Abstract

Two-dimensional image moments with respect to Zernike polynomials are defined, and it is shown how to construct an arbitrarily large number of independent, algebraic combinations of Zernike moments that are invariant to image translation, orientation, and size. This approach is contrasted with the usual method of moments. The general problem of two-dimensional pattern recognition and three-dimensional object recognition is discussed within this framework. A unique reconstruction of an image in either real space or Fourier space is given in terms of a finite number of moments. Examples of applications of the method are given. A coding scheme for image storage and retrieval is discussed.

© 1980 Optical Society of America

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References

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  1. R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976).
    [Crossref]
  2. W. H. Southwell, “Wave-front analyzer using maximum likelihood algorithm,” J. Opt. Soc. Am. 67, 396–399 (1977).
    [Crossref]
  3. S. R. Robinson, “On the problem of phase from intensity measurements,” J. Opt. Soc. Am. 68, 87–92 (1978).
    [Crossref]
  4. A. J. Devaney and R. Childlaw, “On the uniqueness question in the problem of phase retrieval from intensity measurements,” J. Opt. Soc. Am. 68, 1352–1354 (1978).
    [Crossref]
  5. M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).
  6. S. A. Dudani, K. J. Breeding, and R. B. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
    [Crossref]
  7. R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).
  8. N. Bareket and W. L. Wolfe, “Image chopping techniques for fast measurements of irradiance distribution parameters,” Appl. Opt. 18, 389–392 (1979).
    [Crossref] [PubMed]
  9. M. R. Teague, “Automatic image analysis via the method of moments,” Laser Digest, Summer1979, p. 25–43, AFWL, Kirtland AFB, New Mexico (unpublished).
  10. See, for example, N. I. Akhiezer, The Classical Moment Problem and Some Related Question in Analysis (Hafner, New York, 1965).
  11. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  12. A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).
  13. A. D. Myskis, Advanced Mathematics for Engineers (MIR, Moscow, 1975).
  14. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I (Interscience, New York, 1953).
  15. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  17. M. R. Teague, “Optical calculation of image moments,” Appl. Opt. 19, 1353–1356, (1980).
    [Crossref] [PubMed]
  18. D. Casasent and D. Psaltis, “Optical pattern recognition using normalized invariant moments,” SPIE Proc. (to be published).

1980 (1)

1979 (1)

1978 (2)

1977 (2)

W. H. Southwell, “Wave-front analyzer using maximum likelihood algorithm,” J. Opt. Soc. Am. 67, 396–399 (1977).
[Crossref]

S. A. Dudani, K. J. Breeding, and R. B. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
[Crossref]

1976 (1)

1962 (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).

Akhiezer, N. I.

See, for example, N. I. Akhiezer, The Classical Moment Problem and Some Related Question in Analysis (Hafner, New York, 1965).

Bareket, N.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Breeding, K. J.

S. A. Dudani, K. J. Breeding, and R. B. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
[Crossref]

Casasent, D.

D. Casasent and D. Psaltis, “Optical pattern recognition using normalized invariant moments,” SPIE Proc. (to be published).

Childlaw, R.

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I (Interscience, New York, 1953).

Devaney, A. J.

Duda, R. O.

R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Dudani, S. A.

S. A. Dudani, K. J. Breeding, and R. B. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
[Crossref]

Gonsalves, R. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Hart, P. E.

R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I (Interscience, New York, 1953).

Hu, M. K.

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).

McGhee, R. B.

S. A. Dudani, K. J. Breeding, and R. B. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
[Crossref]

Myskis, A. D.

A. D. Myskis, Advanced Mathematics for Engineers (MIR, Moscow, 1975).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Psaltis, D.

D. Casasent and D. Psaltis, “Optical pattern recognition using normalized invariant moments,” SPIE Proc. (to be published).

Robinson, S. R.

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

Southwell, W. H.

Teague, M. R.

M. R. Teague, “Optical calculation of image moments,” Appl. Opt. 19, 1353–1356, (1980).
[Crossref] [PubMed]

M. R. Teague, “Automatic image analysis via the method of moments,” Laser Digest, Summer1979, p. 25–43, AFWL, Kirtland AFB, New Mexico (unpublished).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Wolfe, W. L.

Appl. Opt. (2)

IEEE Trans. Comput. (1)

S. A. Dudani, K. J. Breeding, and R. B. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
[Crossref]

IRE Trans. Inf. Theory (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).

J. Opt. Soc. Am. (4)

Other (10)

R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, 1973).

D. Casasent and D. Psaltis, “Optical pattern recognition using normalized invariant moments,” SPIE Proc. (to be published).

M. R. Teague, “Automatic image analysis via the method of moments,” Laser Digest, Summer1979, p. 25–43, AFWL, Kirtland AFB, New Mexico (unpublished).

See, for example, N. I. Akhiezer, The Classical Moment Problem and Some Related Question in Analysis (Hafner, New York, 1965).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

A. D. Myskis, Advanced Mathematics for Engineers (MIR, Moscow, 1975).

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I (Interscience, New York, 1953).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

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Figures (4)

FIG. 1
FIG. 1

Image ellipse.

FIG. 2
FIG. 2

Reconstruction of the letter E. Top row, left to right: original input image, reconstructed image with up through 2nd-order moments, up through 3rd order, up through 4th order. Succeeding rows in left to right order show effect of adding up through 5th-order through up through 16th-order moments.

FIG. 3
FIG. 3

Reconstruction of letter F. Description of Fig. 2 applies except the effect of adding up through 2nd order through up through 15th order is shown and last entry includes all moments through 18th order. The image for 16th and 17th order were very similar to that for the 15th order.

FIG. 4
FIG. 4

Zernike moments invariants for: (a) a centered, constant irradiance, rectangle of 5 by 19 pixels; (b) a centered 5 by 13 pixel rectangle with a small 3 by 3 pixel square sitting on the left-hand end of the big rectangle; (c) the letter E (same as Fig. 2); (d) the letter F (same as Fig. 3); (e) the letter C. For these figures the Zernike polynomials were normalized to unity inside the unit circle rather than using the norm of Eq. (38).

Tables (2)

Tables Icon

TABLE I Ellipse Tilt Angle for Various Cases of the signs of the second moments.

Tables Icon

TABLE II Average relative pixel errors in the reconstructed irradiance functions of Figs. 2 and 3. Errors are in %.

Equations (96)

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μ 00 = dx dy f ( x , y )
μ 10 = dx dy xf ( x , y ) ,
μ 01 = dx dy yf ( x , y )
μ 20 = dx dy f ( x , y ) x 2
μ 11 = dx dy f ( x , y ) x y ,
μ 02 = dx dy f ( x , y ) y 2
a = ( μ 20 + μ 02 + [ ( μ 20 + μ 02 ) 2 + 4 μ 11 2 ] 1 / 2 μ 00 / 2 ) 1 / 2 ,
b = ( μ 20 + μ 02 [ ( μ 20 μ 02 ) 2 + 4 μ 11 2 ] 1 / 2 μ 00 / 2 ) 1 / 2 ,
ϕ = ( 1 / 2 ) tan 1 ( 2 μ 11 μ 20 μ 02 )
F = μ 00 / π a b
[ μ 20 μ 11 μ 02 ] = [ 1 + cos 2 ϕ 2 sin 2 ϕ 1 cos 2 ϕ 2 sin 2 ϕ 2 cos 2 ϕ sin 2 ϕ 2 1 cos 2 ϕ 2 sin 2 ϕ 1 + cos 2 ϕ 2 ] [ μ 20 μ 11 μ 02 ] ,
a b .
π / 2 tan 1 x π / 2 .
M j k = dx dy f ( x , y ) x j y k .
x ( 1 , + 1 ) , y ( 1 , + 1 ) .
F ( ξ , η ) = dx dy e i 2 π ( ξ x + η y ) f ( x , y ) .
F ( ξ , η ) = dx dy j = 0 ( i 2 π ξ x ) j j ! k = 0 ( i 2 π η y ) k k ! f ( x , y ) = j = 0 k = 0 ( i 2 π ) j + k j ! k ! M j k ξ j η k ,
f ( x , y ) = d ξ d η e i 2 π ( ξ x + η y ) F ( ξ , η ) = d ξ d η e i 2 π ( ξ x + η y ) × j = 0 k = 0 ( i 2 π ) j + k j ! k ! M j k ξ j η k .
g ( x , y ) = g 00 + g 10 x + g 01 y + g 20 x 2 + g 11 x y + g 02 y 2 + g 30 x 3 + g 21 x 2 y + g 12 x y 2 + g 03 y 3 + ,
1 + 1 d x 1 + 1 dy g ( x , y ) x j y k M j k .
[ 1 1 3 1 3 1 3 1 5 1 9 1 3 1 9 1 5 ] [ g 00 g 20 g 02 ] = ( 1 / 4 ) × [ M 00 M 20 M 02 ] ,
[ 1 3 1 5 1 9 1 5 1 7 1 15 1 9 1 15 1 15 ] [ g 10 g 30 g 12 ] = ( 1 / 4 ) × [ M 10 M 30 M 12 ] ,
g 11 = ( 9 / 4 ) M 11 .
16 g ( x , y ) = ( 14 M 00 15 M 20 15 M 02 ) + ( 90 M 10 105 M 30 45 M 12 ) x + ( 90 M 01 105 M 03 45 M 21 ) y + ( 15 M 00 + 45 M 20 ) x 2 + 36 M 11 x y + ( 15 M 00 + 45 M 02 ) y 2 + ( 105 M 10 + 175 M 30 ) x 3 + ( 45 M 01 + 135 M 21 ) x 2 y + ( 45 M 10 + 135 M 12 ) x y 2 + ( 105 M 01 + 175 M 03 ) y 3 .
M j k = dx dy f ( x , y ) x j y k
f ( x , y ) = m = 0 n = 0 λ m n P m ( x ) P n ( y ) ,
1 + 1 d x P m ( x ) P m ( x ) = 2 2 m + 1 δ m m .
λ m n ( 2 m + 1 ) ( 2 n + 1 ) 4 dx dy f ( x , y ) P m ( x ) P n ( y ) .
P m ( x ) = j = 0 m C m j x j ,
λ m n = ( 2 m + 1 ) ( 2 n + 1 ) 4 j = 0 m k = 0 n C m j C n k M j k .
f ( x , y ) N = 0 N max n = 0 N λ N n , n P N n ( x ) P n ( y ) .
F ( x , y ) F ( x , y ) ,
M j k M j k .
x ¯ = M 10 / M 00 , y ¯ = M 01 / M 00 .
μ j k = r = 0 j s = 0 k ( j j ) ( k s ) ( x ¯ ) j r ( y ¯ ) k s M r s .
μ 10 = μ 01 = 0
f ( x , y ) = f ( x / λ , y / λ ) ,
λ = R / R
μ p q = dx dy f ( x / λ , y / λ ) x p y q = λ 2 + p + q μ p q ,
μ p q / ( μ 00 ) ( p + q + 2 ) / 2
μ j k = r = 0 j s = 0 k ( 1 ) k s ( j r ) ( k s ) ( cos ϕ ) j r + s × ( sin ϕ ) k + r s ( μ j + k r s , r + s ) ,
V n l ( x , y ) = V n l ( ρ sin θ , ρ cos θ ) = R n l ( ρ ) exp ( i l θ ) .
dx dy [ V n l ( x , y ) ] * V m k ( x , y ) = π / ( n + 1 ) δ m n , δ k l ,
f ( x , y ) = n l A n l V n l ( ρ , θ ) ,
n | l | is even , | l | < n .
A n l = [ ( n + 1 ) / π ] dx dy f ( x , y ) [ V n l ( ρ , θ ) ] * = ( A n , l ) * .
f ( x , y ) = n l ( C n l cos l θ + S n l sin l θ ) R n l ( ρ ) ,
[ C n l S n l ] = [ ( 2 n + 2 ) / π ] dx dy f ( x , y ) R n l ( ρ ) [ cos l θ sin l θ ] ,
C n 0 = A n 0 = ( 1 / π ) dx dy f ( x , y ) R n 0 ( ρ ) ,
S n 0 = 0 .
C n l = 2 Re ( A n l ) , S n l = 2 Im ( A n l ) , A n l = ( C n l i S n l ) / 2 = ( A n , l ) * .
R n l ( ρ ) = k = 1 n B nlk ρ k ,
A n l = [ ( n + 1 ) / π ] k = l n j = 0 q m = 0 l ( i ) m ( q j ) ( l m ) B nlk × μ k 2 j l + m , 2 j + l m ,
A 00 = μ 00 / π = 1 / π , A 11 = A 1 , 1 = 0 , A 22 = 3 ( μ 02 μ 20 2 i μ 11 ) / π , A 20 = 3 ( 2 μ 20 + 2 μ 02 1 ) / π , A 33 = 4 [ μ 03 3 μ 21 + i ( μ 30 3 μ 12 ) ] / π , A 31 = 12 [ μ 03 + μ 21 i ( μ 30 + μ 12 ) ] / π , A 44 = 5 [ μ 40 6 μ 22 + μ 04 + 4 i ( μ 31 μ 13 ) ] / π , A 42 = 5 { 4 ( μ 04 μ 40 ) + 3 ( μ 20 + μ 02 ) 2 i [ 4 ( μ 31 + μ 13 ) 3 μ 11 ] } / π , A 40 = 5 [ 6 ( μ 40 + 2 μ 22 + μ 04 ) 6 ( μ 20 + μ 02 ) + 1 ] / π .
x = x cos θ 0 y sin θ 0 , y = x sin θ 0 + y cos θ 0 .
f ( ρ , θ ) = f ( ρ , θ θ 0 ) ,
( A n l ) = [ ( n + 1 ) / π ] ρ d ρ d θ f ( ρ , θ θ 0 ) × R n l ( ρ ) exp ( i l θ ) ,
( A n l ) = A n l exp ( i l θ 0 ) .
( C n l ) = C n l cos l θ 0 S n l sin l θ 0 , ( S n l ) = C n l sin l θ 0 + S n l cos l θ 0 .
x = x cos 2 θ 0 y sin 2 θ 0 , y = x sin 2 θ 0 y cos 2 θ 0 .
f ( x , y ) = f ( x , y ) ,
ρ = ρ , θ = 2 θ 0 θ .
( A n l ) = [ ( n + 1 ) / π ] dx dy f ( x , y ) R n l ( ρ ) × exp [ i l ( 2 θ 0 θ ) ]
( A n l ) = ( A n l ) * exp ( i 2 l θ 0 ) ,
( C n l ) = C n l cos 2 l θ 0 + S n l sin 2 l θ 0 , ( S n l ) = + C n l sin 2 l θ 0 S n l cos 2 l θ 0 .
A n l = | A n l | exp ( i ϕ n l ) ,
S 1 = A 20 ,
S 2 = A 22 A 2 , 2 = | A 22 | 2 ,
S 3 = | A 33 | 2 ,
S 4 = | A 31 | 2 .
A 33 ( A 3 , 1 ) 3 = A 33 [ ( A 31 ) * ] 3
( A 33 ) * ( A 31 ) 3 .
S 5 = A 33 [ ( A 31 ) * ] 3 + c . c . = 2 | A 33 | | A 31 | 3 cos ( ϕ 33 3 ϕ 31 ) ,
P 1 = i { A 33 [ ( A 31 ) * ] 3 c . c . } = 2 | A 33 | | A 31 | 3 sin ( ϕ 33 3 ϕ 31 ) .
S 6 = ( A 31 ) 2 ( A 22 ) * + c . c . = 2 | A 31 | 2 A 22 cos ( 2 ϕ 31 ϕ 22 ) .
( A 33 ) 2 [ ( A 22 ) * ] 3 + c . c . = 2 | A 33 | 2 | A 22 | 3 cos ( 2 ϕ 33 3 ϕ 22 )
S 7 = | A 44 | 2 ,
S 8 = | A 42 | 2 ,
S 9 = A 40 .
S 10 = ( A 44 ) * ( A 42 ) 2 + c . c . = 2 | A 44 | | A 42 | 2 cos ( ϕ 44 2 ϕ 42 ) ,
S 11 = A 42 ( A 22 ) * + c . c . = 2 | A 42 | | A 22 | cos ( ϕ 42 ϕ 22 ) .
S 12 = | A 55 | 2 ,
S 13 = | A 53 | 2 ,
S 14 = | A 51 | 2 .
( A 55 ) * 3 ( A 53 ) 5 + c . c . = 2 | A 55 | 3 | A 53 | 5 cos ( 3 ϕ 55 5 ϕ 53 ) , ( A 55 ) * ( A 51 ) 5 + c . c . = 2 | A 55 | | A 51 | 5 cos ( ϕ 55 5 ϕ 51 ) , ( A 53 ) * ( A 51 ) 3 + c . c . = 2 | A 53 | | A 51 | 3 cos ( ϕ 53 3 ϕ 51 ) .
3 ϕ 55 5 ϕ 53 = a , ϕ 55 5 ϕ 51 = b , ϕ 53 3 ϕ 51 = c ,
S 15 = ( A 51 ) * A 31 + c . c . ,
S 16 = ( A 53 ) * A 33 + c . c . ,
S 17 = ( A 55 ) * ( A 31 ) 5 + c . c .
S 1 = A 20 = 3 [ 2 ( μ 20 + μ 02 ) 1 ] / π , S 2 = | A 22 | 2 = 9 [ ( μ 20 μ 02 ) 2 + 4 ( μ 11 ) 2 ] / π 2 .
S 3 = | A 33 | 2 = 16 [ ( μ 03 3 μ 21 ) 2 + ( μ 30 3 μ 12 ) 2 ] / π 2 , S 4 = | A 31 | 2 = 144 [ ( μ 03 + μ 21 ) 2 + ( μ 30 μ 12 ) 2 ] / π 2 , S 5 = ( A 33 ) * ( A 31 ) 3 + c . c . = 13824 π 4 { ( μ 03 3 μ 21 ) ( μ 03 + μ 21 ) × [ ( μ 03 + μ 21 ) 2 3 ( μ 30 + μ 12 ) 2 ] ( μ 30 3 μ 12 ) ( μ 30 + μ 12 ) × [ ( μ 30 + μ 12 ) 2 3 ( μ 03 + μ 21 ) 2 ] } , S 6 = ( A 31 ) 2 A 22 * + c . c . = 864 π 3 { ( μ 02 μ 20 ) [ ( μ 03 + μ 21 ) 2 ( μ 30 + μ 12 ) 2 ] + 4 μ 11 ( μ 03 + μ 21 ) ( μ 30 + μ 12 ) } .
S 7 = | A 44 | 2 = 25 [ ( μ 40 6 μ 22 + μ 04 ) 2 + 16 ( μ 31 μ 13 ) 2 ] / π 2 , S 8 = | A 42 | 2 = 25 { [ 4 ( μ 04 μ 40 ) + 3 ( μ 20 μ 02 ) } 2 + 4 { 4 ( μ 31 + μ 13 ) 3 μ 11 ] 2 } / π 2 , S 9 = A 40 = 5 [ 6 ( μ 40 + 2 μ 22 + μ 04 ) 6 ( μ 20 + μ 02 ) + 1 ] / π , S 10 = ( A 44 ) * ( A 42 ) 2 + c . c . = 250 π 3 ( ( μ 40 6 μ 22 + μ 04 ) × { 4 ( μ 04 μ 40 ) + 3 ( μ 20 μ 02 ) ] 2 4 [ 4 ( μ 31 + μ 13 ) 3 μ 11 ] 2 } 16 [ 4 ( μ 04 μ 40 ) + 3 ( μ 20 μ 02 ) ] × [ 4 ( μ 31 + μ 13 ) 3 μ 11 ] ( μ 31 μ 13 ) ) , S 11 = A 42 ( A 22 ) * + c . c . = 30 π 2 { [ 4 ( μ 04 μ 40 ) + 3 ( μ 20 μ 02 ) ] ( μ 02 μ 20 ) + 4 μ 11 [ 4 ( μ 31 + μ 13 ) 3 μ 11 ] } .
P 1 = i ( A 33 * A 31 3 c . c . ) = 13 824 { ( μ 30 3 μ 12 ) ( μ 03 + μ 21 ) [ ( μ 03 + μ 21 ) 2 3 ( μ 30 + μ 12 ) 2 ] ( μ 03 3 μ 21 ) ( μ 30 + μ 12 ) × [ ( μ 30 + μ 12 ) 2 3 ( μ 03 + μ 21 ) 2 ] } / π 4 , P 2 = i [ ( A 31 ) 2 ( A 22 ) * c . c . ] = 1728 { μ 11 [ ( μ 03 + μ 21 ) 2 ( μ 30 + μ 12 ) 2 ] ( μ 03 + μ 21 ) ( μ 30 + μ 12 ) ( μ 02 μ 20 ) } / π 3 .
S 18 = | A 66 | 2 , S 19 = | A 64 | 2 , S 20 = | A 62 | 2 , S 21 = A 60 , S 22 = ( A 66 ) * ( A 33 ) 2 + c . c . , S 23 = ( A 64 ) * A 44 + c . c . S 24 = ( A 62 ) * A 22 + c . c .
S 25 = | A 77 | 2 , S 26 = | A 75 | 2 , S 27 = | A 73 | 2 , S 28 = | A 71 | 2 , S 29 = ( A 77 ) * ( A 31 ) 7 + c . c . , S 30 = ( A 75 ) * A 55 + c . c . , S 31 = ( A 73 ) * A 33 + c . c . , S 32 = ( A 71 ) * A 31 + c . c .
S 33 = | A 88 | 2 , S 34 = | A 86 | 2 , S 35 = | A 84 | 2 , S 36 = | A 82 | 2 , S 37 = A 80 , S 38 = ( A 88 ) * ( A 44 ) 2 + c . c . , S 39 = ( A 86 ) * A 66 + c . c . , S 40 = ( A 84 ) * A 44 + c . c . , S 41 = ( A 82 ) * A 22 + c . c .