Abstract

A theory of partially coherent imaging is presented. In this theory, a singular matrix P is introduced in a spatial frequency domain. The matrix P can be obtained by stacking pupil functions that are shifted according to the illumination condition. Applying singular-value decomposition to the matrix P generates eigenvalues and eigenfunctions. Using eigenvalues and eigenfunctions, the aerial image can be computed without the transmission cross coefficient (TCC). A notable feature of the matrix P is that the relationship between the matrix P and the TCC matrix T is T=PP, where † represents the Hermitian conjugate. This suggests that the matrix P can be regarded as a fundamental operator in partially coherent imaging.

© 2008 Optical Society of America

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  1. H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London, Ser. A 217, 408-432 (1953).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.
  3. E. Kintner, “Method for the calculation of partially coherent imagery,” Appl. Opt. 17, 2747-2753 (1978).
    [CrossRef] [PubMed]
  4. R. Barakat, “Partially coherent imagery in the presence of aberrations,” Opt. Acta 17, 337-347 (1970).
    [CrossRef]
  5. M. Yeung, “Modeling aerial images in two and three dimensions,” in Proc. Kodak Microelectronics Seminar: Interface '85, Kodak Publ. G-154 (Eastman Kodak, 1986), pp. 115-126.
  6. A. K. Wong, Optical Imaging in Projection Microlithography, A.R.Weeks, Jr., ed. (SPIE, 2005), Chap 8.
    [CrossRef]
  7. H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, Vol. III, E.Wolf, ed. (North-Holland, 1964), Chap. 3.
    [CrossRef]
  8. E. L. O'Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.
  9. J. W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), pp. 109-111.
  10. E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3-6 (1981).
    [CrossRef]
  11. B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21, 3770-3777 (1982).
    [CrossRef]
  12. H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19, 1563-1571 (2002).
    [CrossRef]
  13. R. von Bünau, G. Owen, and R. F. W. Pease, “Depth of focus enhancement in optical lithography,” J. Vac. Sci. Technol. B 10, 3047-3054 (1992).
    [CrossRef]
  14. Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A 11, 2438-2452 (1994).
    [CrossRef]
  15. R. J. Socha and A. R. Neureuther, “Propagation effects of partial coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724-3729 (1996).
    [CrossRef]
  16. R. J. Socha, “Propagation effects of partially coherent light in optical lithography and inspection,” Ph.D. dissertation (Electronics Research Laboratory, University of California, Berkeley, 1997).
  17. N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (Electrical Engineering and Computer Science, University of California, Berkeley, 1998).
  18. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786-805 (1989).
    [CrossRef]
  19. M. Mansuripur, “Certain computational aspects of vector diffraction problems: erratum,” J. Opt. Soc. Am. A 10, 382-383 (1993).
    [CrossRef]
  20. K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78-91 (2003).
    [CrossRef]
  21. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1994), Chap. 2.
  22. M. J. Bastiaans, “Uncertainty principle and informational entropy for partially coherent light,” J. Opt. Soc. Am. A 3, 1243-1246 (1986).
    [CrossRef]
  23. R. Köhle, “Fast TCC algorithm for the model building of high NA lithography simulation,” Proc. SPIE 5754, 918-929 (2005).
    [CrossRef]
  24. A. Starikov, “Effective number of degrees of freedom of partially coherent source,” J. Opt. Soc. Am. 72, 1538-1544 (1982).
    [CrossRef]
  25. C. Zuniga and E. Tejnil, “Heuristics for truncating the number of optical kernels in Hopkins image calculation for model-based OPC treatment,” Proc. SPIE 6520, 65203l-1-14 (2007).

2007

C. Zuniga and E. Tejnil, “Heuristics for truncating the number of optical kernels in Hopkins image calculation for model-based OPC treatment,” Proc. SPIE 6520, 65203l-1-14 (2007).

2005

R. Köhle, “Fast TCC algorithm for the model building of high NA lithography simulation,” Proc. SPIE 5754, 918-929 (2005).
[CrossRef]

2003

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78-91 (2003).
[CrossRef]

2002

1996

R. J. Socha and A. R. Neureuther, “Propagation effects of partial coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724-3729 (1996).
[CrossRef]

1994

1993

1992

R. von Bünau, G. Owen, and R. F. W. Pease, “Depth of focus enhancement in optical lithography,” J. Vac. Sci. Technol. B 10, 3047-3054 (1992).
[CrossRef]

1989

1986

1982

A. Starikov, “Effective number of degrees of freedom of partially coherent source,” J. Opt. Soc. Am. 72, 1538-1544 (1982).
[CrossRef]

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21, 3770-3777 (1982).
[CrossRef]

1981

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3-6 (1981).
[CrossRef]

1978

1970

R. Barakat, “Partially coherent imagery in the presence of aberrations,” Opt. Acta 17, 337-347 (1970).
[CrossRef]

1953

H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London, Ser. A 217, 408-432 (1953).
[CrossRef]

Adam, K.

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78-91 (2003).
[CrossRef]

Barakat, R.

R. Barakat, “Partially coherent imagery in the presence of aberrations,” Opt. Acta 17, 337-347 (1970).
[CrossRef]

Bastiaans, M. J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.

Cobb, N.

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78-91 (2003).
[CrossRef]

Cobb, N. B.

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (Electrical Engineering and Computer Science, University of California, Berkeley, 1998).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1994), Chap. 2.

Gamo, H.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, Vol. III, E.Wolf, ed. (North-Holland, 1964), Chap. 3.
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), pp. 109-111.

Granik, Y.

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78-91 (2003).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London, Ser. A 217, 408-432 (1953).
[CrossRef]

Kailath, T.

Kintner, E.

Köhle, R.

R. Köhle, “Fast TCC algorithm for the model building of high NA lithography simulation,” Proc. SPIE 5754, 918-929 (2005).
[CrossRef]

Kutay, M. A.

Mansuripur, M.

Neureuther, A. R.

R. J. Socha and A. R. Neureuther, “Propagation effects of partial coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724-3729 (1996).
[CrossRef]

O'Neill, E. L.

E. L. O'Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.

Owen, G.

R. von Bünau, G. Owen, and R. F. W. Pease, “Depth of focus enhancement in optical lithography,” J. Vac. Sci. Technol. B 10, 3047-3054 (1992).
[CrossRef]

Ozaktas, H. M.

Pati, Y. C.

Pease, R. F. W.

R. von Bünau, G. Owen, and R. F. W. Pease, “Depth of focus enhancement in optical lithography,” J. Vac. Sci. Technol. B 10, 3047-3054 (1992).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1994), Chap. 2.

Rabbani, M.

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21, 3770-3777 (1982).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21, 3770-3777 (1982).
[CrossRef]

Socha, R. J.

R. J. Socha and A. R. Neureuther, “Propagation effects of partial coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724-3729 (1996).
[CrossRef]

R. J. Socha, “Propagation effects of partially coherent light in optical lithography and inspection,” Ph.D. dissertation (Electronics Research Laboratory, University of California, Berkeley, 1997).

Starikov, A.

Tejnil, E.

C. Zuniga and E. Tejnil, “Heuristics for truncating the number of optical kernels in Hopkins image calculation for model-based OPC treatment,” Proc. SPIE 6520, 65203l-1-14 (2007).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1994), Chap. 2.

Torres, A.

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78-91 (2003).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1994), Chap. 2.

von Bünau, R.

R. von Bünau, G. Owen, and R. F. W. Pease, “Depth of focus enhancement in optical lithography,” J. Vac. Sci. Technol. B 10, 3047-3054 (1992).
[CrossRef]

Wolf, E.

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3-6 (1981).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.

Wong, A. K.

A. K. Wong, Optical Imaging in Projection Microlithography, A.R.Weeks, Jr., ed. (SPIE, 2005), Chap 8.
[CrossRef]

Yeung, M.

M. Yeung, “Modeling aerial images in two and three dimensions,” in Proc. Kodak Microelectronics Seminar: Interface '85, Kodak Publ. G-154 (Eastman Kodak, 1986), pp. 115-126.

Yüksel, S.

Zuniga, C.

C. Zuniga and E. Tejnil, “Heuristics for truncating the number of optical kernels in Hopkins image calculation for model-based OPC treatment,” Proc. SPIE 6520, 65203l-1-14 (2007).

Appl. Opt.

E. Kintner, “Method for the calculation of partially coherent imagery,” Appl. Opt. 17, 2747-2753 (1978).
[CrossRef] [PubMed]

B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21, 3770-3777 (1982).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

R. J. Socha and A. R. Neureuther, “Propagation effects of partial coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724-3729 (1996).
[CrossRef]

R. von Bünau, G. Owen, and R. F. W. Pease, “Depth of focus enhancement in optical lithography,” J. Vac. Sci. Technol. B 10, 3047-3054 (1992).
[CrossRef]

Opt. Acta

R. Barakat, “Partially coherent imagery in the presence of aberrations,” Opt. Acta 17, 337-347 (1970).
[CrossRef]

Opt. Commun.

E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3-6 (1981).
[CrossRef]

Proc. R. Soc. London, Ser. A

H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London, Ser. A 217, 408-432 (1953).
[CrossRef]

Proc. SPIE

K. Adam, Y. Granik, A. Torres, and N. Cobb, “Improved modeling performance with an adapted vectorial formulation of the Hopkins imaging equation,” Proc. SPIE 5040, 78-91 (2003).
[CrossRef]

R. Köhle, “Fast TCC algorithm for the model building of high NA lithography simulation,” Proc. SPIE 5754, 918-929 (2005).
[CrossRef]

C. Zuniga and E. Tejnil, “Heuristics for truncating the number of optical kernels in Hopkins image calculation for model-based OPC treatment,” Proc. SPIE 6520, 65203l-1-14 (2007).

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1994), Chap. 2.

R. J. Socha, “Propagation effects of partially coherent light in optical lithography and inspection,” Ph.D. dissertation (Electronics Research Laboratory, University of California, Berkeley, 1997).

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (Electrical Engineering and Computer Science, University of California, Berkeley, 1998).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.

M. Yeung, “Modeling aerial images in two and three dimensions,” in Proc. Kodak Microelectronics Seminar: Interface '85, Kodak Publ. G-154 (Eastman Kodak, 1986), pp. 115-126.

A. K. Wong, Optical Imaging in Projection Microlithography, A.R.Weeks, Jr., ed. (SPIE, 2005), Chap 8.
[CrossRef]

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, Vol. III, E.Wolf, ed. (North-Holland, 1964), Chap. 3.
[CrossRef]

E. L. O'Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), pp. 109-111.

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Figures (9)

Fig. 1
Fig. 1

Target object and the illumination shape. (a) The target object is two clear apertures 100 nm square in an opaque background. (b) Schematic view of illumination shape; each pixel shows the mutually incoherent point source.

Fig. 2
Fig. 2

Diffracted light amplitude computed from the object illustrated in Fig. 1a under normal incident illumination. The maximum value is normalized to unity. The white circle indicates the pupil.

Fig. 3
Fig. 3

The P-operator computed under the ideal conditions. The i th row and j th column of the discrete pupil function is stacked on the [ ( i 1 ) × M + j ] th element by the stacking operator Y. A white cell is 1 and a black cell is 0.

Fig. 4
Fig. 4

Eigenvalue computed from the P-operator shown in Fig. 3. After sorting the eigenvalues in descending order, the eigenvalues are normalized for λ 1 to be unity. The explicit eigenvalues up to λ 5 are listed.

Fig. 5
Fig. 5

Eigenfunctions computed from the P-operator shown in Fig. 3. (a) First eigenfunction Φ 1 , (b) second eigenfunction Φ 2 , (c) third eigenfunction Φ 3 .

Fig. 6
Fig. 6

Eigenfunctions of the aerial image. (a) First eigenfunction F T [ a ̂ Φ 1 ] . (b) The second eigenfunction is a pure imaginary number so that the imaginary part of F T [ a ̂ Φ 2 ] is plotted. (c) The third eigenfunction is the pure imaginary number so that the imaginary part of F T [ a ̂ Φ 3 ] is plotted.

Fig. 7
Fig. 7

Final aerial image computed by Eq. (32). The maximum value is normalized to unity.

Fig. 8
Fig. 8

Approximate aerial image with nine kinds of eigenvalues and eigenfunctions. The maximum value is normalized to unity.

Fig. 9
Fig. 9

Difference of Fig. 8 and Fig. 7; approximate aerial image minus complete aerial image.

Equations (44)

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

I ( x , y ) = S ( f , g ) F T [ a ̂ ( f f , g g ) P ( f , g ) ] 2 d f d g ,
I ( x , y ) = i , j S ( f i , g j ) F T [ a ̂ ( f f i , g g j ) P ( f , g ) ] 2 .
ϕ 1 D = [ exp ( i 2 π f 1 x ) exp ( i 2 π f 2 x ) exp ( i 2 π f M x ) ] T ,
a ̂ 1 D = [ a ̂ 1 exp ( i 2 π f 1 x ) a ̂ 2 exp ( i 2 π f 2 x ) a ̂ 7 exp ( i 2 π f 7 x ) ] T ,
( f 1 f 2 f 3 f 4 f 5 f 6 f 7 ) = ( 2 4 3 2 3 0 2 3 4 3 2 ) .
P 1 D = ( 0 0 1 1 1 0 0 ) .
D 1 D = P 1 1 D a ̂ 1 D ,
I 1 ( x ) = D 1 D D 1 D = a ̂ 1 D ( P 1 1 D ) P 1 1 D a ̂ 1 D .
I ( x , y ) = i , j S ( f i , g j ) F T [ a ̂ ( f , g ) P ( f + f i , g + g j ) ] 2 .
I 2 ( x ) = a ̂ 1 D ( P 2 1 D ) P 2 1 D a ̂ 1 D ,
P 2 1 D = ( 0 1 1 1 0 0 0 ) .
I ( x ) = I 1 ( x ) + I 2 ( x ) = a ̂ 1 D ( P 1 1 D ) P 1 1 D a ̂ 1 D + a ̂ 1 D ( P 2 1 D ) P 2 1 D a ̂ 1 D .
I ( x ) = a ̂ 1 D ( P 1 D ) P 1 D a ̂ 1 D ,
P 1 D = ( P 1 1 D P 2 1 D ) = ( 0 0 1 1 1 0 0 0 1 1 1 0 0 0 ) .
ϕ ( f , g ) = ( exp [ i 2 π ( f 1 x + g 1 y ) ] exp [ i 2 π ( f 1 x + g 2 y ) ] exp [ i 2 π ( f 1 x + g M y ) ] exp [ i 2 π ( f 2 x + g 1 y ) ] exp [ i 2 π ( f 2 x + g 2 y ) ] exp [ i 2 π ( f 1 x + g M y ) ] exp [ i 2 π ( f M x + g 1 y ) ] exp [ i 2 π ( f M x + g 2 y ) ] exp [ i 2 π ( f M x + g M y ) ] ) .
a ̂ ( f , g ) = ( a ̂ 1 , 1 exp [ i 2 π ( f 1 x + g 1 y ) ] a ̂ 1 , 2 exp [ i 2 π ( f 2 x + g 1 y ) ] a ̂ 1 , 7 exp [ i 2 π ( f 1 x + g 7 y ) ] a ̂ 2 , 1 exp [ i 2 π ( f 2 x + g 1 y ) ] a ̂ 2 , 2 exp [ i 2 π ( f 2 x + g 2 y ) ] a ̂ 2 , 7 exp [ i 2 π ( f 2 x + g 7 y ) ] a ̂ 7 , 1 exp [ i 2 π ( f 7 x + g 1 y ) ] a ̂ 7 , 2 exp [ i 2 π ( f 7 x + g 2 y ) ] a ̂ 7 , 7 exp [ i 2 π ( f 7 x + g 7 y ) ] ) ,
P ( f , g ) = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] .
a ̂ = Y [ a ̂ ( f , g ) ] ,
D = P 1 a ̂ ,
I 1 ( x , y ) = D D = a ̂ P 1 P 1 a ̂ .
I ( x , y ) = I 1 ( x , y ) + I 2 ( x , y ) = a ̂ P 1 P 1 a ̂ + a ̂ P 2 P 2 a ̂ = a ̂ P P a ̂ ,
P = ( P 1 P 2 ) .
P = ( Y [ P ( f f 1 , g g 1 ) ] T Y [ P ( f f 2 , g g 2 ) ] T Y [ P ( f f N , g g N ) ] T ) = ( P 1 P 2 P N ) ,
I ( x , y ) = a ̂ P P a ̂ .
P ( f , g ) = circ ( f , g ) o ( f , g ) exp [ i 2 π W ( f , g ) ] .
P = ( S 1 Y [ P ( f f 1 , g g 1 ) ] T S 2 Y [ P ( f f 2 , g g 2 ) ] T S N Y [ P ( f f N , g g N ) ] T ) ,
P l = ( S 1 Y [ P l ( f f 1 , g g 1 ) P ( f f 1 , g g 1 ) ] T S 2 Y [ P l ( f f 2 , g g 2 ) P ( f f 2 , g g 2 ) ] T S N Y [ P l ( f f N , g g N ) P ( f f N , g g N ) ] T ) ,
P full = ( P x P y P z ) .
I ( x , y ) = a ̂ P P a ̂ .
P = U Λ V .
I ( x , y ) = a ̂ V Λ U U Λ V a ̂ = a ̂ V Λ 2 V a ̂ .
I ( x , y ) = i = 1 N λ i 2 F T [ a ̂ ( f , g ) Φ i ( f , g ) ] 2 ,
I ( x , y ) = TCC ( f , g , f , g ) a ̂ ( f , g ) a ̂ * ( f , g ) exp { i 2 π [ ( f f ) x + ( g g ) y ] } d f d g d f d g ,
TCC ( f , g , f , g ) = S ( f , g ) P ( f + f , g + g ) P * ( f + f , g + g ) d f d g .
I ( x ) = i , j [ a ̂ * ( f j ) exp ( i 2 π f j x ) ] TCC 1 D ( f i , f j ) [ a ̂ ( f i ) exp ( i 2 π f i x ) ] = i , j [ a ̂ ( f j ) exp ( i 2 π f j x ) ] * TCC 1 D ( f i , f j ) [ a ̂ ( f i ) exp ( i 2 π f i x ) ] .
I ( x ) = j a ̂ * ( f j ) [ i TCC 1 D ( f j , f i ) a ̂ ( f i ) ] = a ̂ T 1 D a ̂ ,
T 1 D = ( P 1 D ) P 1 D .
T = P P .
rank ( T ) = min [ rank ( P ) , rank ( P ) ] = min [ N , N ] = N .
A = [ a b c d e f g h i ] .
Y [ A ] = ( a b c d e f g h i ) ,
= B ,
Y [ A ] T = ( a b c d e f g h i ) = B T .
Y 1 [ B ] = A .

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