Abstract

Several approaches to the design of reduced-resolution synthetic discriminant functions (SDF's) using multiresolution wavelet analysis (MWA) techniques are investigated. In the first approach, reduced-resolution approximations of a full-resolution SDF are obtained by MWA. In the second approach, reduced-resolution approximations of the training-image Fourier transforms are obtained by MWA, and a reduced-resolution SDF is obtained directly by training on these. For both approaches, reduced-resolution MICE-SDF filters were designed with MWA and conventional down-sampling techniques. Simulations showed that filters designed by the second approach with MWA techniques permitted reductions in the number of filter pixels from 128 × 128 to 32 × 32, while still satisfying the design constraints. In comparison, the performances of 32 × 32 filters designed by conventional down-sampling techniques were significantly degraded.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. G. Temen, C. F. Hester, “Design considerations for pattern recognition demonstration for transition of optical processing to systems (TOPS),” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 16–20 (1992).
  2. S. D. Lindell, W. B. Hahn, “Overview of the Martin Marietta transfer of optical processing to systems (TOPS) optical correlation program,” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 21–30 (1992).
  3. S. P. Kozaitis, W. E. Foor, “Optical correlation using reduced resolution filters,” Opt. Eng. 31, 1929–1935 (1992).
    [CrossRef]
  4. D. L. Flannery, S. C. Gustafson, “Adaptive optical correlation using neural network approaches,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.CR40, 25–45 (1992).
  5. P. C. Miller, “Optimum reduced-resolution phase-only filters for extended target recognition,” Opt. Eng. 32, 2890–2898 (1993).
    [CrossRef]
  6. M. W. Farn, J. W. Goodman, “Optimal maximum correlation filter for arbitrarily constrained devices,” Appl. Opt. 28, 3362–3366 (1989).
    [CrossRef]
  7. C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
    [CrossRef] [PubMed]
  8. D. Casasent, G. Ravichandran, “Advanced distortion-invariant minimum average correlation energy (MACE) filters,” Appl. Opt. 31, 1109–1116 (1992).
    [CrossRef] [PubMed]
  9. G. Ravichandran, D. Casasent, “Minimum noise and correlation energy optical correlation filter,” Appl. Opt. 31, 1823–1833 (1992).
    [CrossRef] [PubMed]
  10. G. Ravichandran, D. Casasent, “Noise and discrimination performance of the MINACE optical correlation filter,” in Automatic Object Recognition, F. A. Sadjadi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1471, 223–248 (1991).
  11. S. G. Mallat, “Atheory 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, New York, 1992), Chap. 1, pp. 1–22;An Introduction to WaveletsChap. 5, pp. 119–122.
  13. B. V. K. Vijaya Kumar, “Minimum-variance synthetic discriminant functins,” J. Opt. Soc. Am. A 3, 1579–1583 (1986).
    [CrossRef]

1993

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

1992

1989

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

M. W. Farn, J. W. Goodman, “Optimal maximum correlation filter for arbitrarily constrained devices,” Appl. Opt. 28, 3362–3366 (1989).
[CrossRef]

1986

1980

Casasent, D.

Chui, C. K.

C. K. Chui, An Introduction to Wavelets (Academic, New York, 1992), Chap. 1, pp. 1–22;An Introduction to WaveletsChap. 5, pp. 119–122.

Farn, M. W.

Flannery, D. L.

D. L. Flannery, S. C. Gustafson, “Adaptive optical correlation using neural network approaches,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.CR40, 25–45 (1992).

Foor, W. E.

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

Goodman, J. W.

Gustafson, S. C.

D. L. Flannery, S. C. Gustafson, “Adaptive optical correlation using neural network approaches,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.CR40, 25–45 (1992).

Hahn, W. B.

S. D. Lindell, W. B. Hahn, “Overview of the Martin Marietta transfer of optical processing to systems (TOPS) optical correlation program,” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 21–30 (1992).

Hester, C. F.

C. F. Hester, D. Casasent, “Multivariant technique for multiclass pattern recognition,” Appl. Opt. 19, 1758–1761 (1980).
[CrossRef] [PubMed]

M. G. Temen, C. F. Hester, “Design considerations for pattern recognition demonstration for transition of optical processing to systems (TOPS),” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 16–20 (1992).

Kozaitis, S. P.

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

Lindell, S. D.

S. D. Lindell, W. B. Hahn, “Overview of the Martin Marietta transfer of optical processing to systems (TOPS) optical correlation program,” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 21–30 (1992).

Mallat, S. G.

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

Miller, P. C.

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

Ravichandran, G.

G. Ravichandran, D. Casasent, “Minimum noise and correlation energy optical correlation filter,” Appl. Opt. 31, 1823–1833 (1992).
[CrossRef] [PubMed]

D. Casasent, G. Ravichandran, “Advanced distortion-invariant minimum average correlation energy (MACE) filters,” Appl. Opt. 31, 1109–1116 (1992).
[CrossRef] [PubMed]

G. Ravichandran, D. Casasent, “Noise and discrimination performance of the MINACE optical correlation filter,” in Automatic Object Recognition, F. A. Sadjadi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1471, 223–248 (1991).

Temen, M. G.

M. G. Temen, C. F. Hester, “Design considerations for pattern recognition demonstration for transition of optical processing to systems (TOPS),” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 16–20 (1992).

Vijaya Kumar, B. V. K.

Appl. Opt.

IEEE Trans. Pattern Anal. Mach. Intell.

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

J. Opt. Soc. Am. A

Opt. Eng.

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

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

Other

D. L. Flannery, S. C. Gustafson, “Adaptive optical correlation using neural network approaches,” in Optical Pattern Recognition, J. L. Horner, B. Javidi, eds., Proc. Soc. Photo-Opt. Instrum. Eng.CR40, 25–45 (1992).

M. G. Temen, C. F. Hester, “Design considerations for pattern recognition demonstration for transition of optical processing to systems (TOPS),” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 16–20 (1992).

S. D. Lindell, W. B. Hahn, “Overview of the Martin Marietta transfer of optical processing to systems (TOPS) optical correlation program,” in Optical Pattern Recognition III, D. P. Casasent, T. Chao, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1701, 21–30 (1992).

G. Ravichandran, D. Casasent, “Noise and discrimination performance of the MINACE optical correlation filter,” in Automatic Object Recognition, F. A. Sadjadi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1471, 223–248 (1991).

C. K. Chui, An Introduction to Wavelets (Academic, New York, 1992), Chap. 1, pp. 1–22;An Introduction to WaveletsChap. 5, pp. 119–122.

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 (10)

Fig. 1
Fig. 1

Discrete approximations of a function (the function is actually a slice through a 2-D MICE SDF) at resolutions (a) j = 0 and (c) j = −1. Also shown are interpolations of (a) and (c) with the scaling functions given by (b) relation (18) and (d) relation (20).

Fig. 2
Fig. 2

Graphical representations of (a) ϕ(x) and ϕ2−1(x) and (b) r(m).

Fig. 3
Fig. 3

Schematic diagram of simplified MWA example function space that shows the wavelet and down-sampling approximations, f2−1 and, f 2 1 respectively, of f0 in V2−1, and their respective difference functions, g2−1 and g 2 1 (some vectors are slightly displaced for clarity).

Fig. 4
Fig. 4

(a) Graph showing the first approach to reduced-resolution SDF design, where the full-resolution filter H0 is approximated by H2−1 (wavelet) and H 2 1 (down sampling). (b) Graph showing the second approach, where the training images Xi0 are approximated by Xi2−1 (wavelet) and X i 2 1 (down sampling). The reduced-resolution training images are then used to train a reduced-resolution SDF (some vectors are slightly displaced for clarity).

Fig. 5
Fig. 5

Gray-scale plots of tank training images at aspect angles of (a) 0°, (b) 50°, and (c) 90°.

Fig. 6
Fig. 6

Graphs showing variation of (a) the NCP and (b) the PCE with training-image number for filters H0, H12−1, H12−2, H 1 2 1 , and H 1 2 2 .

Fig. 7
Fig. 7

Same as Fig. 6 but for H0, H22−1, H22−2, H 2 2 1 , and H 2 2 2 .

Fig. 8
Fig. 8

Three-dimensional intensity plots of the correlation plane obtained with training images 0 [(a) and (b)], 5 [(c) and (d)] and 9 [(e) and (f)] as inputs. The filters used are H22−2 [(a), (c), and (e)] and H 2 2 1 [(b, (d), and (f)].

Fig. 9
Fig. 9

Graphical representation of ϕ(x) and Ψ2−1(x).

Fig. 10
Fig. 10

Schematic diagram showing the relationships between the vector spaces V0, V2−1 and W2−1 and the basis vectors ϕ(x), ϕ(x − 1), ϕ2−1(x), and Ψ2−1(x).

Tables (1)

Tables Icon

Table 1 NP50 versus Training Image for All Filters

Equations (51)

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

x i = [ x i ( 1 ) , x i ( 2 ) , , x i ( d ) ] T ,
X i = [ X i ( 1 ) , X i ( 2 ) , , X i ( d ) ] T ,
H + X i = u i ,
H + X = u ,
H = A 1 X ( X + A 1 X ) 1 u .
Ψ 2 j ( x 2 j k , y 2 j l ) = 2 j Ψ ( 2 j x k , 2 j y l ) ,
L 2 ( R 2 ) = W 2 1 W 2 0 W 2 1 ,
f ( x , y ) = + g 2 1 ( x , y ) + g 2 0 ( x , y ) + g 2 1 ( x , y ) + ,
V 2 j = W 2 j 3 W 2 j 2 W 2 j 1 ,
ϕ 2 j ( x 2 j k , y 2 j l ) = 2 j ϕ ( 2 j x k , 2 j y l )
V 2 1 V 2 0 V 2 1 .
V 2 j + 1 = V 2 j W 2 j .
f 2 j + 1 ( x , y ) = f 2 j ( x , y ) + g 2 j ( x , y ) ,
d 2 j ( x , y ) V 2 j , d 2 j ( x , y ) f ( x , y ) f 2 j ( x , y ) f ( x , y ) ,
f 2 j ( x , y ) = k , l = f ( u , v ) , ϕ 2 j ( u 2 j k , v 2 j l ) × ϕ 2 j ( x 2 j k , y 2 j l ) ,
f ( u , v ) , ϕ 2 j ( u 2 j k , v 2 j l ) = f ( u , v ) , ϕ 2 j ( u 2 j k , v 2 j l ) d u d v , k , l Z ,
f ( x , y ) , ϕ 2 j ( x 2 j k , y 2 j l ) = n , m = r ( m 2 k , n 2 l × f ( x , y ) , ϕ 2 j + 1 ( x 2 j 1 m , y 2 j 1 n ) ,
m , n Z , r ( m , n ) = ϕ 2 1 ( u , v ) , ϕ ( u m , v n )
ϕ ( x ) : = { 1 for 0 x < 1 0 otherwise ,
Ψ ( x ) : = { 1 for 0 x < ½ 1 for ½ x < 1 0 otherwise ,
ϕ 2 1 ( x ) : = { 2 1 / 2 for 0 x < 2 0 otherwise ,
f 0 ( x ) = x 0 ϕ ( x ) + x 1 ϕ ( x 1 ) .
f 2 1 ( x ) = [ x 0 / ( 2 1 / 2 ) + x 1 / ( 2 1 / 2 ) ] ϕ 2 1 ( x ) ,
g 2 1 ( x ) = [ x 0 / ( 2 1 / 2 ) x 1 / ( 2 1 / 2 ) ] Ψ 2 1 ( x ) .
f 2 1 ( x ) = x 0 ϕ ( x ) + x 0 ϕ ( x 1 ) = ( 2 1 / 2 x 0 ) ϕ 2 1 ( x )
ϕ ( x , y ) : = { 1 for 0 x , y < 1 0 otherwise ,
ϕ 2 1 ( x , y ) : = { 2 1 for 0 x , y < 2 0 otherwise .
r ( m , n ) = { 2 1 for m , n = 0 , 1 0 otherwise .
T ( k , k ) = max [ D 1 ( k , k ) , D 2 ( k , k ) , D N ( k , k ) ] ,
H 0 = H 1 2 1 + Δ H 1 2 1 .
H 1 2 1 + X i 0 = ( H 0 + Δ H 1 2 1 + ) X i 0 = u i Δ H 1 2 1 + X i 0 ,
H 1 2 2 + X i 0 = ( H 0 + Δ H 1 2 1 + Δ H 1 2 2 + ) X i 0 = u i Δ H 1 2 1 + X i 0 Δ H 1 2 2 + X i 0 .
H 2 2 1 + X i 2 1 = u i .
X i 0 = X i 2 1 + Δ X i 2 1 .
H 2 2 1 + X i 0 = H 2 2 1 + ( X i 2 1 + Δ X i 2 1 ) .
H 2 2 1 + Δ X i 2 1 = 0 .
H 2 2 1 + X i 0 = H 2 2 1 + X i 2 1 = u i .
ϕ 2 J ( x 2 j k , y 2 j l ) = ϕ ( 2 j x k , 2 j y l ) ,
H 1 2 1
H 1 2 2
H 2 2 1
H 2 2 2
ϕ 2 j ( x 2 j k , y 2 j l ) = n , m = + ϕ 2 j ( u 2 j k , v 2 j l ) , ϕ 2 j + 1 ( u 2 j 1 m , v 2 j 1 n ) × ϕ 2 j + 1 ( x 2 j 1 m , y 2 j 1 n ) .
ϕ 2 j ( u 2 j k , v 2 j l ) , ϕ 2 j + 1 ( u 2 j 1 m , v 2 j 1 n ) = ϕ 2 1 ( ( ( u , v ) , ϕ [ u ( m 2 k ) , v ( n 2 l ) ] .
f ( x , y ) , ϕ 2 j ( x 2 j k , y 2 j l ) = n , m = + ϕ 2 1 ( u , v ) , ϕ [ u ( m 2 k ) , v ( n 2 l ) ] × f ( x , y ) , ϕ 2 j + 1 ( x 2 j 1 m , y 2 j 1 n ) .
m , n Z , r ( m , n ) = ϕ 2 1 ( u , v ) , ϕ ( u m , v n ) ,
f ( x , y ) , ϕ 2 j ( x 2 j k , y 2 j l ) = n , m = + r ( m 2 k , n 2 l × f ( x , y ) , ϕ 2 j + 1 ( x 2 j 1 m , y 2 j 1 n ) ,
m , n Z , s ( m , n ) = Ψ 2 1 ( x , y ) , ϕ ( x m , y n ) .
Ψ 2 1 ( x ) : = { 2 1 / 2 for 0 x < 1 2 1 / 2 for 0 x < 2 0 otherwise .
ϕ 2 1 ( x ) = 2 1 / 2 ϕ ( x ) + 2 1 / 2 ϕ ( x 1 ) .
Ψ 2 1 ( x ) = 2 1 / 2 ϕ ( x ) 2 1 / 2 ϕ ( x 1 ) ,

Metrics