Abstract

Two reconstruction methods that recognize smoothness to be a priori information and use numerically space-limited basis functions have been developed. The first method is a modification of the well-known convolution method and uses such basis functions for the projections. The second method is a continuous algebraic reconstruction technique that employs consistent basis functions for the projections as well as the distribution and makes use of other a priori information like the constraints on the domain as well as the range of the distribution in a rigorous way. The efficacy of these methods has been demonstrated using a limited number of projections synthetically generated from a distribution phantom.

© 1988 Optical Society of America

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References

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  1. J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Sachsische Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262 (1917).
  2. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).
  3. R. M. Lewitt, “Reconstruction Algorithms: Transform Methods,” Proc. IEEE 71, 390 (1983).
    [CrossRef]
  4. Y. Censor, “Finite Series-Expansion Reconstruction Methods,” Proc. IEEE 71, 409 (1983).
    [CrossRef]
  5. P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular Absorption Diagnosis of a Turbulent Argon-Methane Jet,” J. Energy 4, 70 (1980).
    [CrossRef]
  6. S. R. Ray, H. G. Semerjian, “Laser Tomography for Simultaneous Concentration and Temperature Measurement in Reacting Flows,” Prog. Astronaut. Aeronaut. 92, 301 (1983).
  7. C. M. Vest, Ivan Prikryl, “Tomography by Iterative Convolution: Empirical Study and Application to Interferometry,” Appl. Opt. 23, 2433 (1984).
    [CrossRef] [PubMed]
  8. D. W. Sweeney, C. M. Vest, “Reconstruction of Three-Dimensional Refractive Index Fields from Multidirectional Interferometric Data,” Appl. Opt. 12, 2649 (1973).
    [CrossRef] [PubMed]
  9. L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1975).
  10. A. Macovski, “Physical Problems of Computerized Tomography,” Proc. IEEE 71, 373 (1983).
    [CrossRef]
  11. G. T. Herman, Image Reconstructions from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).
  12. The Kaiser window, which is an approximation to prolate spheroidal wave functions which are closed form solutions to the mathematical problem of finding a time-limited (or space-limited in the present case) function whose Fourier transform best approximates a band-limited function, has been identified as an ideal candidate for a window (Rabiner and Gold9); and its application to reconstruction problems has previously been suggested by Lewitt.3
  13. It should be noted that the modified inversion formula is not equivalent to the standard CBP inversion formula [Eq. (7)] with W(R) in Eq. (8) replaced by the Kaiser window.
  14. G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Natl. Acad. Sci. USA 68, 2236 (1970).
    [CrossRef]
  15. Profile error is defined as ||frecons − f||/||f − f˜Ω ||, where f˜Ω = ∫Ωf(x,y)dxdy/∫Ωdxdy.
  16. Maximum deviation is defined as maxΩ|frecons − f|.
  17. This method essentially involves implementing Eq. (4) by explicitly computing the forward and inverse Fourier transforms. Its usual implementation is error prone due to interpolation in the Fourier space between polar and Cartesian grids.
  18. R. C. Hansen, “A One-Parameter Aperture Distribution with Narrow Beamwidth and Low Sidelobes,” IEEE Trans. Antennas Propag. AP-24, 477 (1976).
    [CrossRef]
  19. This is equivalent to what is referred to as hexagonal sampling in signal processing.20
  20. D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1984).
  21. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U.P., Baltimore, 1985).
  22. In the context of ARTs, Censor (1983) characterizes typical projection matrices to be ~105 × 105 with <1% nonzero elements. In our test problems, C is ~102 × 102 with ~30% nonzero elements.
  23. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974).
  24. For example, some of the 2-D basis function coefficients representing certain regions of the domain of the distribution may be known to be more important than the others.
  25. C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, Boston, 1984).
  26. Contributions can be stored as GJK(∈RJ×K), which is used in f¯ = GJKf, where f˜ (∈RJ) is the discrete representation of f(x,y) at J locations.
  27. For example, S. No. 4 of Table I with a D matrix size of 102 × 97 and ninety-seven non-negativity constraints required ≈30s for this step on an IBM 4341 machine.
  28. The quantity 1/σi has been factored out from the coefficients as well as their standard deviations.
  29. The projection data are free of external errors, but the use of numerically space-limited basis functions introduces internal errors whose standard deviation is estimated to be ≈1.5 × 10−4.

1984 (1)

1983 (4)

R. M. Lewitt, “Reconstruction Algorithms: Transform Methods,” Proc. IEEE 71, 390 (1983).
[CrossRef]

Y. Censor, “Finite Series-Expansion Reconstruction Methods,” Proc. IEEE 71, 409 (1983).
[CrossRef]

S. R. Ray, H. G. Semerjian, “Laser Tomography for Simultaneous Concentration and Temperature Measurement in Reacting Flows,” Prog. Astronaut. Aeronaut. 92, 301 (1983).

A. Macovski, “Physical Problems of Computerized Tomography,” Proc. IEEE 71, 373 (1983).
[CrossRef]

1980 (1)

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular Absorption Diagnosis of a Turbulent Argon-Methane Jet,” J. Energy 4, 70 (1980).
[CrossRef]

1976 (1)

R. C. Hansen, “A One-Parameter Aperture Distribution with Narrow Beamwidth and Low Sidelobes,” IEEE Trans. Antennas Propag. AP-24, 477 (1976).
[CrossRef]

1973 (1)

1970 (1)

G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Natl. Acad. Sci. USA 68, 2236 (1970).
[CrossRef]

1917 (1)

J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Sachsische Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262 (1917).

Censor, Y.

Y. Censor, “Finite Series-Expansion Reconstruction Methods,” Proc. IEEE 71, 409 (1983).
[CrossRef]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

Dudgeon, D. E.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1984).

Emmerman, P. J.

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular Absorption Diagnosis of a Turbulent Argon-Methane Jet,” J. Energy 4, 70 (1980).
[CrossRef]

Gold, B.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1975).

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U.P., Baltimore, 1985).

Goulard, R.

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular Absorption Diagnosis of a Turbulent Argon-Methane Jet,” J. Energy 4, 70 (1980).
[CrossRef]

Groetsch, C. W.

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, Boston, 1984).

Hansen, R. C.

R. C. Hansen, “A One-Parameter Aperture Distribution with Narrow Beamwidth and Low Sidelobes,” IEEE Trans. Antennas Propag. AP-24, 477 (1976).
[CrossRef]

Hanson, R. J.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974).

Herman, G. T.

G. T. Herman, Image Reconstructions from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).

Lakshminarayanan, A. V.

G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Natl. Acad. Sci. USA 68, 2236 (1970).
[CrossRef]

Lawson, C. L.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974).

Lewitt, R. M.

R. M. Lewitt, “Reconstruction Algorithms: Transform Methods,” Proc. IEEE 71, 390 (1983).
[CrossRef]

Macovski, A.

A. Macovski, “Physical Problems of Computerized Tomography,” Proc. IEEE 71, 373 (1983).
[CrossRef]

Mersereau, R. M.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1984).

Prikryl, Ivan

Rabiner, L. R.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1975).

Radon, J.

J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Sachsische Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262 (1917).

Ramachandran, G. N.

G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Natl. Acad. Sci. USA 68, 2236 (1970).
[CrossRef]

Ray, S. R.

S. R. Ray, H. G. Semerjian, “Laser Tomography for Simultaneous Concentration and Temperature Measurement in Reacting Flows,” Prog. Astronaut. Aeronaut. 92, 301 (1983).

Santoro, R. J.

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular Absorption Diagnosis of a Turbulent Argon-Methane Jet,” J. Energy 4, 70 (1980).
[CrossRef]

Semerjian, H. G.

S. R. Ray, H. G. Semerjian, “Laser Tomography for Simultaneous Concentration and Temperature Measurement in Reacting Flows,” Prog. Astronaut. Aeronaut. 92, 301 (1983).

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular Absorption Diagnosis of a Turbulent Argon-Methane Jet,” J. Energy 4, 70 (1980).
[CrossRef]

Sweeney, D. W.

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U.P., Baltimore, 1985).

Vest, C. M.

Appl. Opt. (2)

Ber. Sachsische Akad. Wiss. Leipzig Math. Phys. Kl. (1)

J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Sachsische Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262 (1917).

IEEE Trans. Antennas Propag. (1)

R. C. Hansen, “A One-Parameter Aperture Distribution with Narrow Beamwidth and Low Sidelobes,” IEEE Trans. Antennas Propag. AP-24, 477 (1976).
[CrossRef]

J. Energy (1)

P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular Absorption Diagnosis of a Turbulent Argon-Methane Jet,” J. Energy 4, 70 (1980).
[CrossRef]

Proc. IEEE (3)

R. M. Lewitt, “Reconstruction Algorithms: Transform Methods,” Proc. IEEE 71, 390 (1983).
[CrossRef]

Y. Censor, “Finite Series-Expansion Reconstruction Methods,” Proc. IEEE 71, 409 (1983).
[CrossRef]

A. Macovski, “Physical Problems of Computerized Tomography,” Proc. IEEE 71, 373 (1983).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Natl. Acad. Sci. USA 68, 2236 (1970).
[CrossRef]

Prog. Astronaut. Aeronaut. (1)

S. R. Ray, H. G. Semerjian, “Laser Tomography for Simultaneous Concentration and Temperature Measurement in Reacting Flows,” Prog. Astronaut. Aeronaut. 92, 301 (1983).

Other (19)

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1975).

Profile error is defined as ||frecons − f||/||f − f˜Ω ||, where f˜Ω = ∫Ωf(x,y)dxdy/∫Ωdxdy.

Maximum deviation is defined as maxΩ|frecons − f|.

This method essentially involves implementing Eq. (4) by explicitly computing the forward and inverse Fourier transforms. Its usual implementation is error prone due to interpolation in the Fourier space between polar and Cartesian grids.

G. T. Herman, Image Reconstructions from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).

The Kaiser window, which is an approximation to prolate spheroidal wave functions which are closed form solutions to the mathematical problem of finding a time-limited (or space-limited in the present case) function whose Fourier transform best approximates a band-limited function, has been identified as an ideal candidate for a window (Rabiner and Gold9); and its application to reconstruction problems has previously been suggested by Lewitt.3

It should be noted that the modified inversion formula is not equivalent to the standard CBP inversion formula [Eq. (7)] with W(R) in Eq. (8) replaced by the Kaiser window.

This is equivalent to what is referred to as hexagonal sampling in signal processing.20

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1984).

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U.P., Baltimore, 1985).

In the context of ARTs, Censor (1983) characterizes typical projection matrices to be ~105 × 105 with <1% nonzero elements. In our test problems, C is ~102 × 102 with ~30% nonzero elements.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974).

For example, some of the 2-D basis function coefficients representing certain regions of the domain of the distribution may be known to be more important than the others.

C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, Boston, 1984).

Contributions can be stored as GJK(∈RJ×K), which is used in f¯ = GJKf, where f˜ (∈RJ) is the discrete representation of f(x,y) at J locations.

For example, S. No. 4 of Table I with a D matrix size of 102 × 97 and ninety-seven non-negativity constraints required ≈30s for this step on an IBM 4341 machine.

The quantity 1/σi has been factored out from the coefficients as well as their standard deviations.

The projection data are free of external errors, but the use of numerically space-limited basis functions introduces internal errors whose standard deviation is estimated to be ≈1.5 × 10−4.

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Figures (14)

Fig. 1
Fig. 1

Coordinate system—Radon transform.

Fig. 2
Fig. 2

Kaiser window for C0 = 5.0, varying C0H0 values.

Fig. 3
Fig. 3

One-dimensional inverse Fourier transform of the Kaiser window for C0 = 5.0, varying C0H0 values. Each curve is normalized with ψ(0).

Fig. 4
Fig. 4

Distribution phantom—original.

Fig. 5
Fig. 5

Comparison between modified CBP with C0H0 = 1.6 and standard CBP with the Ramachandran-Lakshminarayanan discrete convolving kernel.14 Both methods have been implemented using linear interpolation in θ. Number of views, N = 12, and number of projections per view, M = 10. Reconstructed and original distributions sectioned at y = 0.0.

Fig. 6
Fig. 6

Effect of view orientation relative to the grid on the spacing between 1-D basis functions.

Fig. 7
Fig. 7

Characteristic angles and pitch lengths for a triangular grid with Ng = 6 and total number of grid points in the domain K = 97. Because of the symmetry about l = 7, ϕl(°) = 60° − ϕ14−l for l = 8, …, 12.

Fig. 8
Fig. 8

Variation of subcondition number of C with rank deficiency for six angular views with two different experiment designs, both involving 102 projections. See table above for the view angles, projection spacings, and number of projections used in each of the three 60° segments for the designated experiment designs. The legend above reflects the composition of each of the 60° segments and the total number of projections employed in each case. Basis function parameters: C0H0 = 1.6; C0 = 4.2.

Fig. 9
Fig. 9

Variation of subcondition number of C with rank deficiency for varying number of views in different optimized experiment designs. See table above for the view angles, projection spacings, and number of projections used in each of the three 60° segments for the designated experiment designs. The legend above reflects the composition of each of the 60° segments and the total number of projections employed in each case. Design (ii) is identical to that in Fig. 8. Basis function parameters: C0H0 = 1.6; C0 = 4.2.

Fig. 10
Fig. 10

Variation of subcondition number of C with rank deficiency for varying basis function parameters. All curves correspond to experiment design (ii) in Fig. 9.

Fig. 11
Fig. 11

Reconstructed distribution and error in reconstruction for solution by the FDDI method incorporating the use of a priori information. Projections are generated using the phantom in Fig. 4 with other parameters as in S. No. 4 of Table I. Error is defined as freconsf and has been plotted to the same scale as the reconstructed profile.

Fig. 12
Fig. 12

Comparison between FDDI and CBP reconstructions using 105 projections in three views. FDDI reconstruction is obtained using experiment design (i) in Fig. 9 with non-negativity constraints imposed on a 20 × 20 square grid with 317 points inside the domain. CBP reconstruction is obtained as in Fig. 5 but with M = 35 and N = 3.

Fig. 13
Fig. 13

Residual norm vs solution norm for a range of values of the regularization parameter λ with other parameters as in S. No. 3 of Table I.

Fig. 14
Fig. 14

Variation of the coefficients u i T p with index i for projections generated using experiment designs with three and six angular views: (a) distribution phantom in Fig. 4 and experiment design (i) of Fig. 9; (b) same distribution and experiment design (ii) of Fig. 9. For both (a) and (b) the ui values are determined from the SVD of the corresponding D matrices. Also, the values of r marked in these plots have been taken from their respective residual norm–solution norm plots based on the criteria established for obtaining the stable solution in each case.

Tables (1)

Tables Icon

Table I Comparison of Reconstruction Errors—FDDI and CBP Methods Using 102 Projections in Six Views

Equations (54)

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d I ν d t = - k ν I ν ,
p ( s , θ ) = - + f ( s cos θ - t sin θ , s sin θ + t cos θ ) d t = - + - + f ( x , y ) δ ( s - x cos θ - y sin θ ) d x d y = R f ,
f ( x , y ) = R - 1 p = - 1 2 π 2 lim 0 1 q 0 2 π × p ( x cos θ + y sin θ + q , θ ) d θ d q ,
f = F 2 - 1 [ F 1 ( R f ) ] ,
f ( x , y ) = 0 π - + p ^ ( R , θ ) × exp { ι 2 π R ( x cos θ + y sin θ ) } R d R d θ ,
p ˜ ( s , θ ) = Ω p ( s , θ ) q ( s - s ) d s ,
f B ( x , y ) = 0 π p ˜ ( x cos θ + y sin θ , θ ) d θ ,
q ( s ) = - 1 / 2 Δ s + 1 / 2 Δ s R W ( R ) exp ( ι 2 π R s ) d R .
f ( x , y ) f ^ ( x , y ) = j = 1 n x j b j ( x , y ) ,
y i = R i f R i f ^ = j = 1 n x j a i j ,
a i j = R i b j ( x , y ) .
W ( R ) = { I 0 { 2 π H 0 ( C 0 2 - R 2 ) 0.5 } I 0 { 2 π H 0 C 0 } if R < C 0 0 otherwise ,
p ( s , θ ) = m = 1 M p m ( θ ) ψ ( s - m Δ s ) ,
ψ ( s ) = - C 0 + C 0 W ( R ) exp ( ι 2 π R s ) d R = { 2 C 0 I 0 ( 2 π H 0 C 0 ) sin { 2 π C 0 ( s 2 - H 0 2 ) 0.5 } 2 π C 0 ( s 2 - H 0 2 ) 0.5 if s H 0 2 C 0 I 0 ( 2 π H 0 C 0 ) sinh { 2 π C 0 ( H 0 2 - s 2 ) 0.5 } 2 π C 0 ( H 0 2 - s 2 ) 0.5 if s H 0 .
f s ( x , y ) = 0 π m = 1 M p m ( θ ) Ψ ( x cos θ + y sin θ - m Δ s ) d θ ,
Ψ ( s ) = - C 0 + C 0 R W ( R ) exp ( ι 2 π R s ) d R .
p ( s , θ ) = m = 1 M p m ( θ ) ψ ( s - s m ) ,
f ( x , y ) = k = 1 K f k b ( x - x k , y - y k ) .
m = 1 M p m ( θ ) exp ( - ι 2 π R s m ) W ( R ) = k = 1 K f k exp { - ι 2 π R ( x k cos θ + y k sin θ ) } W ( R ) ,
b ( r ) = 0 π - C 0 + C 0 exp ( ι 2 R r cos θ ) W ( R ) R d R d θ = { 2 π C 0 2 I 0 ( 2 π H 0 C 0 ) J 1 { 2 π C 0 ( r 2 - H 0 2 ) 0.5 } 2 π C 0 ( r 2 - H 0 2 ) 0.5 if r H 0 2 π C 0 2 I 0 ( 2 π H 0 C 0 ) I 1 { 2 π C 0 ( H 0 2 - r 2 ) 0.5 } 2 π C 0 ( H 0 2 - r 2 ) 0.5 if r H 0 ,
m = 1 M p m ( θ ) exp ( - ι 2 π R s m ) = k = 1 K f k exp { - ι 2 π R ( x k cos θ + y k sin θ ) } .
p m ( θ ) = k { x k cos θ + y k sin θ = s m } f k ,             m = 1 , , M .
p ( s i , θ j ) = m = 1 M j p m ( θ j ) ψ ( s i - s m ) ,             j = 1 , , N             i = 1 , , M j ,
p m ( θ j ) = k { x k cos θ j + y k sin θ j = s m } f k ,             j = 1 , , N             m = 1 , , M j ,
A p ^ = p ,
A j p ^ j = p j ,             j = 1 , , N ,
A j { a p q [ j ] R M j × M j ,             a p q [ j ] = ψ ( s p - s q ) , p ^ j = { p 1 ( θ j ) , , p M j ( θ j ) } T , p j = { p ( s 1 , θ j ) , , p ( s M j , θ j ) } T .
B f = p ^ ,
B { b p q } R M N × K , b p q = { 1 if x q cos θ p + y q sin θ p = s p 0 otherwise , f R K , with f = { f 1 , , f K } T .
C f = p ,
κ ( C ) κ ( A ) · κ ( B ) ,
C = U Σ V T ,
f LS ( r ) = i = 1 r ( u i T p / σ i ) v i ,
C f - p 2 + λ 2 f 2 .
( C + λ 2 C + T ) f p ,
f LS ( λ ) = i = 1 K u i T p σ i σ i 2 + λ 2 v i .
f LS ( ) - f LS f LS κ ( C ) e p + O ( 2 ) .
min C f - p subject to f Q ,
f ( x , y ) = k = 1 K f k b [ ( x - x k ) 2 + ( y - y k ) 2 ] .
C f = p + p Ω ,
c p q ¬ = 0 if Ω q b Ω ,
D f = p ,
f min f f max ,
f min ( x l , y l ) k = 1 K f k b [ ( x l - x k ) 2 + ( y l - y k ) 2 ] , k = 1 K f k b [ ( x l - x k ) 2 + ( y l - y k ) 2 ] f max ( x l , y l ) .
G f [ G - G ] f h [ f ^ min - f ^ max ] ,
minimize D f - p subject to G f h .
minimize z subject to G ˜ z h ˜ .
minimize E u - v subject to u 0 ,
z ^ = G ˜ T u ^ / ( 1 - h ˜ T u ^ ) .
D = U Σ V T [ U 1 : U 2 ] [ Σ s 0 ] V T , K M N - K
G ˜ = G V s - 1 ,             h ˜ = h - G V s - 1 U 1 T p ,
f LSI = V s - 1 ( z ^ + U 1 T p ) .
f LSI = f LS + V Σ s - 1 z ^ .
f LSI ( λ ) = f LS ( λ ) + V { Σ s ( λ ) } - 1 z ^ ,

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