Abstract

The Fredholm equation representing light scattering by an ensemble of uniform dielectric spheres is inverted to obtain the particle size distribution responsible for the scattering features. The method of deconvolution involves a constrained expansion of the solution in Schmidt-Hilbert eigenfunctions of the scattering kernels. That solution is obtained which minimizes the sum of the squared residual errors subject to a trial function constraint. The method is, thus, dualistic to the well-known Phillips-Twomey method of constrained linear inversion for the solution by matrix techniques of a Fredholm equation of the first kind in the presence of error. The method is implemented in doubly iterative fashion, and test deconvolutions containing various levels of error are presented.

© 1989 Optical Society of America

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References

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  1. J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1964), pp. 292–297.
  2. C. D. Capps, R. L. Henning, G. M. Hess, “Analytic Inversion of Remote-Sensing Data,” Appl. Opt. 21, 3581 (1982).
    [CrossRef] [PubMed]
  3. S. Twomey, H. B. Howell, “Some Aspects of the Optical Estimation of Microstructure in Fog and Cloud,” Appl. Opt. 6, 2125 (1967).
    [CrossRef] [PubMed]
  4. G. Arfkin, Mathematical Methods for Physicists (Academic, New York, 1970).
  5. G. Viera, M. A. Box, “Information Content Analysis of Aerosol Remote-Sensing Experiments Using Singular Function Theory. 1: Extinction Measurements,” Appl. Opt. 26, 1312 (1987).
    [CrossRef] [PubMed]
  6. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1979).
  7. M. Bertero, C. De Mol, E. R. Pike, “Particle Size Distributions from Spectral Turbidity: A Singular-System Analysis,” Inverse Problems 2, 249 (1986).
    [CrossRef]
  8. D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84 (1962).
    [CrossRef]
  9. S. Twomey, “On the Numerical Solution of Fredholm Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97 (1963).
    [CrossRef]
  10. S. Twomey, “The Application of Numerical Filtering to the Solution of Integral Equations Encountered in Indirect Sensing Measurement,” J. Franklin Inst. 279, 95 (1965).
    [CrossRef]
  11. M. D. King, “Sensitivity of Constrained Linear Inversion to the Selection of the Lagrange Multiplier,” J. Atmos. Sci. 39, 1356 (1982).
    [CrossRef]
  12. M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol Size Distributions Obtained by Inversion of Spectral Optical Measurements,” J. Atmos. Sci. 35, 2153 (1978).
    [CrossRef]
  13. P. Chylek, “Partial-Wave Resonances and the Ripple Structure in the Mie Normalized Extinction Cross Section,” J. Opt. Soc. Am. 66, 285 (1976).
    [CrossRef]
  14. C. F. Bohren, T. J. Nevitt, “Absorption by a Sphere: a Simple Approximation,” Appl. Opt. 22, 774 (1983).
    [CrossRef] [PubMed]
  15. B. P. Curry, “Improvements in the Constrained Eigenfunction Expansion Method to Invert the Particle Size Distribution from Light Scattering Data,” in Proceedings, 1986 CRDEC Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, Ed., Aberdeen Proving Ground, MD, to be published.
  16. N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative Study of Inversion Techniques. Part I: Accuracy and Stability,” J. Appl. Meteorol. 18, 543 (1978).
    [CrossRef]
  17. N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative Study of Inversion Techniques. Part II: Resolving Power, Conservation of Normalization, and Superposition Principles,” J. Appl. Meteorol. 18, 556 (1978).
    [CrossRef]
  18. S. Twomey, “Information Content in Remote Sensing,” Appl. Opt. 13, 942 (1974).
    [CrossRef] [PubMed]
  19. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  20. J. R. Bottiger, “Intercomparison of Some Inversion Methods on Systems of Spherical Particles,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, (Deepak Publishing, Hampton, VA, 1985).
  21. B. P. Curry, E. L. Kiech, “Determination of the Particle Size Distribution from Blind Inversion of Synthetic Data,” in Proceedings, 1985 Chemical Research, Development, and Engineering Center, Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, Ed., Aberdeen Proving Ground, MD, CRDEC-SP-86019 (July1986).
  22. W. A. Pearce, “Aerosol Size Distribution Determination from Scattering Matrix Data,” Applied Systems Department Report 00781, EG&G Washington Analytical Services Center (1982).
  23. R. L. Henning, “Comparative Evaluation of Optical Inversion Techniques,” Report D180-26400-1, Boeing Aerospace Co., Seattle (1981).

1987 (1)

1986 (1)

M. Bertero, C. De Mol, E. R. Pike, “Particle Size Distributions from Spectral Turbidity: A Singular-System Analysis,” Inverse Problems 2, 249 (1986).
[CrossRef]

1983 (1)

1982 (2)

C. D. Capps, R. L. Henning, G. M. Hess, “Analytic Inversion of Remote-Sensing Data,” Appl. Opt. 21, 3581 (1982).
[CrossRef] [PubMed]

M. D. King, “Sensitivity of Constrained Linear Inversion to the Selection of the Lagrange Multiplier,” J. Atmos. Sci. 39, 1356 (1982).
[CrossRef]

1978 (3)

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol Size Distributions Obtained by Inversion of Spectral Optical Measurements,” J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative Study of Inversion Techniques. Part I: Accuracy and Stability,” J. Appl. Meteorol. 18, 543 (1978).
[CrossRef]

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative Study of Inversion Techniques. Part II: Resolving Power, Conservation of Normalization, and Superposition Principles,” J. Appl. Meteorol. 18, 556 (1978).
[CrossRef]

1976 (1)

1974 (1)

1967 (1)

1965 (1)

S. Twomey, “The Application of Numerical Filtering to the Solution of Integral Equations Encountered in Indirect Sensing Measurement,” J. Franklin Inst. 279, 95 (1965).
[CrossRef]

1963 (1)

S. Twomey, “On the Numerical Solution of Fredholm Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

1962 (1)

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

Arfkin, G.

G. Arfkin, Mathematical Methods for Physicists (Academic, New York, 1970).

Bertero, M.

M. Bertero, C. De Mol, E. R. Pike, “Particle Size Distributions from Spectral Turbidity: A Singular-System Analysis,” Inverse Problems 2, 249 (1986).
[CrossRef]

Bohren, C. F.

Bottiger, J. R.

J. R. Bottiger, “Intercomparison of Some Inversion Methods on Systems of Spherical Particles,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, (Deepak Publishing, Hampton, VA, 1985).

Box, M. A.

Byrne, D. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol Size Distributions Obtained by Inversion of Spectral Optical Measurements,” J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

Capps, C. D.

Chahine, M. T.

J. R. Bottiger, “Intercomparison of Some Inversion Methods on Systems of Spherical Particles,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, (Deepak Publishing, Hampton, VA, 1985).

Chylek, P.

Curry, B. P.

B. P. Curry, “Improvements in the Constrained Eigenfunction Expansion Method to Invert the Particle Size Distribution from Light Scattering Data,” in Proceedings, 1986 CRDEC Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, Ed., Aberdeen Proving Ground, MD, to be published.

B. P. Curry, E. L. Kiech, “Determination of the Particle Size Distribution from Blind Inversion of Synthetic Data,” in Proceedings, 1985 Chemical Research, Development, and Engineering Center, Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, Ed., Aberdeen Proving Ground, MD, CRDEC-SP-86019 (July1986).

De Mol, C.

M. Bertero, C. De Mol, E. R. Pike, “Particle Size Distributions from Spectral Turbidity: A Singular-System Analysis,” Inverse Problems 2, 249 (1986).
[CrossRef]

Deepak, A.

J. R. Bottiger, “Intercomparison of Some Inversion Methods on Systems of Spherical Particles,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, (Deepak Publishing, Hampton, VA, 1985).

Fleming, H. E.

J. R. Bottiger, “Intercomparison of Some Inversion Methods on Systems of Spherical Particles,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, (Deepak Publishing, Hampton, VA, 1985).

Henning, R. L.

C. D. Capps, R. L. Henning, G. M. Hess, “Analytic Inversion of Remote-Sensing Data,” Appl. Opt. 21, 3581 (1982).
[CrossRef] [PubMed]

R. L. Henning, “Comparative Evaluation of Optical Inversion Techniques,” Report D180-26400-1, Boeing Aerospace Co., Seattle (1981).

Herman, B. M.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol Size Distributions Obtained by Inversion of Spectral Optical Measurements,” J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

Hess, G. M.

Howell, H. B.

Joseph, J. H.

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative Study of Inversion Techniques. Part I: Accuracy and Stability,” J. Appl. Meteorol. 18, 543 (1978).
[CrossRef]

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative Study of Inversion Techniques. Part II: Resolving Power, Conservation of Normalization, and Superposition Principles,” J. Appl. Meteorol. 18, 556 (1978).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Kiech, E. L.

B. P. Curry, E. L. Kiech, “Determination of the Particle Size Distribution from Blind Inversion of Synthetic Data,” in Proceedings, 1985 Chemical Research, Development, and Engineering Center, Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, Ed., Aberdeen Proving Ground, MD, CRDEC-SP-86019 (July1986).

King, M. D.

M. D. King, “Sensitivity of Constrained Linear Inversion to the Selection of the Lagrange Multiplier,” J. Atmos. Sci. 39, 1356 (1982).
[CrossRef]

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol Size Distributions Obtained by Inversion of Spectral Optical Measurements,” J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

Mathews, J.

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1964), pp. 292–297.

Mekler, Y.

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative Study of Inversion Techniques. Part I: Accuracy and Stability,” J. Appl. Meteorol. 18, 543 (1978).
[CrossRef]

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative Study of Inversion Techniques. Part II: Resolving Power, Conservation of Normalization, and Superposition Principles,” J. Appl. Meteorol. 18, 556 (1978).
[CrossRef]

Nevitt, T. J.

Pearce, W. A.

W. A. Pearce, “Aerosol Size Distribution Determination from Scattering Matrix Data,” Applied Systems Department Report 00781, EG&G Washington Analytical Services Center (1982).

Phillips, D. L.

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

Pike, E. R.

M. Bertero, C. De Mol, E. R. Pike, “Particle Size Distributions from Spectral Turbidity: A Singular-System Analysis,” Inverse Problems 2, 249 (1986).
[CrossRef]

Reagan, J. A.

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol Size Distributions Obtained by Inversion of Spectral Optical Measurements,” J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

Twomey, S.

S. Twomey, “Information Content in Remote Sensing,” Appl. Opt. 13, 942 (1974).
[CrossRef] [PubMed]

S. Twomey, H. B. Howell, “Some Aspects of the Optical Estimation of Microstructure in Fog and Cloud,” Appl. Opt. 6, 2125 (1967).
[CrossRef] [PubMed]

S. Twomey, “The Application of Numerical Filtering to the Solution of Integral Equations Encountered in Indirect Sensing Measurement,” J. Franklin Inst. 279, 95 (1965).
[CrossRef]

S. Twomey, “On the Numerical Solution of Fredholm Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1979).

Viera, G.

Walker, R. L.

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1964), pp. 292–297.

Wolfson, N.

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative Study of Inversion Techniques. Part I: Accuracy and Stability,” J. Appl. Meteorol. 18, 543 (1978).
[CrossRef]

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative Study of Inversion Techniques. Part II: Resolving Power, Conservation of Normalization, and Superposition Principles,” J. Appl. Meteorol. 18, 556 (1978).
[CrossRef]

Appl. Opt. (5)

Inverse Problems (1)

M. Bertero, C. De Mol, E. R. Pike, “Particle Size Distributions from Spectral Turbidity: A Singular-System Analysis,” Inverse Problems 2, 249 (1986).
[CrossRef]

J. Appl. Meteorol. (2)

N. Wolfson, J. H. Joseph, Y. Mekler, “Comparative Study of Inversion Techniques. Part I: Accuracy and Stability,” J. Appl. Meteorol. 18, 543 (1978).
[CrossRef]

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative Study of Inversion Techniques. Part II: Resolving Power, Conservation of Normalization, and Superposition Principles,” J. Appl. Meteorol. 18, 556 (1978).
[CrossRef]

J. Assoc. Comput. Mach. (2)

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

S. Twomey, “On the Numerical Solution of Fredholm Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature,” J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

J. Atmos. Sci. (2)

M. D. King, “Sensitivity of Constrained Linear Inversion to the Selection of the Lagrange Multiplier,” J. Atmos. Sci. 39, 1356 (1982).
[CrossRef]

M. D. King, D. M. Byrne, B. M. Herman, J. A. Reagan, “Aerosol Size Distributions Obtained by Inversion of Spectral Optical Measurements,” J. Atmos. Sci. 35, 2153 (1978).
[CrossRef]

J. Franklin Inst (1)

S. Twomey, “The Application of Numerical Filtering to the Solution of Integral Equations Encountered in Indirect Sensing Measurement,” J. Franklin Inst. 279, 95 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (9)

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

J. R. Bottiger, “Intercomparison of Some Inversion Methods on Systems of Spherical Particles,” in Advances in Remote Sensing Retrieval Methods, A. Deepak, H. E. Fleming, M. T. Chahine, (Deepak Publishing, Hampton, VA, 1985).

B. P. Curry, E. L. Kiech, “Determination of the Particle Size Distribution from Blind Inversion of Synthetic Data,” in Proceedings, 1985 Chemical Research, Development, and Engineering Center, Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, Ed., Aberdeen Proving Ground, MD, CRDEC-SP-86019 (July1986).

W. A. Pearce, “Aerosol Size Distribution Determination from Scattering Matrix Data,” Applied Systems Department Report 00781, EG&G Washington Analytical Services Center (1982).

R. L. Henning, “Comparative Evaluation of Optical Inversion Techniques,” Report D180-26400-1, Boeing Aerospace Co., Seattle (1981).

J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1964), pp. 292–297.

G. Arfkin, Mathematical Methods for Physicists (Academic, New York, 1970).

B. P. Curry, “Improvements in the Constrained Eigenfunction Expansion Method to Invert the Particle Size Distribution from Light Scattering Data,” in Proceedings, 1986 CRDEC Scientific Conference on Obscuration and Aerosol Research, R. H. Kohl, Ed., Aberdeen Proving Ground, MD, to be published.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1979).

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Figures (4)

Fig. 1
Fig. 1

Deconvolution of trial distribution G using 18 synthetic scattering inputs.

Fig. 2
Fig. 2

Deconvolution of trial distribution H using 22 synthetic scattering inputs.

Fig. 3
Fig. 3

Deconvolution of trial distribution I using 30 synthetic scattering inputs.

Fig. 4
Fig. 4

Deconvolution of trial distribution J using 24 synthetic scattering inputs.

Tables (6)

Tables Icon

Table I Rank of Anticorrelatlon and Relative Information Content of Polarization Pairs

Tables Icon

Table II OptimaI lnput Subsets(Wavelengths)

Tables Icon

Table III Deconvolution Relative Residual Error

Tables Icon

Table IV Deconvolution Figure of Error

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Table V Number of Eigenvalues Which Exceed the Squared Residual Errors/Squared Norm

Tables Icon

Table VI Deconvolution Figure of Merit (Based on Eigenvalues Which Exceed Square Error/Square Norm)

Equations (42)

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G ( y i ) + E i = 0 f ( x ) K ( x , y i ) d x .
i   =   1 N E i 2
N i j N ( y i , y j ) = 0 K ( x , y i ) K ( x , y j ) d x .
Φ i ( x ) = λ i 1 / 2 k   =   1 N U i ( y k ) K ( x , y k ) ,
0 Φ i ( x ) Φ j ( x ) d x = δ ( i , j )  ,
i Φ i ( x ) Φ i ( x ) = δ ( x x ) ,
K ( x , y i ) = j   =   1 N λ j 1 / 2 Φ j ( x ) U j ( y i )  .
K + ( x , y i ) = j   =   1 N λ j 1 / 2 Φ j ( x ) U j ( y i ) ,
k = 1 N K + ( x , y k ) K ( x , y k ) = δ ( x x )  .
f 0 ( x ) = k = 1 N K + ( x , y k ) G ( y k ) = i = 1 N k = 1 N λ i 1 / 2 Φ i ( x ) U i ( y k ) G ( y k )  .
f ( x ) = i   =   1 N C i Φ i   ( x )
f ( x ) = i   =   1 N K ( x , y i ) 0 K ( x , y i ) F ( x ) d x .
f ( x ) = i j N i j 1 K ( x , y j ) [ G ( y i ) + E i ] = i 0 M 1 ( x , x ) K ( x , y i ) d x [ G ( y i ) + E i ] ,
M ( x , x ) = i   =   1 N K ( x , y i ) K ( x , y i )
f = M 1 K T ( G + E )  .
f ( x ) f 0 ( x ) = i = 1 N k = 1 N λ i 1 / 2 Φ i ( x ) U i ( y k ) E ( y k )  .
0 | f ( x ) f 0 ( x ) | 2 d x 0 | f ( x ) | 2 d x = i = 1 N λ i 1 k = 1 N U i ( y k ) | E ( y k ) | 2 / i = 1 N C i 2 .
0 | f ( x ) f 0 ( x ) | 2 d x 0 | f ( x ) | 2 d x i = 1 N λ i 1 k = 1 N E ( y k ) 2 / i N C i 2 .
Q = γ 0 H ( x ) 2 d x + i   =   1 N E ( y i ) 2 .
H ( x ) = d 2 f ( x ) d x 2 ,
H ( x ) = f ( x ) f T ( x ) ,
C i = j C j 0 ( δ i j + γ H i j λ j ) 1 ,
H i j = 0 d 2 Φ i ( x ) d x 2 d 2 Φ j ( x ) d x 2 d x ,
C i 0 = λ i 1 / 2 k   =   1 N U i ( y k ) G ( y k ) .
C i = ( γ λ i C i T + C i 0 ) / ( 1 + γ λ i ) ,
C i T = 0 | f T ( x ) | Φ i   ( x ) d x .
f ( x ) f 0 ( x ) = i   =   1 N   Φ i   ( x ) ( 1 + γ λ i )   [ γ λ i ( C i T C i 0 ) + λ i 1 / 2   k   =   1 N   U i   ( y k ) δ G ( y k ) ]  ,
C i 0 = λ i 1 / 2   k   =   1 N U i   ( y k ) G 0 ( y k )  .
E ( y i ) = j   =   1 N C j λ j 1 / 2 U j   ( y i ) G ( y i )  .
E ( y i ) = j   =   1 N   U j   ( y i ) ( 1 + γ λ j )   [ γ λ j 1 / 2 ( C j T C j 0 ) + k   =   1 N U j   ( y k ) δ G ( y k ) ]  .
0 | f ( x ) f 0 ( x )   | 2 d x RRV i   =   1 N { [ γ λ i ( c i T C i 0 ) ( 1 + γ λ i ) ] 2 + 1 λ i k   =   1 N [ Δ G ( y k ) ] 2 ( 1 + γ λ i ) 2 }  ,
i j 1 / 2 2 Q C i C j .
| Δ f ( x ) | 2 = i α i i 1 Φ i 2 ( x )  .
α 1 = β ( I + γ β ) 1 ,
β i j = k   =   1 N λ i 1 / 2 U i   ( y k ) [ Δ G ( y k ) ] 2 U j   ( y k ) λ j 1 / 2 .
| Δ f ( x ) | 2 = i   =   1 N ( λ i 1 ) Φ i ( x ) 2 { k   =   1 N U i 2 ( y k ) [ Δ G ( y k ) ] 2 ( 1 + γ λ i ) k   =   1 N U i 2 ( y k ) [ Δ G ( y k ) ] 2 } .
K w ( x , y i ) = K ( x , y i ) Δ G ( y i ) .
α i j 1 = δ i j γ + γ i w ,
| Δ f w ( x )   | 2 = i   =   1 N ( λ i w ) 1 Φ i w ( x ) 2 1 + γ γ i w .
G ( y i ) + E ( y i ) = 1 N 0 0 K ( x , y i ) f ( x ) d x ,
N 0 = i   =   1 N A i Δ G ( y i ) j   =   1 N C j T λ j 1 / 2 U j   ( y i )  ,
X ( y i , y j ) = 1 N i j ( N i i N j j ) 1 / 2 .

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