methods of Optimization (Wiley, 1987).

Gersho, A.

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, The Kluwer International Series in Engineering and Computer Science (Kluwer, 1992).
[CrossRef]

Gray, R. M.

Y. Linde, A. Buzo, and R. M. Gray, " An algorithm for vector quantization," IEEE Trans. Commun. Theory COMM-28, 84- 95 ( 1980).
[CrossRef]

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, The Kluwer International Series in Engineering and Computer Science (Kluwer, 1992).
[CrossRef]

Haggan-Ozaki, V.

H. Peng, T. Ozaki, V. Haggan-Ozaki, and Y. Toyoda, " A parameter optimization method for radial basis function type models," IEEE Trans. Neural Net. 14, 432- 438 ( 2003).
[CrossRef]

Hardie, R. C.

Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image restoration," IEEE Signal Process. Lett. 12, 613- 616 ( 2005).
[CrossRef]

K. E. Barner, A. M. Sarhan, and R. C. Hardie, " Partition-based weighted sum filters for image restoration," IEEE Trans. Image Process. 8, 740- 745 ( 1999).
[CrossRef]

K. E. Barner, R. C. Hardie, and G. R. Arce, " On the permutation and quantization partitioning of RN and the filtering problem," presented at the 28th Annual Conference on Information Sciences and Systems, Princeton, N.J., March 1994.

Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image deconvolution," presented at the 2005 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2005), Philadelphia, Pa., 18-23 March 2005.

Hunt, B. R.

D. G. Sheppard, K. Panchapakesan, A. Bilgin, B. R. Hunt, and M. W. Marcellin, " Lapped nonlinear interpolative vector quantization and image super-resolution," IEEE Trans. Image Process. 9, 295- 298 ( 2000).
[CrossRef]

Johnson, T.

L. Ungar, T. Johnson, and R. de Veaux, " Radial basis function neural networks for process control," presented at the Rutger's Conference on Computer Integrated Manufacturing in the Process Industries, New Brunswick, N.J., 25-26 April 1994.

Katsaggelos, A. K.

R. Nakagaki and A. K. Katsaggelos, " A VQ-based blind image restoration algorithm," IEEE Trans. Image Process. 12, 1044- 1053 ( 2003).
[CrossRef]

Lightstone, M.

E. Abreu, M. Lightstone, S. Mitra, and K. Arakawa, " A new efficient approach for the removal of impulsive noise from highly corrupted images," IEEE Trans. Image Process. 5, 1012- 1025 ( 1996).
[CrossRef] [PubMed]

Lin, J. -H.

K. E. Barner, G. R. Arce, and J. -H. Lin, " On the performance of stack filters and vector detection in image restoration," Circuits Syst. Signal Process. 11, 153- 169 ( 1992).
[CrossRef]

Lin, Y.

Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image restoration," IEEE Signal Process. Lett. 12, 613- 616 ( 2005).
[CrossRef]

Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image deconvolution," presented at the 2005 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2005), Philadelphia, Pa., 18-23 March 2005.

Linde, Y.

Y. Linde, A. Buzo, and R. M. Gray, " An algorithm for vector quantization," IEEE Trans. Commun. Theory COMM-28, 84- 95 ( 1980).
[CrossRef]

Marcellin, M. W.

D. G. Sheppard, K. Panchapakesan, A. Bilgin, B. R. Hunt, and M. W. Marcellin, " Lapped nonlinear interpolative vector quantization and image super-resolution," IEEE Trans. Image Process. 9, 295- 298 ( 2000).
[CrossRef]

Mitra, S.

E. Abreu, M. Lightstone, S. Mitra, and K. Arakawa, " A new efficient approach for the removal of impulsive noise from highly corrupted images," IEEE Trans. Image Process. 5, 1012- 1025 ( 1996).
[CrossRef] [PubMed]

Nakagaki, R.

R. Nakagaki and A. K. Katsaggelos, " A VQ-based blind image restoration algorithm," IEEE Trans. Image Process. 12, 1044- 1053 ( 2003).
[CrossRef]

Ozaki, T.

H. Peng, T. Ozaki, V. Haggan-Ozaki, and Y. Toyoda, " A parameter optimization method for radial basis function type models," IEEE Trans. Neural Net. 14, 432- 438 ( 2003).
[CrossRef]

Panchapakesan, K.

D. G. Sheppard, K. Panchapakesan, A. Bilgin, B. R. Hunt, and M. W. Marcellin, " Lapped nonlinear interpolative vector quantization and image super-resolution," IEEE Trans. Image Process. 9, 295- 298 ( 2000).
[CrossRef]

Peng, H.

H. Peng, T. Ozaki, V. Haggan-Ozaki, and Y. Toyoda, " A parameter optimization method for radial basis function type models," IEEE Trans. Neural Net. 14, 432- 438 ( 2003).
[CrossRef]

Sarhan, A. M.

K. E. Barner, A. M. Sarhan, and R. C. Hardie, " Partition-based weighted sum filters for image restoration," IEEE Trans. Image Process. 8, 740- 745 ( 1999).
[CrossRef]

Shao, M.

M. Shao and K. E. Barner, " Optimization of partition based weighted sum filters," presented at the IEEE EURASIP Nonlinear Signal and Image Processing (NSIP) Workshop, Baltimore, Md., 3-6 June 2001.

M. Shao and K. E. Barner, " Optimization of partition-based weighted sum filters and their application to image denoising," IEEE Trans. Image Process. (to be published).
[PubMed]

Sheppard, D. G.

D. G. Sheppard, K. Panchapakesan, A. Bilgin, B. R. Hunt, and M. W. Marcellin, " Lapped nonlinear interpolative vector quantization and image super-resolution," IEEE Trans. Image Process. 9, 295- 298 ( 2000).
[CrossRef]

Sidebotham, T. H.

T. H. Sidebotham, The A to Z of Mathematics, A Basic Guide (Wiley, 2002).

Toyoda, Y.

H. Peng, T. Ozaki, V. Haggan-Ozaki, and Y. Toyoda, " A parameter optimization method for radial basis function type models," IEEE Trans. Neural Net. 14, 432- 438 ( 2003).
[CrossRef]

Ungar, L.

L. Ungar, T. Johnson, and R. de Veaux, " Radial basis function neural networks for process control," presented at the Rutger's Conference on Computer Integrated Manufacturing in the Process Industries, New Brunswick, N.J., 25-26 April 1994.

Wu, H. R.

T. Chen and H. R. Wu, " Application of partition-based median type filters for suppressing noise in images," IEEE Trans. Image Processing 10, 829- 836 ( 2001).
[CrossRef]

Circuits Syst. Signal Process. (1)

K. E. Barner, G. R. Arce, and J. -H. Lin, " On the performance of stack filters and vector detection in image restoration," Circuits Syst. Signal Process. 11, 153- 169 ( 1992).
[CrossRef]

IEEE Signal Process. Lett. (1)

Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image restoration," IEEE Signal Process. Lett. 12, 613- 616 ( 2005).
[CrossRef]

IEEE Trans. Commun. Theory (1)

Y. Linde, A. Buzo, and R. M. Gray, " An algorithm for vector quantization," IEEE Trans. Commun. Theory COMM-28, 84- 95 ( 1980).
[CrossRef]

IEEE Trans. Image Process. (5)

K. E. Barner, A. M. Sarhan, and R. C. Hardie, " Partition-based weighted sum filters for image restoration," IEEE Trans. Image Process. 8, 740- 745 ( 1999).
[CrossRef]

D. G. Sheppard, K. Panchapakesan, A. Bilgin, B. R. Hunt, and M. W. Marcellin, " Lapped nonlinear interpolative vector quantization and image super-resolution," IEEE Trans. Image Process. 9, 295- 298 ( 2000).
[CrossRef]

E. Abreu, M. Lightstone, S. Mitra, and K. Arakawa, " A new efficient approach for the removal of impulsive noise from highly corrupted images," IEEE Trans. Image Process. 5, 1012- 1025 ( 1996).
[CrossRef] [PubMed]

R. Nakagaki and A. K. Katsaggelos, " A VQ-based blind image restoration algorithm," IEEE Trans. Image Process. 12, 1044- 1053 ( 2003).
[CrossRef]

M. Shao and K. E. Barner, " Optimization of partition-based weighted sum filters and their application to image denoising," IEEE Trans. Image Process. (to be published).
[PubMed]

IEEE Trans. Image Processing (1)

T. Chen and H. R. Wu, " Application of partition-based median type filters for suppressing noise in images," IEEE Trans. Image Processing 10, 829- 836 ( 2001).
[CrossRef]

IEEE Trans. Neural Net. (1)

H. Peng, T. Ozaki, V. Haggan-Ozaki, and Y. Toyoda, " A parameter optimization method for radial basis function type models," IEEE Trans. Neural Net. 14, 432- 438 ( 2003).
[CrossRef]

IEEE Trans. Signal Process. (1)

K. E. Barner and G. R. Arce, " Permutation filters: a class of non-linear filters based on set permutations," IEEE Trans. Signal Process. 42, 782- 798 ( 1994).
[CrossRef]

Other (8)

L. Ungar, T. Johnson, and R. de Veaux, " Radial basis function neural networks for process control," presented at the Rutger's Conference on Computer Integrated Manufacturing in the Process Industries, New Brunswick, N.J., 25-26 April 1994.

T. H. Sidebotham, The A to Z of Mathematics, A Basic Guide (Wiley, 2002).

R. Fletcher, Practical Methods of Optimization (Wiley, 1987).

Y. Lin, R. C. Hardie, and K. E. Barner, " Subspace partition weighted sum filters for image deconvolution," presented at the 2005 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2005), Philadelphia, Pa., 18-23 March 2005.

M. Shao and K. E. Barner, " Optimization of partition based weighted sum filters," presented at the IEEE EURASIP Nonlinear Signal and Image Processing (NSIP) Workshop, Baltimore, Md., 3-6 June 2001.

K. E. Barner, R. C. Hardie, and G. R. Arce, " On the permutation and quantization partitioning of RN and the filtering problem," presented at the 28th Annual Conference on Information Sciences and Systems, Princeton, N.J., March 1994.

K. R. Castleman, Digital Image Processing (Prentice-Hall, 1995).

A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, The Kluwer International Series in Engineering and Computer Science (Kluwer, 1992).
[CrossRef]

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 (8)

Fig. 1
Fig. 1

(Color online) Voronoi regions formed by the VQ partitioning process.

Fig. 2
Fig. 2

(Color online) Contours of the Gaussian partition functions illustrating the soft-partitioning process.

Fig. 3
Fig. 3

Desired training aerial image (434 × 491).

Fig. 4
Fig. 4

Center portions (150 × 150) of (a) a noisy car image, (b) a Wiener filter output, (c) a desired car image, (d) a PWS filter output, and (e) a Soft-PWS filter output with updated w, c, and s (N = 5 × 5, M = 50).

Fig. 5
Fig. 5

Center portions (150 × 150) of (a) a noisy building image, (b) a Wiener filter output, (c) a desired building image, (d) a PWS filter output, and (e) a Soft-PWS filter output with updated w, c, and s (N = 5 × 5, M = 50).

Fig. 6
Fig. 6

(Color online) Mean-square-error results for various noise levels.

Fig. 7
Fig. 7

(Color online) Mean-square-error results for various observation window size.

Fig. 8
Fig. 8

(Color online) Mean-square-error results for various numbers of partitions.

Tables (8)

Tables Icon

Table 1 Optimization Overview when the Cyclic Coordinate Descent Method is Used

Tables Icon

Table 2 Quasi-Newton Optimization Method Overview

Tables Icon

Table 3 Steepest Descent Method Optimization Overview

Tables Icon

Table 4 Mean-Square-Error Results for Car Images

Tables Icon

Table 5 Mean-Square-Error Results for Building Test Images

Tables Icon

Table 6 Mean-Square-Error Results for Tool Test Images

Tables Icon

Table 7 Mean-Square-Error Results for Aerial Images

Tables Icon

Table 8 Filter Training Time and MSE Results

Equations (47)

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

x ( n ) = [ x 1 ( n ) , x 2 ( n ) , , x N ( n ) ] T ,
p ( x ) = arg min i x c i 2 .
F PWS ( x ) = w p ( x ) T x ,
w i * = R i     1 p i ,
F Soft-PWS ( x ) = i = 1 M w i     T x p ^ i ( x ) ,
p ^ i ( x ) = exp ( x c i 2 s i ) k = 1 M exp ( x c k 2 s k )
F Soft-PWS ( x ) = i = 1 M p ^ i ( x ) w i     T x = [ i = 1 M p ^ i ( x ) w i     T ] x = w RBF T x ,
w RBF T = i = 1 M p ^ i ( x ) w i     T .
J ( w , c , s ) = E { ( d F Soft-PWS ) 2 }
= E { d 2 2 d F Soft-PWS + F Soft-PWS 2 }
= E { d 2 } 2 E { d F Soft-PWS } + E { F Soft-PWS 2 }
= σ 2 2 E { d F Soft-PWS } + E { F Soft-PWS 2 } .
F Soft - PWS ( x ) = i = 1 M w i T x ˜ i , = w T x ˜ ,
J ( w , c , s ) = σ 2 2 E { d w T x ˜ } + E { ( w T x ˜ ) 2 } = σ 2 2 w T E { d x ˜ } + w T E { x ˜ x ˜ T } w .
J ( w , c , s ) = σ 2 2 w T p ˜ + w T R ˜ w .
w J ( w , c , s ) = 2 p ˜ + 2 R ˜ w .
w Soft-PWS = R ˜ 1 p ˜ .
p ^ i ( x ) = exp ( β i ) γ ,
β i = - x c i 2 s i ,
γ = k = 1 M exp ( β k ) .
F Soft-PWS ( x ) = i = 1 M exp ( β i ) γ w i     T x .
c i J ( w , c , s ) = 2 E { d c i F Soft-PWS } + 2 E { F Soft-PWS c i F Soft-PWS }
= 2 E { ( d F Soft-PWS ) c i F Soft-PWS } .
c i F Soft-PWS = c i [ i = 1 M exp ( β i ) w i     T x γ ] = c i [ i = 1 M exp ( β i ) w i     T x ( n ) ] γ c i γ i = 1 M exp ( β i ) w i     T x γ 2 = c i [ exp ( β i ) w i     T x ] c i γ i = 1 M exp ( β i ) γ w i     T x γ = c i exp ( β i ) w i     T x c i γ F Soft-PWS γ .
c i exp ( β i ) = β i exp ( β i ) c i β i = exp ( β i ) c i ( x c i 2 s i ) = exp ( β i ) [ 2 ( x c i ) s i ] = 2 exp ( β i ) ( x c i ) s i .
c i γ = c i [ k = 1 M exp ( β k ) ] = c i exp ( β i ) .
c i F Soft-PWS = c i exp ( β i ) w i     T x c i exp ( β i ) F Soft-PWS γ = c i exp ( β i ) ( w i     T x F Soft-PWS ) γ = 2 exp ( β i ) ( x c i ) ( w i     T x F Soft-PWS ) s i γ = 2 ( x c i ) s i ( w i     T x F Soft-PWS ) p ^ i ( x ) .
c i J ( w , c , s ) = 2 E { ( d F Soft-PWS ) 2 ( x c i ) s i × ( w i     T x F Soft-PWS ) p ^ i ( x ) }
= 4 E { ( d F Soft-PWS ) ( x c i ) s i × ( w i     T x F Soft-PWS ) × exp ( x c i 2 s i ) k = 1 M exp ( x c k 2 s k ) } .
J ( w , c , s ) s i = 2 E { d   F Soft-PWS s i } + 2 E { F Soft-PWS F Soft-PWS s i }
= 2 E { ( d F Soft-PWS ) F Soft-PWS s i } .
F Soft-PWS s i =
[ i = 1 M exp ( β i ) w i     T x ] γ s i γ i = 1 M exp ( β i ) w i     T x s i γ 2
= [ exp ( β i ) w i     T x ] s i γ i = 1 M exp ( β i ) γ w i     T x s i γ
= exp ( β i ) w i     T x s i γ F Soft-PWS s i γ .
exp ( β i ) s i = exp ( β i ) β i β i s i
= exp ( β i ) ( x c i 2 s i ) s i
= exp ( β i ) ( x c i 2 s i     2 )
= exp ( β i ) x c i 2 s i     2 .
γ s i = [ k = 1 M exp ( β k ) ] s i
= exp ( β i ) s i .
F Soft-PWS s i = exp ( β i ) w i     T x s i exp ( β i ) F Soft-PWS s i γ
= exp ( β i ) ( w i     T x F Soft-PWS ) s i γ
= exp ( β i ) x c i 2 ( w i     T x F Soft-PWS ) s i     2 γ
= x c i 2 s i     2 ( w i     T x F Soft-PWS ) p ^ i ( x ) .
J ( w , c , s ) s i = 2 E { ( d F Soft-PWS ) x c i 2 s i     2 × ( w i     T x F Soft-PWS ) p ^ i ( x ) }
= 2 E { ( d F Soft-PWS ) x c i 2 s i     2 × ( w i     T x F Soft-PWS ) × exp ( - x c i 2 s i ) k = 1 M exp ( - x c k 2 s k ) } .

Metrics