Abstract

The concept of multiresolution optical correlators is formally introduced. A mathematical analysis is performed for a generalized multiresolution correlator that emphasizes the roles of both input and filter spatial light modulator resolutions. Conditions are derived for overlapping and nonoverlapping correlation orders. A simulation is performed in which it is shown that the predicted performance of composite binary-phase-only filters designed by the conventional design procedure is different from the actual performance when they are implemented in a real optical correlator. The training of filters on multiresolution approximations of high-resolution discrete Fourier transforms generated by multiresolution wavelet analysis (MWA) techniques is proposed. An analysis is performed that shows that training on MWA approximations results in filters whose performance is the same in a real correlator as that predicted by the design procedure. This analysis is confirmed by simulation. Further simulations show that the performance of reduced-resolution filters designed by MWA techniques is markedly superior to the performance of those designed by conventional means. Finally, an analysis is performed that explains why the ratio of zero- to higher-order correlation peak intensities is much greater for the former than the latter.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. A. Serati, T. K. Ewing, R. A. Serati, K. M. Johnson, D. M. Simon, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE 1959, 55–68 (1993).
  2. J. L. Horner, P. D. Gianino, “Applying the phase-only filter concept to the synthetic discriminant function correlation filter,” Appl. Opt. 24, 851–855 (1985).
    [CrossRef] [PubMed]
  3. G. Gheen, E. Washwell, C. Huang, “Problems facing optical correlators,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE 1772, 96–102 (1992).
  4. S. P. Kozaitis, W. E. Foor, “Optical correlation using reduced resolution filters,” Opt. Eng. 31, 1929–1935 (1992).
    [CrossRef]
  5. D. L. Flannery, S. C. Gustafson, “Adaptive optical correlation using neural network approaches,” in Optical Pattern Recognition, J. L. Horner, B. Javadi, eds., Vol. CR40 of SPIE Critical Review Series (SPIE Optical Engineering Press, Bellingham, Wash., 1992), pp. 25–45.
  6. P. C. Miller, “Optimum reduced-resolution phase-only filters for extended target recognition,” Opt. Eng. 32, 2890–2898 (1993).
    [CrossRef]
  7. P. C. Miller, “Reduced-resolution synthetic-discriminant-function design by multiresolution wavelet analysis,” Appl. Opt. 34, 865–878 (1995).
    [CrossRef] [PubMed]
  8. P. D. Gianino, C. L. Woods, “General treatment of spatial light modulator dead-zone effects on optical correlation. II. Mathematical analysis,” Appl. Opt. 32, 6536–6541 (1993).
    [CrossRef] [PubMed]
  9. S. Lindell, “tops optical correlation program,” in Transition of Optical Processors into Systems 1993, L. E. Garn, L. L. Graceffo, eds., Proc. SPIE 1958, 7–18 (1993).
  10. D. A. Jared, D. J. Ennis, “Inclusion of filter modulation in synthetic-discriminant-function construction,” Appl. Opt. 28, 232–239 (1989).
    [CrossRef] [PubMed]
  11. S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
    [CrossRef]
  12. C. K. Chui, An Introduction to Wavelets (Academic, San Diego, 1992).

1995 (1)

1993 (2)

1992 (1)

S. P. Kozaitis, W. E. Foor, “Optical correlation using reduced resolution filters,” Opt. Eng. 31, 1929–1935 (1992).
[CrossRef]

1989 (2)

D. A. Jared, D. J. Ennis, “Inclusion of filter modulation in synthetic-discriminant-function construction,” Appl. Opt. 28, 232–239 (1989).
[CrossRef] [PubMed]

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

1985 (1)

Chui, C. K.

C. K. Chui, An Introduction to Wavelets (Academic, San Diego, 1992).

Ennis, D. J.

Ewing, T. K.

S. A. Serati, T. K. Ewing, R. A. Serati, K. M. Johnson, D. M. Simon, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE 1959, 55–68 (1993).

Flannery, D. L.

D. L. Flannery, S. C. Gustafson, “Adaptive optical correlation using neural network approaches,” in Optical Pattern Recognition, J. L. Horner, B. Javadi, eds., Vol. CR40 of SPIE Critical Review Series (SPIE Optical Engineering Press, Bellingham, Wash., 1992), pp. 25–45.

Foor, W. E.

S. P. Kozaitis, W. E. Foor, “Optical correlation using reduced resolution filters,” Opt. Eng. 31, 1929–1935 (1992).
[CrossRef]

Gheen, G.

G. Gheen, E. Washwell, C. Huang, “Problems facing optical correlators,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE 1772, 96–102 (1992).

Gianino, P. D.

Gustafson, S. C.

D. L. Flannery, S. C. Gustafson, “Adaptive optical correlation using neural network approaches,” in Optical Pattern Recognition, J. L. Horner, B. Javadi, eds., Vol. CR40 of SPIE Critical Review Series (SPIE Optical Engineering Press, Bellingham, Wash., 1992), pp. 25–45.

Horner, J. L.

Huang, C.

G. Gheen, E. Washwell, C. Huang, “Problems facing optical correlators,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE 1772, 96–102 (1992).

Jared, D. A.

Johnson, K. M.

S. A. Serati, T. K. Ewing, R. A. Serati, K. M. Johnson, D. M. Simon, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE 1959, 55–68 (1993).

Kozaitis, S. P.

S. P. Kozaitis, W. E. Foor, “Optical correlation using reduced resolution filters,” Opt. Eng. 31, 1929–1935 (1992).
[CrossRef]

Lindell, S.

S. Lindell, “tops optical correlation program,” in Transition of Optical Processors into Systems 1993, L. E. Garn, L. L. Graceffo, eds., Proc. SPIE 1958, 7–18 (1993).

Mallat, S. G.

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

Miller, P. C.

P. C. Miller, “Reduced-resolution synthetic-discriminant-function design by multiresolution wavelet analysis,” Appl. Opt. 34, 865–878 (1995).
[CrossRef] [PubMed]

P. C. Miller, “Optimum reduced-resolution phase-only filters for extended target recognition,” Opt. Eng. 32, 2890–2898 (1993).
[CrossRef]

Serati, R. A.

S. A. Serati, T. K. Ewing, R. A. Serati, K. M. Johnson, D. M. Simon, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE 1959, 55–68 (1993).

Serati, S. A.

S. A. Serati, T. K. Ewing, R. A. Serati, K. M. Johnson, D. M. Simon, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE 1959, 55–68 (1993).

Simon, D. M.

S. A. Serati, T. K. Ewing, R. A. Serati, K. M. Johnson, D. M. Simon, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE 1959, 55–68 (1993).

Washwell, E.

G. Gheen, E. Washwell, C. Huang, “Problems facing optical correlators,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE 1772, 96–102 (1992).

Woods, C. L.

Appl. Opt. (4)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

Opt. Eng. (2)

S. P. Kozaitis, W. E. Foor, “Optical correlation using reduced resolution filters,” Opt. Eng. 31, 1929–1935 (1992).
[CrossRef]

P. C. Miller, “Optimum reduced-resolution phase-only filters for extended target recognition,” Opt. Eng. 32, 2890–2898 (1993).
[CrossRef]

Other (5)

S. A. Serati, T. K. Ewing, R. A. Serati, K. M. Johnson, D. M. Simon, “Programmable 128 × 128 ferroelectric-liquid-crystal spatial-light-modulator compact correlator,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. SPIE 1959, 55–68 (1993).

S. Lindell, “tops optical correlation program,” in Transition of Optical Processors into Systems 1993, L. E. Garn, L. L. Graceffo, eds., Proc. SPIE 1958, 7–18 (1993).

D. L. Flannery, S. C. Gustafson, “Adaptive optical correlation using neural network approaches,” in Optical Pattern Recognition, J. L. Horner, B. Javadi, eds., Vol. CR40 of SPIE Critical Review Series (SPIE Optical Engineering Press, Bellingham, Wash., 1992), pp. 25–45.

C. K. Chui, An Introduction to Wavelets (Academic, San Diego, 1992).

G. Gheen, E. Washwell, C. Huang, “Problems facing optical correlators,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE 1772, 96–102 (1992).

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 (16)

Fig. 1
Fig. 1

Schematic diagram of a modern optical correlator consisting of a classical 4-f FT-lens arrangement with pixellated SLM’s, a pixellated CCD, and a low-pass filter.

Fig. 2
Fig. 2

Edge-enhanced binary training images for (a) in-class ship 1, (b) out-of-class ship 2, and (c) ship 3.

Fig. 3
Fig. 3

Plot of log10 D versus N as calculated with both simulation 64 and simulation 256. The filters were designed by the use of conventional 64 × 64 pixel DFT’s of the training images.

Fig. 4
Fig. 4

Plots of (a) log10 D versus N as calculated with simulations 64 and 256 for filters trained on 64 × 64 pixel MWA approximations of the training image FT’s, and (b) the average in-class correlation peak intensity versus N as calculated with simulation 256 for 64 × 64 pixel filters trained on the MWA and downsampling approximations.

Fig. 5
Fig. 5

Plot of log10 D versus N as calculated with the 32 × 32 pixel simulation and simulation 256. The filters were designed with conventional 32 × 32 pixel DFT’s of the training images.

Fig. 6
Fig. 6

Plots of (a) log10 D versus N as calculated with the 32 × 32 pixel simulation and simulation 256 for filters trained on 32 × 32 pixel MWA approximations of the training-image FT’s, and (b) the in-class average correlation peak intensity versus N as calculated with simulation 256 for 32 × 32 pixel filters trained on the MWA and downsampling approximations.

Fig. 7
Fig. 7

Plot of the correlation-plane intensity and a slice through the plot: (a) a 3-D plot of the correlation-plane intensity, calculated with simulation 256, for which the input was an in-class ship 1 target and the filter was trained on downsampled DFT’s with a value of N = 6, and (b) a 1-D slice through the center of (a).

Fig. 8
Fig. 8

Plot of the correlation-plane intensity and a slice through the plot: (a) a 3-D plot of the correlation-plane intensity, calculated with simulation 256, for which the input was an in-class ship 1 target and the filter was trained on MWA approximations with a value of N = 6, and (b) a 1-D slice through the center of (a).

Fig. 9
Fig. 9

(a) Schematic diagram of an N-point DFT with a resolution of u 0/2 j and a rect function of width u 0/2 j −2. (b) An (N/4)-point discrete MWA approximation. (c) An N-point discrete simulation of a continuous MWA approximation.

Fig. 10
Fig. 10

Schematic diagram illustrating how the various terms in M h 2 j−2 (x) contribute to the (a) zero-order, (b) first-order, and (c) second-order impulse responses.

Fig. 11
Fig. 11

Schematic diagram illustrating how the various terms in D h 2 j−2 (x) contribute to the (a) zero-order, (b) first-order, and (c) second-order impulse responses.

Fig. 12
Fig. 12

Plots of (a) s ˜ (x), (b) the real component, and (c) the imaginary component of the impulse response M h 2 j−2 (x), and of (d) the real component and (e) the imaginary component of the impulse response D h 2 j−2 (x).

Fig. 13
Fig. 13

Correlation plots obtained with (a) M h 2 j−2 (x) and (b) D h 2 j−2 (x).

Fig. 14
Fig. 14

Enlarged plot of the region of the sinc curve (solid curve), the sinc curve multiplied by the cosine term (dashed curve), and the sinc curve multiplied by the sine term (dotted curve), around the node x = L/4. The sinc curve multiplied by the cosine term is reflected around the point x = L/4 for ease of comparison.

Fig. 15
Fig. 15

Plots of B versus x d around the node at x = L/4, corresponding to each of the curves in Fig. 14. (The legend is the same as that for Fig. 14.)

Fig. 16
Fig. 16

Plots of log(B) versus x d around the node at x = L/2 corresponding to each of the curves in Fig. 14. (The legend is the same as that for Fig. 14.)

Equations (74)

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

c ( x , y ) = I ( u , v ) F ( u , v ) exp [ 2 π i ( x u + y v ) ] d u d v , = i ( x , y ) f ( x , y ) ,
i ( x , y ) = [ n = - 2 j - 1 2 j - 1 - 1 m = - 2 j - 1 2 j - 1 - 1 s n , m δ ( x - n L 2 j , y - m L 2 j ) ] rect { [ x - ( L / 2 j + 1 ) L / 2 j } rect { [ y - ( L / 2 j + 1 ) ] L / 2 j }
rect ( x a ) = { 1 when - a / 2 x < a / 2 0 ' elsewhere ,
s n , m = y = m ( L / 2 j ) ( m + 1 ) ( L / 2 j ) x = n ( L / 2 j ) ( n + 1 ) ( L / 2 j ) s ( x , y ) d x d y ,
s n , m = [ s ( x , y ) ( rect { [ x + ( L / 2 j + 1 ) ] ( L / 2 j ) } × rect { [ y + ( L / 2 j + 1 ) ] ( L / 2 j ) } ) ] ( n L 2 j , m L 2 j ) .
I ( u , v ) = { [ S ( u , v ) exp ( i π u L 2 j ) L 2 j sinc ( u L 2 j ) × exp ( i π v L 2 j ) L 2 j sinc ( v L 2 j ) ] [ ( 2 j L ) 2 n = - m = - δ ( u - n 2 j L , v - m 2 j L ) L sinc ( u L ) L sinc ( v L ) ] } exp ( - i π u L 2 j ) × L 2 j sinc ( u L 2 j ) exp ( - i π v L 2 j ) L 2 j sinc ( v L 2 j ) ,
sinc ( α ) = sin ( π α ) π α .
u max = x max λ f ,
2 j - 1 L = ( L / 2 ) λ f f = L 2 λ 2 j .
I ˜ ( u , v ) = I ( u , v ) rect ( u L 2 j ) rect ( v L 2 j ) .
i ˜ ( x , y ) = i ( x , y ) 2 j L sinc ( x 2 j L ) 2 j L sinc ( y 2 j L ) .
F ( u , v ) = [ n = - 2 k - 1 2 k - 1 - 1 m = - 2 k - 1 2 k - 1 - 1 H n , m δ ( u - n L λ f 2 k , v - m L λ f 2 k ) ] ( rect { λ f 2 k [ u - ( L / λ f 2 k + 1 ) ] L } × rect { λ f 2 k [ v - ( L / λ f 2 k + 1 ) ] L } ) ,
F ( u , v ) = { [ H ( u , v ) n = - m = - δ ( u - n L λ f 2 k , v - m L λ f 2 k ) ] × rect ( u λ f L ) rect ( v λ f L ) } ( rect { λ f 2 k [ u - ( L / λ f 2 k + 1 ) ] L } × rect { λ f 2 k [ v - ( L / λ f 2 k + 1 ) ] L } ) ,
f ( x , y ) = { [ h ( x , y ) ( λ f 2 k L ) 2 n = - m = - × δ ( x - n λ f 2 k L , y - m λ f 2 k L ) ] L λ f sinc ( x L λ f ) L λ f sinc ( y L λ f ) } L λ f 2 k exp ( i π x L λ f 2 k ) × sinc ( x L λ f 2 k ) L λ f 2 k exp ( i π y L λ f 2 k ) sinc ( y L λ f 2 k ) ,
f ( x , y ) = { [ ( λ f 2 k L ) 2 n = - m = - × h ( x - n λ f 2 k L , y - m λ f 2 k L ) ] L λ f sinc ( x L λ f ) L λ f sinc ( y L λ f ) } × L λ f 2 k exp ( i π x L λ f 2 k ) sinc ( x L λ f 2 k ) L λ f 2 k × exp ( i π y L λ f 2 k ) sinc ( y L λ f 2 k ) .
c ( 0 , 0 ) = I ˜ ( u , v ) { [ n = - 2 j - 1 2 j - 1 - 1 m = - 2 j - 1 2 j - 1 - 1 × H n , m δ ( u - n L λ f 2 j , v - m L λ f 2 j ) ] ( rect { λ f 2 j [ u - ( L / λ f 2 j + 1 ) ] L } × rect { λ f 2 j [ v - ( L / λ f 2 j + 1 ) ] L } ) } d u d v .
S n , m = S ˜ ( n L , m L ) ,
S ˜ ( u , v ) = FT { [ s ( x , y ) ( rect { [ x + ( L / 2 j + 1 ) ] ( L / 2 j ) } × rect { [ y + ( L / 2 j + 1 ) ] ( L / 2 j ) } ) ] n = - 2 j - 1 2 j - 1 - 1 m = - 2 j - 1 2 j - 1 - 1 × δ ( x - n L 2 j , y - m L 2 j ) } .
c = n = - 2 j - 1 2 j - 1 - 1 m = - 2 j - 1 2 j - 1 - 1 S n , m H n , m .
c = l = 1 d S l H l ,             where d = 2 j × 2 j = S · H ,
H = BPOF { n = 1 N a n S n } ,
S n · H = c n ,
a n i + 1 = a n i + β [ c n - c 1 ( m n i m 1 i ) ] ,
D = min 1 n N / 3 m n 2 max N / 3 < n N m n 2 ,
{ 0 } V 2 - 1 V 2 0 V 2 1 L 2 ( R 2 ) ,
ϕ 2 j , k , l ( x , y ) = 2 j ϕ ( 2 j x - k , 2 j y - l ) .
{ 0 } W 2 - 1 W 2 0 W 2 1 L 2 ( R 2 ) ,
ψ 2 j , k , l ( x , y ) = 2 j ψ ( 2 j x - k , 2 j y - l ) .
V 2 j = W 2 j - 3 W 2 j - 2 W 2 j - 1 .
V 2 j + 1 = V 2 j W 2 j .
S 2 j ( u , v ) = k , l = - S ( u , v ) , ϕ 2 j , k , l ( u , v ) [ ϕ 2 j , k , l ( u , v ) ] ,
S ( u , v ) , ϕ 2 j , k , l ( u , v ) = S ( u , v ) ϕ 2 j , k , l ( u , v ) d u d v ,
Δ S 2 j ( u , v ) = k , l = - S ( u , v ) , ψ 2 j , k , l ( u , v ) [ ψ 2 j , k , l ( u , v ) ] ,
S 2 j + 1 ( u , v ) = S 2 j ( u , v ) + Δ S 2 j ( u , v ) .
Δ S 2 j ( u , v ) , S 2 j ( u , v ) = 0.
[ S 2 j + 1 ( u , v ) - Y 2 j ( u , v ) ] [ Y 2 j ( u , v ) ] > 0 ,
S 2 j + 1 ( u , v ) , S 2 j ( u , v ) | | S 2 j ( u , v ) | | > S 2 j + 1 ( u , v ) , Y 2 j ( u , v ) | | Y 2 j ( u , v ) | | ,
H M 2 j = BPOF { n a n I S M n 2 j } .
S M n 2 j · H M 2 j = m M n ,
H M 2 j [ ( k / 2 j + 2 ) , ( l / 2 j + 2 ) ] = H M 2 j ( u , v ) × k l δ [ u - ( k / 2 j + 2 ) , v - ( l / 2 j + 2 ) ] .
S n 2 j + 2 · H M 2 j + 2 2 j = p M n ,
[ ( S n 2 j + 2 - S M n 2 j + 2 2 j ) + S M n 2 j + 2 2 j ] · H M 2 j + 2 2 j = p M n .
( S n 2 j + 2 - S M n 2 j + 2 2 j ) · H M 2 j + 2 2 j + S M n 2 j + 2 2 j · H M 2 j + 2 2 j = p M n .
( S n 2 j + 2 - S M n 2 j + 2 2 j ) · H M 2 j + 2 2 j = 0.
p M n = S M n 2 j + 2 2 j · H M 2 j + 2 2 j m M n ,
H D 2 j = BPOF { n b n I S D n 2 j } .
S D n 2 j · H D 2 j = m D n .
S n 2 j + 2 · H D 2 j + 2 2 j = p D n .
( S n 2 j + 2 - S D n 2 j + 2 2 j ) · H D 2 j + 2 2 j + S D n 2 j + 2 2 j · H D 2 j + 2 2 j = p D n .
( S n 2 j + 2 - S D n 2 j + 2 2 j ) · H D 2 j + 2 2 j > 0.
p D n m D n .
R = [ 1 2 1 2 1 2 1 2 ] .
s ˜ ( x ) = n = - 2 j - 1 2 j - 1 - 1 s n δ ( x - n L 2 j ) ,
S D 2 j = S ˜ ( u ) n = - 2 j - 1 2 j - 1 - 1 δ ( u - n u 0 2 j ) = n = - 2 j - 1 2 j - 1 - 1 S D 2 j , n δ ( u - n u 0 2 j ) ,
s D 2 j ( - x ) * = IFT { S D 2 j * } = [ s ˜ ( - x ) * 2 j L sinc ( 2 j L x ) ] L n = - δ ( x - n L ) ,
S D 2 j - 2 = S ˜ ( u ) n = - 2 j - 3 2 j - 3 - 1 δ ( u - n u 0 2 j - 2 ) .
s D 2 j - 2 ( - x ) * = IFT { S D 2 j - 2 * } = [ s ˜ ( - x ) * 2 j L sinc ( 2 j L x ) ] L 4 n = - δ ( x - n L 4 ) .
FT { n = - δ ( x - n a ) } = 1 a n = - δ ( u - n a )
S M 2 j - 2 = { S D 2 j rect [ u + ( u 0 / 2 ( j - 2 ) + 1 ) - ( u 0 / 2 j + 1 ) ( u 0 / 2 j - 2 ) ] } × n = - 2 j - 3 2 j - 3 - 1 δ ( u - n u 0 2 j - 2 ) .
S M 2 j - 2 = { S D 2 j rect [ u + ( u 0 / 2 ( j - 2 ) + 1 ) - ( u 0 / 2 j + 1 ) ( u 0 / 2 j - 2 ) ] } × n = - δ ( u - n u 0 2 j - 2 ) .
S M 2 j - 2 ( u ) = [ ( S D 2 j rect { u + [ u 0 / 2 ( j - 2 ) + 1 ] - ( u 0 / 2 j + 1 ) ( u 0 / 2 j - 2 ) } ) × n = - δ ( u - n u 0 2 j - 2 ) ] rect ( u - [ u 0 / 2 ( j - 2 ) + 1 ] + ( u 0 / 2 j + 1 ) ( u 0 / 2 j - 2 ) ) ,
h M 2 j - 2 ( x ) = [ ( s D 2 j ( - x ) * exp { 2 π i u 0 [ - 1 2 ( j - 2 ) + 1 + 1 2 j + 1 ] x } × u 0 2 j - 2 sinc ( u 0 x 2 j - 2 ) ) 2 j - 2 u 0 n = - δ ( x - n 2 j - 2 u 0 ) ] × exp { 2 π i u 0 [ 1 2 ( j - 2 ) + 1 - 1 2 j + 1 ] x } u 0 2 j - 2 sinc ( u 0 x 2 j - 2 ) .
h M 2 j - 2 ( x ) = { [ s D 2 j ( - x ) * exp ( - i 3 π x L ) 4 L sinc ( 4 x L ) ] L 4 n = - δ ( x - n L 4 ) } × exp ( i 3 π x L ) 4 L sinc ( 4 x L ) .
h M 2 j - 2 0 ( x ) = [ s D 2 j ( - x ) * exp ( - i 3 π x L ) sinc ( 4 x L ) ] , × exp ( i 3 π x L ) 4 L sinc ( 4 x L )
h M 2 j - 2 0 ( x ) = s D 2 j ( - x ) * 4 L sinc 2 ( 4 x L ) .
[ s D 2 j ( - x ) * ] sinc ( 4 x L ) [ s ˜ ( - x ) * 2 j L sin c ( 2 j L x ) ] × sinc ( 4 x L ) .
h M 2 j - 2 1 ( x ) = s D 2 j ( - x + L 4 ) * exp ( i 3 π 4 ) sinc [ 4 ( x - L / 4 ) L ] 4 L × sinc ( 4 x L ) = ( - 1 2 + i 1 2 ) × { s D 2 j ( - x + L 4 ) * sinc [ 4 ( x - L / 4 ) L ] } 4 L × sinc ( 4 x L ) .
h M 2 j - 2 2 ( x ) = s D 2 j ( - x + L 2 ) * exp ( i 3 π 2 ) sinc [ 4 ( x - L / 2 ) L ] 4 L × sinc ( 4 x L ) = - i { s D 2 j ( - x + L 2 ) * × sinc [ 4 ( x - L / 2 ) L ] } 4 L sinc ( 4 x L ) .
S D 2 j - 2 ( u ) = [ S ˜ ( u ) n = - 2 j - 3 2 j - 3 - 1 δ ( u - n u 0 2 j - 2 ) ] rect { u - [ u 0 / 2 ( j - 2 ) + 1 ] + ( u 0 / 2 j + 1 ) u 0 / 2 j - 2 } ,
h D 2 j - 2 ( x ) = { [ s ˜ ( - x ) * 2 j L sinc ( 2 j x L ) ] L 4 - δ ( x - n L 4 ) } exp ( i 3 π x L ) 4 L sinc ( 4 x L ) .
h D 2 j - 2 0 ( x ) = [ s ˜ ( - x ) * 2 j L sinc ( 2 j x L ) ] exp ( i 3 π x L ) sinc ( 4 x L ) = [ s ˜ ( - x ) * 2 j L sinc ( 2 j x L ) ] × [ cos ( 3 π x L ) + i sin ( 3 π x L ) ] sinc ( 4 x L ) .
h D 2 j - 2 1 ( x ) = { s ˜ ( - x ) * 2 j L sinc [ 2 j L ( x - L 4 ) ] } × [ cos ( 3 π x L ) + i sin ( 3 π x L ) ] sinc ( 4 x L ) .
h D 2 j - 2 2 ( x ) = { s ˜ ( - x ) * 2 j L sinc [ 2 j L ( x - L 2 ) ] } × [ cos ( 3 π x L ) + i sin ( 3 π x L ) ] sinc ( 4 x L ) .
B ( x d ) = | f ( α - x d ) + f ( α + x d ) f ( α - x d ) - f ( α + x d ) | ,

Metrics