One-dimensional logarithmic harmonic synthetic ... - OSA Publishing
April 1, 1998 / Vol. 23, No. 7 / OPTICS LETTERS
537
One-dimensional logarithmic harmonic synthetic discriminant function filters for shift-, scale-, and projection-invariant pattern recognition Jianping Yao and Guy Lebreton Groupe d’Etudes Signaux et Systemes, ` Universite´ de Toulon et du Var, B.P. 132, 83957 La Garde Cedex, France Received October 16, 1997 We introduce a new approach for shift-, scale-, and projection-invariant pattern recognition that combines the harmonic expansion and the synthetic discriminant function approaches by use of a synthetic discriminant function filter with equal-order one-dimensional logarithmic harmonic components. Because projection invariance in one direction is guaranteed by the harmonics, the required number of training images is much fewer than with classical synthetic discriminant function filters. 1998 Optical Society of America OCIS codes: 070.4550, 070.5010, 100.4550, 100.5010, 100.6740.
Different solutions for implementing shift-, scale-, and projection-invariant pattern recognition have been proposed by various authors. Since all these solutions require a priori knowledge of the projection axis, the invariance is restricted to projections along two orthogonal directions, which is called two-dimensional (2D) scale invariance here. Following the idea of preliminary coordinate transforms from Casasent and Psaltis,1 Mendlovic et al.2 demonstrated an optical implementation of the 2D Mellin transform without shift invariance. More recently, to include shift invariance, Mendlovic et al.3 proposed a composite harmonic filter. The reference pattern is f irst decomposed into its logarithmic radial harmonic components, and a single harmonic component is selected for scale invariance. The single component is then decomposed into its one-dimensional (1D) logarithmic harmonic components (for the projection invariance), from which a single harmonic component is kept as the correlation filter. From this two-stage decomposition, one simultaneously obtains shift, scale, and projection invariance. Since the single harmonic component from the two-stage decomposition contains a very small part of the energy of the reference pattern, its correlation performance for discrimination and its noise robustness are very poor. Another approach, proposed by Szoplik, uses an anamorphic correlator,4 which yields a limited range of scale invariance with a matched f ilter. However, angular scanning with a wedge mask is required for sequential exploration of partial matching at different scales. The synthetic discriminant function (SDF) approach, which can potentially solve any distortion problem,5,6 was investigated for invariance to one geometric distortion parameter.7 However, invariance to several parameters simultaneously would require an excessive number of training images for each pattern in a multiclass problem at the cost of lower discrimination capability and signal-to-noise ratio.8 The new f ilter proposed here, the 1D logarithmic harmonic SDF (1DLHSDF) f ilter, combines the harmonic expansion and the SDF approaches. The new approach differs from the rotation-invariant composite filters 0146-9592/98/070537-03$15.00/0
proposed by Arsenault,9 in which SDF constraints are used to improve the discrimination capability of the circular harmonic filters but not to provide an additional parameter of invariance. We obtained the 1DLHSDF by giving the SDF f ilter equal-order 1D logarithmic harmonic components that provide for the projection invariance along a given axis. The number of training images for the SDF is thus reduced to the projection parameter along the axis that is orthogonal to the direction of harmonic expansion. The 1D logarithmic harmonic expansion used here is a modif ied version of the expansion proposed by Mendlovic et al.10 to account for the pseudo-period T def ined by the domain of integration: ` X fN s y; x0 , y0 djxjis2p/T dN 21/2 , (1) f sx, y; x0 , y0 d N2`
with fN s y; x0 , y0 d
1 Z 2A f sx, y; x0 , y0 d 2T 2B 3 s2xd2is2p/T dN 21/2 dx 1 3
Z
B
A
1 2T
f sx, y; x0 , y0 dx2is2p/T dN 21/2 dx , (2)
where sx0 , y0 d is the expansion center, N is the harmonic order, B is the finite size of the pattern to be detected in the x direction, and A is the minimum size used in def ining the expansion, which must satisfy the orthogonality condition ln B 2 ln A T .
(3)
By selecting one harmonic from the expansion, one obtains a 1D logarithmic harmonic filter with shift and projection invariance: hsx, yd 2 fN s ydjxjis2p/T dN 21/2 .
(4)
The 1DLHSDF for a p-class problem is computed as a linear combination of 1D logarithmic harmonic f ilters 1998 Optical Society of America
538
OPTICS LETTERS / Vol. 23, No. 7 / April 1, 1998
with the same harmonic order N from q (properly spaced) projected versions of p patterns: sN d
h1DLHSDF sx, yd
M X
sN d
ai hi sx, yd,
M pq ,
(5)
i1
with a constraint that makes the correlation output at the origin equal for all harmonic components belonging to the same class and zero for other patterns. We obtain the complex coefficients ai in Eq. (5) by solving a matrix equation for their conjugates: 3 2 Cf1 hsNd 1DLHSDF 7 6 7 6 C sNd 6 f2 h1DLHSDF 7 7 6 .. 7 6 7 6 . 5 4 CfM hsNd 1DLHSDF
2
C sNd 6 f1 h1 6 C sNd 6 f2 h1 6 .. 6 6 . 4 CfM hsNd 1
Cf1 hsNd 2 Cf2 hsNd 2 .. . CfM hsNd 2
32 3 · · · Cf1 hsNd ap1 M 7 6 p7 · · · Cf2 hsNd 7 6a 7 M 76 76 .2 7 , .. .. 76 . 7 74 . 7 . . 5 5 apM · · · CfM hsNd
(6)
M
where the input vector is formed from the correlation constraints and the matrix is formed with the central values from all correlations. If the input pattern is a projected version of the pattern fi sx, yd, say, fi sxyb, yd, the central value of correlation of the 1DLHSDF filter can be expressed as a linear combination of individual correlations: sbd
Cf hsNd i
1DLHSDF
Shift, scale, and projection invariance were tested on four versions of the F18 aircraft pattern, as shown in Fig. 1(a), with the following sx, yd projection factors: a1, (1, 1); a2, (0.9, 0.6); a3, (0.5, 0.5); a4, (0.8, 0.5). It is easily seen that the 1IDLHSDF filter is invariant to shift, scale, and projection and that the outputs are sharp correlation peaks. It is important to note that, despite size differences, the correlation peak remains almost constant for all versions of input. This property is important for implementing a threshold level for pattern detection. Discrimination capability was tested on four versions of a TOR aircraft pattern, as shown in Fig. 2(a), with the same scale and projection factors as in Fig. 1(a). The correlation results with the 1DLHSDF are shown in Fig. 2(b). A comparison of the responses for the true class (Fig. 1) and the false class (Fig. 2), plotted on the same scale, demonstrates good discrimination capability. In conclusion, a novel filter has been described that performs pattern recognition with simultaneous invariance to shift, to scale changes, and to projection along a known direction. The 1D logarithmic expansion reduces the number of training images for the SDF to only one parameter of geometric distortion (vertical projection). The simulations demonstrated invariance of the correlation peak and good discrimination capability. In the simulations the SDF was computed as a linear combination of the training images with constraints on only the peak value, but all
i h ap1 Cfi hsNd 1 a2p Cfi hsNd 1 . . . 1 apM Cfi hsNd 1
2
µ ∂ p 2p 3 b exp 2i N ln b T µ ∂ p 2p , N ln b Cfi hsNd b exp 2i 1DLHSDF T
M
(7)
which demonstrates that the 1DLHSDF filter remains projection invariant along the axis of the harmonic expansion if all harmonic components used in making the SDF f ilter have equal harmonic order. Since projection invariance along the orthogonal axis is provided by the projected components of the SDF, 2D scale-invariant pattern recognition is achieved. To test this invariance we performed simulations for a simple two-class problem. For fast computations edge-detected objects are used, but we showed11 that edge detection gives results equivalent to those for correlation with a full-object phase-only filter (for which high-pass filtering provides the edge enhancement). All calculations are restricted to windows of 64 3 64 pixels. The 1DLHSDF is formed from 1D logarithmic harmonic components expanded along the horizontal axis, with harmonic order N 3 generated from four projected versions along the vertical axis (with projection factors 1, 0.8, 0.6, and 0.4), for F18 (true-class) and TOR (false-class) aircraft patterns.
Fig. 1. Shift-, scale-, and projection-invariance test for the IDLHSDF. (a) F18 aircraft and three scaled projected versions used as input. ( b) Correlation between the 1DLHSDF and the true-class inputs from (a).
April 1, 1998 / Vol. 23, No. 7 / OPTICS LETTERS
539
bustness, minimum average correlation energy13 for best correlation of peak sharpness and reduction of sidelobes, or a compromise between them with optimal trade-off filters,14 which allows one to obtain the desired correlation performance. References
Fig. 2. Discrimination capability test for the 1DLHSDF filter. (a) TOR aircraft and three scaled project versions used as input. (b) Correlation between the 1DLHSDF and the false-class inputs from (a).
optimization methods for SDF f ilters can be used with the 1DLHSDF: minimum variance12 for best noise ro-
1. D. Casasent and D. Psaltis, Appl. Opt. 15, 1795 (1976). 2. D. Mendlovic, N. Konforti, and E. Maron, Appl. Opt. 28, 4982 (1989). 3. D. Mendlovic, Z. Zalevsky, I. Kiryuschev, and G. Lebreton, Appl. Opt. 34, 310 (1995). 4. T. Szoplik, J. Opt. Soc. Am. A 9, 1419 (1984). 5. C. F. Hester and D. Casasent, Appl. Opt. 19, 1758 (1980). 6. D. Casasent, Appl. Opt. 23, 1620 (1984). 7. J. Riggins and S. Butler, Opt. Eng. 23, 721 (1984). 8. B. V. K. Vijaya Kumar and E. Pochapsky, J. Opt. Soc. Am. A 3, 777 (1986). 9. H. H. Arsenault, Proc. SPIE 613, 239 (1986). 10. D. Mendlovic, N. Konforti, and E. Marom, Appl. Opt. 29, 4784 (1990). 11. J. Yao, ‘‘Algorithms for optical distortion invariant pattern recognition,’’ Ph.D. dissertation (Universit´e de Toulon et du Var, La Garde, France, 1997). 12. B. V. K. Vijaya Kumar, J. Opt. Soc. Am. A 3, 1579 (1986). 13. A. Mahalanobis and B. V. K. Vijaya Kumar, Appl. Opt. 26, 3633 (1987). 14. Ph. R´efr´egier, Opt. Lett. 15, 854 (1990).
537
One-dimensional logarithmic harmonic synthetic discriminant function filters for shift-, scale-, and projection-invariant pattern recognition Jianping Yao and Guy Lebreton Groupe d’Etudes Signaux et Systemes, ` Universite´ de Toulon et du Var, B.P. 132, 83957 La Garde Cedex, France Received October 16, 1997 We introduce a new approach for shift-, scale-, and projection-invariant pattern recognition that combines the harmonic expansion and the synthetic discriminant function approaches by use of a synthetic discriminant function filter with equal-order one-dimensional logarithmic harmonic components. Because projection invariance in one direction is guaranteed by the harmonics, the required number of training images is much fewer than with classical synthetic discriminant function filters. 1998 Optical Society of America OCIS codes: 070.4550, 070.5010, 100.4550, 100.5010, 100.6740.
Different solutions for implementing shift-, scale-, and projection-invariant pattern recognition have been proposed by various authors. Since all these solutions require a priori knowledge of the projection axis, the invariance is restricted to projections along two orthogonal directions, which is called two-dimensional (2D) scale invariance here. Following the idea of preliminary coordinate transforms from Casasent and Psaltis,1 Mendlovic et al.2 demonstrated an optical implementation of the 2D Mellin transform without shift invariance. More recently, to include shift invariance, Mendlovic et al.3 proposed a composite harmonic filter. The reference pattern is f irst decomposed into its logarithmic radial harmonic components, and a single harmonic component is selected for scale invariance. The single component is then decomposed into its one-dimensional (1D) logarithmic harmonic components (for the projection invariance), from which a single harmonic component is kept as the correlation filter. From this two-stage decomposition, one simultaneously obtains shift, scale, and projection invariance. Since the single harmonic component from the two-stage decomposition contains a very small part of the energy of the reference pattern, its correlation performance for discrimination and its noise robustness are very poor. Another approach, proposed by Szoplik, uses an anamorphic correlator,4 which yields a limited range of scale invariance with a matched f ilter. However, angular scanning with a wedge mask is required for sequential exploration of partial matching at different scales. The synthetic discriminant function (SDF) approach, which can potentially solve any distortion problem,5,6 was investigated for invariance to one geometric distortion parameter.7 However, invariance to several parameters simultaneously would require an excessive number of training images for each pattern in a multiclass problem at the cost of lower discrimination capability and signal-to-noise ratio.8 The new f ilter proposed here, the 1D logarithmic harmonic SDF (1DLHSDF) f ilter, combines the harmonic expansion and the SDF approaches. The new approach differs from the rotation-invariant composite filters 0146-9592/98/070537-03$15.00/0
proposed by Arsenault,9 in which SDF constraints are used to improve the discrimination capability of the circular harmonic filters but not to provide an additional parameter of invariance. We obtained the 1DLHSDF by giving the SDF f ilter equal-order 1D logarithmic harmonic components that provide for the projection invariance along a given axis. The number of training images for the SDF is thus reduced to the projection parameter along the axis that is orthogonal to the direction of harmonic expansion. The 1D logarithmic harmonic expansion used here is a modif ied version of the expansion proposed by Mendlovic et al.10 to account for the pseudo-period T def ined by the domain of integration: ` X fN s y; x0 , y0 djxjis2p/T dN 21/2 , (1) f sx, y; x0 , y0 d N2`
with fN s y; x0 , y0 d
1 Z 2A f sx, y; x0 , y0 d 2T 2B 3 s2xd2is2p/T dN 21/2 dx 1 3
Z
B
A
1 2T
f sx, y; x0 , y0 dx2is2p/T dN 21/2 dx , (2)
where sx0 , y0 d is the expansion center, N is the harmonic order, B is the finite size of the pattern to be detected in the x direction, and A is the minimum size used in def ining the expansion, which must satisfy the orthogonality condition ln B 2 ln A T .
(3)
By selecting one harmonic from the expansion, one obtains a 1D logarithmic harmonic filter with shift and projection invariance: hsx, yd 2 fN s ydjxjis2p/T dN 21/2 .
(4)
The 1DLHSDF for a p-class problem is computed as a linear combination of 1D logarithmic harmonic f ilters 1998 Optical Society of America
538
OPTICS LETTERS / Vol. 23, No. 7 / April 1, 1998
with the same harmonic order N from q (properly spaced) projected versions of p patterns: sN d
h1DLHSDF sx, yd
M X
sN d
ai hi sx, yd,
M pq ,
(5)
i1
with a constraint that makes the correlation output at the origin equal for all harmonic components belonging to the same class and zero for other patterns. We obtain the complex coefficients ai in Eq. (5) by solving a matrix equation for their conjugates: 3 2 Cf1 hsNd 1DLHSDF 7 6 7 6 C sNd 6 f2 h1DLHSDF 7 7 6 .. 7 6 7 6 . 5 4 CfM hsNd 1DLHSDF
2
C sNd 6 f1 h1 6 C sNd 6 f2 h1 6 .. 6 6 . 4 CfM hsNd 1
Cf1 hsNd 2 Cf2 hsNd 2 .. . CfM hsNd 2
32 3 · · · Cf1 hsNd ap1 M 7 6 p7 · · · Cf2 hsNd 7 6a 7 M 76 76 .2 7 , .. .. 76 . 7 74 . 7 . . 5 5 apM · · · CfM hsNd
(6)
M
where the input vector is formed from the correlation constraints and the matrix is formed with the central values from all correlations. If the input pattern is a projected version of the pattern fi sx, yd, say, fi sxyb, yd, the central value of correlation of the 1DLHSDF filter can be expressed as a linear combination of individual correlations: sbd
Cf hsNd i
1DLHSDF
Shift, scale, and projection invariance were tested on four versions of the F18 aircraft pattern, as shown in Fig. 1(a), with the following sx, yd projection factors: a1, (1, 1); a2, (0.9, 0.6); a3, (0.5, 0.5); a4, (0.8, 0.5). It is easily seen that the 1IDLHSDF filter is invariant to shift, scale, and projection and that the outputs are sharp correlation peaks. It is important to note that, despite size differences, the correlation peak remains almost constant for all versions of input. This property is important for implementing a threshold level for pattern detection. Discrimination capability was tested on four versions of a TOR aircraft pattern, as shown in Fig. 2(a), with the same scale and projection factors as in Fig. 1(a). The correlation results with the 1DLHSDF are shown in Fig. 2(b). A comparison of the responses for the true class (Fig. 1) and the false class (Fig. 2), plotted on the same scale, demonstrates good discrimination capability. In conclusion, a novel filter has been described that performs pattern recognition with simultaneous invariance to shift, to scale changes, and to projection along a known direction. The 1D logarithmic expansion reduces the number of training images for the SDF to only one parameter of geometric distortion (vertical projection). The simulations demonstrated invariance of the correlation peak and good discrimination capability. In the simulations the SDF was computed as a linear combination of the training images with constraints on only the peak value, but all
i h ap1 Cfi hsNd 1 a2p Cfi hsNd 1 . . . 1 apM Cfi hsNd 1
2
µ ∂ p 2p 3 b exp 2i N ln b T µ ∂ p 2p , N ln b Cfi hsNd b exp 2i 1DLHSDF T
M
(7)
which demonstrates that the 1DLHSDF filter remains projection invariant along the axis of the harmonic expansion if all harmonic components used in making the SDF f ilter have equal harmonic order. Since projection invariance along the orthogonal axis is provided by the projected components of the SDF, 2D scale-invariant pattern recognition is achieved. To test this invariance we performed simulations for a simple two-class problem. For fast computations edge-detected objects are used, but we showed11 that edge detection gives results equivalent to those for correlation with a full-object phase-only filter (for which high-pass filtering provides the edge enhancement). All calculations are restricted to windows of 64 3 64 pixels. The 1DLHSDF is formed from 1D logarithmic harmonic components expanded along the horizontal axis, with harmonic order N 3 generated from four projected versions along the vertical axis (with projection factors 1, 0.8, 0.6, and 0.4), for F18 (true-class) and TOR (false-class) aircraft patterns.
Fig. 1. Shift-, scale-, and projection-invariance test for the IDLHSDF. (a) F18 aircraft and three scaled projected versions used as input. ( b) Correlation between the 1DLHSDF and the true-class inputs from (a).
April 1, 1998 / Vol. 23, No. 7 / OPTICS LETTERS
539
bustness, minimum average correlation energy13 for best correlation of peak sharpness and reduction of sidelobes, or a compromise between them with optimal trade-off filters,14 which allows one to obtain the desired correlation performance. References
Fig. 2. Discrimination capability test for the 1DLHSDF filter. (a) TOR aircraft and three scaled project versions used as input. (b) Correlation between the 1DLHSDF and the false-class inputs from (a).
optimization methods for SDF f ilters can be used with the 1DLHSDF: minimum variance12 for best noise ro-
1. D. Casasent and D. Psaltis, Appl. Opt. 15, 1795 (1976). 2. D. Mendlovic, N. Konforti, and E. Maron, Appl. Opt. 28, 4982 (1989). 3. D. Mendlovic, Z. Zalevsky, I. Kiryuschev, and G. Lebreton, Appl. Opt. 34, 310 (1995). 4. T. Szoplik, J. Opt. Soc. Am. A 9, 1419 (1984). 5. C. F. Hester and D. Casasent, Appl. Opt. 19, 1758 (1980). 6. D. Casasent, Appl. Opt. 23, 1620 (1984). 7. J. Riggins and S. Butler, Opt. Eng. 23, 721 (1984). 8. B. V. K. Vijaya Kumar and E. Pochapsky, J. Opt. Soc. Am. A 3, 777 (1986). 9. H. H. Arsenault, Proc. SPIE 613, 239 (1986). 10. D. Mendlovic, N. Konforti, and E. Marom, Appl. Opt. 29, 4784 (1990). 11. J. Yao, ‘‘Algorithms for optical distortion invariant pattern recognition,’’ Ph.D. dissertation (Universit´e de Toulon et du Var, La Garde, France, 1997). 12. B. V. K. Vijaya Kumar, J. Opt. Soc. Am. A 3, 1579 (1986). 13. A. Mahalanobis and B. V. K. Vijaya Kumar, Appl. Opt. 26, 3633 (1987). 14. Ph. R´efr´egier, Opt. Lett. 15, 854 (1990).
