Abstract

In this paper, we show that our proposed hybrid optoelectronic correlator (HOC), which correlates images using spatial light modulators (SLMs), detectors, and field-programmable gate arrays (FPGAs), is capable of detecting objects in a scale and rotation invariant manner, along with the shift invariance feature, by incorporating polar Mellin transform (PMT). For realistic images, we cut out a small circle at the center of the Fourier transform domain, as required for PMT, and illustrate how this process corresponds to correlating images with real and imaginary parts. Furthermore, we show how to carry out shift, rotation, and scale invariant detection of multiple matching objects simultaneously, a process previously thought to be incompatible with PMT-based correlators. We present results of numerical simulations to validate the concepts.

© 2014 Optical Society of America

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References

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  1. A. V. Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
    [CrossRef]
  2. A. Heifetz, G. S. Pati, J. T. Shen, J.-K. Lee, M. S. Shahriar, C. Phan, and M. Yamomoto, “Shift-invariant real-time edge-enhanced Vander Lugt correlator using video-rate compatible photorefractive polymer,” Appl. Opt. 45, 6148–6153 (2006).
  3. A. Heifetz, J. T. Shen, J.-K. Lee, R. Tripathi, and M. S. Shahriar, “Translation-invariant object recognition system using an optical correlator and a super-parallel holographic RAM,” Opt. Eng. 45, 025201 (2006).
    [CrossRef]
  4. J. Khoury, M. C. Golomb, P. Gianino, and C. Woods, “Photorefractive two-beam-coupling nonlinear joint-transform correlator,” J. Opt. Soc. Am. B 11, 2167–2174 (1994).
    [CrossRef]
  5. Q. Tang and B. Javidi, “Multiple-object detection with a chirp-encoded joint transform correlator,” Appl. Opt. 32, 5079–5088 (1993).
    [CrossRef]
  6. T. H. Barnes, K. Matsuda, T. Eiju, K. Matsumoto, and F. Johnson, “Joint transform correlator using a phase-only spatial light modulator,” Jpn. J. Appl. Phys. 29, L1293–L1296 (1990).
    [CrossRef]
  7. M. S. Shahriar, R. Tripathi, M. Kleinschmit, J. Donoghue, W. Weathers, M. Huq, and J. T. Shen, “Superparallel holographic correlator for ultrafast database searches,” Opt. Lett. 28, 525–527 (2003).
    [CrossRef]
  8. M. S. Monjur, S. Tseng, R. Tripathi, J. Donoghue, and M. S. Shahriar, “Hybrid optoelectronic correlator architecture for shift invariant target recognition,” J. Opt. Soc. Am. A 31, 41–47 (2014).
    [CrossRef]
  9. B. Javidi and C. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. 27, 663–665 (1988).
    [CrossRef]
  10. D. Casasent and D. Psaltis, “Scale-invariant optical correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
    [CrossRef]
  11. D. Casasent and D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [CrossRef]
  12. D. Casasent and D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
    [CrossRef]
  13. C.-Y. Lin, M. Wu, J. A. Bloom, I. J. Cox, M. L. Miller, and Y. M. Lui, “Rotation, scale and translation resilient watermarking for images,” IEEE Trans. Image Process. 10, 767–782 (2001).
    [CrossRef]
  14. J. Esteve-Taboada, J. García, and C. Ferreira, “Rotation-invariant optical recognition of three-dimensional objects,” Appl. Opt. 39, 5998–6005 (2000).
    [CrossRef]
  15. D. Sazbon, Z. Zalevsky, E. Rivlina, and D. Mendlovic, “Using Fourier-Mellin-based correlators and their fractional versions in navigational tasks,” Pattern Recogn. 35, 2993–2999 (2002).
    [CrossRef]
  16. D. Asselin and H. H. Arsenault, “Rotation and scale invariance with polar and log-polar coordinate transformation,” Opt. Commun. 104, 391–404 (1994).
    [CrossRef]
  17. J. Rosen and J. Shamir, “Scale invariant pattern recognition with logarithmic radial harmonic filter,” Appl. Opt. 28, 240–244 (1989).
    [CrossRef]
  18. Recall that a PMT image is plotted as a function of ρ and θ. In the θ direction, any image will cover the whole range from θ to 2π. As a result, two PMT images in two boxes adjacent in the vertical direction will tend to merge into each other. This problem is circumvented by scaling each PMT image to 90% of its actual size, thus creating a guard band. This step does not affect the outcome of the correlator process.

2014 (1)

2006 (2)

A. Heifetz, G. S. Pati, J. T. Shen, J.-K. Lee, M. S. Shahriar, C. Phan, and M. Yamomoto, “Shift-invariant real-time edge-enhanced Vander Lugt correlator using video-rate compatible photorefractive polymer,” Appl. Opt. 45, 6148–6153 (2006).

A. Heifetz, J. T. Shen, J.-K. Lee, R. Tripathi, and M. S. Shahriar, “Translation-invariant object recognition system using an optical correlator and a super-parallel holographic RAM,” Opt. Eng. 45, 025201 (2006).
[CrossRef]

2003 (1)

2002 (1)

D. Sazbon, Z. Zalevsky, E. Rivlina, and D. Mendlovic, “Using Fourier-Mellin-based correlators and their fractional versions in navigational tasks,” Pattern Recogn. 35, 2993–2999 (2002).
[CrossRef]

2001 (1)

C.-Y. Lin, M. Wu, J. A. Bloom, I. J. Cox, M. L. Miller, and Y. M. Lui, “Rotation, scale and translation resilient watermarking for images,” IEEE Trans. Image Process. 10, 767–782 (2001).
[CrossRef]

2000 (1)

1994 (2)

D. Asselin and H. H. Arsenault, “Rotation and scale invariance with polar and log-polar coordinate transformation,” Opt. Commun. 104, 391–404 (1994).
[CrossRef]

J. Khoury, M. C. Golomb, P. Gianino, and C. Woods, “Photorefractive two-beam-coupling nonlinear joint-transform correlator,” J. Opt. Soc. Am. B 11, 2167–2174 (1994).
[CrossRef]

1993 (1)

1990 (1)

T. H. Barnes, K. Matsuda, T. Eiju, K. Matsumoto, and F. Johnson, “Joint transform correlator using a phase-only spatial light modulator,” Jpn. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

1989 (1)

1988 (1)

1977 (1)

D. Casasent and D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

1976 (2)

D. Casasent and D. Psaltis, “Scale-invariant optical correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

D. Casasent and D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef]

1964 (1)

A. V. Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

Arsenault, H. H.

D. Asselin and H. H. Arsenault, “Rotation and scale invariance with polar and log-polar coordinate transformation,” Opt. Commun. 104, 391–404 (1994).
[CrossRef]

Asselin, D.

D. Asselin and H. H. Arsenault, “Rotation and scale invariance with polar and log-polar coordinate transformation,” Opt. Commun. 104, 391–404 (1994).
[CrossRef]

Barnes, T. H.

T. H. Barnes, K. Matsuda, T. Eiju, K. Matsumoto, and F. Johnson, “Joint transform correlator using a phase-only spatial light modulator,” Jpn. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Bloom, J. A.

C.-Y. Lin, M. Wu, J. A. Bloom, I. J. Cox, M. L. Miller, and Y. M. Lui, “Rotation, scale and translation resilient watermarking for images,” IEEE Trans. Image Process. 10, 767–782 (2001).
[CrossRef]

Casasent, D.

D. Casasent and D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

D. Casasent and D. Psaltis, “Scale-invariant optical correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

D. Casasent and D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef]

Cox, I. J.

C.-Y. Lin, M. Wu, J. A. Bloom, I. J. Cox, M. L. Miller, and Y. M. Lui, “Rotation, scale and translation resilient watermarking for images,” IEEE Trans. Image Process. 10, 767–782 (2001).
[CrossRef]

Donoghue, J.

Eiju, T.

T. H. Barnes, K. Matsuda, T. Eiju, K. Matsumoto, and F. Johnson, “Joint transform correlator using a phase-only spatial light modulator,” Jpn. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Esteve-Taboada, J.

Ferreira, C.

García, J.

Gianino, P.

Golomb, M. C.

Heifetz, A.

A. Heifetz, J. T. Shen, J.-K. Lee, R. Tripathi, and M. S. Shahriar, “Translation-invariant object recognition system using an optical correlator and a super-parallel holographic RAM,” Opt. Eng. 45, 025201 (2006).
[CrossRef]

A. Heifetz, G. S. Pati, J. T. Shen, J.-K. Lee, M. S. Shahriar, C. Phan, and M. Yamomoto, “Shift-invariant real-time edge-enhanced Vander Lugt correlator using video-rate compatible photorefractive polymer,” Appl. Opt. 45, 6148–6153 (2006).

Huq, M.

Javidi, B.

Johnson, F.

T. H. Barnes, K. Matsuda, T. Eiju, K. Matsumoto, and F. Johnson, “Joint transform correlator using a phase-only spatial light modulator,” Jpn. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Khoury, J.

Kleinschmit, M.

Kuo, C.

Lee, J.-K.

A. Heifetz, G. S. Pati, J. T. Shen, J.-K. Lee, M. S. Shahriar, C. Phan, and M. Yamomoto, “Shift-invariant real-time edge-enhanced Vander Lugt correlator using video-rate compatible photorefractive polymer,” Appl. Opt. 45, 6148–6153 (2006).

A. Heifetz, J. T. Shen, J.-K. Lee, R. Tripathi, and M. S. Shahriar, “Translation-invariant object recognition system using an optical correlator and a super-parallel holographic RAM,” Opt. Eng. 45, 025201 (2006).
[CrossRef]

Lin, C.-Y.

C.-Y. Lin, M. Wu, J. A. Bloom, I. J. Cox, M. L. Miller, and Y. M. Lui, “Rotation, scale and translation resilient watermarking for images,” IEEE Trans. Image Process. 10, 767–782 (2001).
[CrossRef]

Lugt, A. V.

A. V. Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

Lui, Y. M.

C.-Y. Lin, M. Wu, J. A. Bloom, I. J. Cox, M. L. Miller, and Y. M. Lui, “Rotation, scale and translation resilient watermarking for images,” IEEE Trans. Image Process. 10, 767–782 (2001).
[CrossRef]

Matsuda, K.

T. H. Barnes, K. Matsuda, T. Eiju, K. Matsumoto, and F. Johnson, “Joint transform correlator using a phase-only spatial light modulator,” Jpn. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Matsumoto, K.

T. H. Barnes, K. Matsuda, T. Eiju, K. Matsumoto, and F. Johnson, “Joint transform correlator using a phase-only spatial light modulator,” Jpn. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Mendlovic, D.

D. Sazbon, Z. Zalevsky, E. Rivlina, and D. Mendlovic, “Using Fourier-Mellin-based correlators and their fractional versions in navigational tasks,” Pattern Recogn. 35, 2993–2999 (2002).
[CrossRef]

Miller, M. L.

C.-Y. Lin, M. Wu, J. A. Bloom, I. J. Cox, M. L. Miller, and Y. M. Lui, “Rotation, scale and translation resilient watermarking for images,” IEEE Trans. Image Process. 10, 767–782 (2001).
[CrossRef]

Monjur, M. S.

Pati, G. S.

Phan, C.

Psaltis, D.

D. Casasent and D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

D. Casasent and D. Psaltis, “Scale-invariant optical correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

D. Casasent and D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[CrossRef]

Rivlina, E.

D. Sazbon, Z. Zalevsky, E. Rivlina, and D. Mendlovic, “Using Fourier-Mellin-based correlators and their fractional versions in navigational tasks,” Pattern Recogn. 35, 2993–2999 (2002).
[CrossRef]

Rosen, J.

Sazbon, D.

D. Sazbon, Z. Zalevsky, E. Rivlina, and D. Mendlovic, “Using Fourier-Mellin-based correlators and their fractional versions in navigational tasks,” Pattern Recogn. 35, 2993–2999 (2002).
[CrossRef]

Shahriar, M. S.

Shamir, J.

Shen, J. T.

Tang, Q.

Tripathi, R.

Tseng, S.

Weathers, W.

Woods, C.

Wu, M.

C.-Y. Lin, M. Wu, J. A. Bloom, I. J. Cox, M. L. Miller, and Y. M. Lui, “Rotation, scale and translation resilient watermarking for images,” IEEE Trans. Image Process. 10, 767–782 (2001).
[CrossRef]

Yamomoto, M.

Zalevsky, Z.

D. Sazbon, Z. Zalevsky, E. Rivlina, and D. Mendlovic, “Using Fourier-Mellin-based correlators and their fractional versions in navigational tasks,” Pattern Recogn. 35, 2993–2999 (2002).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Image Process. (1)

C.-Y. Lin, M. Wu, J. A. Bloom, I. J. Cox, M. L. Miller, and Y. M. Lui, “Rotation, scale and translation resilient watermarking for images,” IEEE Trans. Image Process. 10, 767–782 (2001).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. V. Lugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Jpn. J. Appl. Phys. (1)

T. H. Barnes, K. Matsuda, T. Eiju, K. Matsumoto, and F. Johnson, “Joint transform correlator using a phase-only spatial light modulator,” Jpn. J. Appl. Phys. 29, L1293–L1296 (1990).
[CrossRef]

Opt. Commun. (2)

D. Casasent and D. Psaltis, “Scale-invariant optical correlation using Mellin transforms,” Opt. Commun. 17, 59–63 (1976).
[CrossRef]

D. Asselin and H. H. Arsenault, “Rotation and scale invariance with polar and log-polar coordinate transformation,” Opt. Commun. 104, 391–404 (1994).
[CrossRef]

Opt. Eng. (1)

A. Heifetz, J. T. Shen, J.-K. Lee, R. Tripathi, and M. S. Shahriar, “Translation-invariant object recognition system using an optical correlator and a super-parallel holographic RAM,” Opt. Eng. 45, 025201 (2006).
[CrossRef]

Opt. Lett. (1)

Pattern Recogn. (1)

D. Sazbon, Z. Zalevsky, E. Rivlina, and D. Mendlovic, “Using Fourier-Mellin-based correlators and their fractional versions in navigational tasks,” Pattern Recogn. 35, 2993–2999 (2002).
[CrossRef]

Proc. IEEE (1)

D. Casasent and D. Psaltis, “New optical transforms for pattern recognition,” Proc. IEEE 65, 77–84 (1977).
[CrossRef]

Other (1)

Recall that a PMT image is plotted as a function of ρ and θ. In the θ direction, any image will cover the whole range from θ to 2π. As a result, two PMT images in two boxes adjacent in the vertical direction will tend to merge into each other. This problem is circumvented by scaling each PMT image to 90% of its actual size, thus creating a guard band. This step does not affect the outcome of the correlator process.

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Figures (22)

Fig. 1.
Fig. 1.

Illustration of the architecture of an HOC. PMT, polar Mellin transform; PBS, polarizing beam splitter; BS, beam splitter; HWP, half-wave plate; PZT, piezoelectric transducer.

Fig. 2.
Fig. 2.

Final stage of the HOC.

Fig. 3.
Fig. 3.

Illustration of the resolving power of the HOC architecture for 1D identical images.

Fig. 4.
Fig. 4.

Illustration of the resolving power of the HOC architecture for 2D identical images.

Fig. 5.
Fig. 5.

(a) Flow diagram for transforming of query/reference image to the log-polar domain. (b) Schematic illustration of the architecture for implementing PMT.

Fig. 6.
Fig. 6.

(a) V(x,y) is the amplitude of the FT of an image with a flower shape hole in it. (b) Corresponding polar function, F(r,θ) is shown. (c) Polar-logarithmic function, G(ρ,θ) is shown [here, r0=0.1].

Fig. 7.
Fig. 7.

(a) Shows VA(x,y), which is the FT of an image. (b)  Shows the magnitude of an image whose FT is VA(x,y). (c) Shows the phase of the actual image whose FT is VA(x,y).

Fig. 8.
Fig. 8.

(a) Final output signal when two identical images are inputs to the HOC. The PMT version is shown in Fig. 6(c). (b) Output signal after thresholding shows that there is a peak in the center when match between two objects is found.

Fig. 9.
Fig. 9.

(a) We consider an artificial case, where VA(x,y) is the FT of an image. (b) Corresponding polar distribution FA(r,θ). (c) Corresponding log-polar distribution GA(ρ,θ) (d) VB(x,y) is smaller in area than VA(x,y) by a factor of 4. (e) Corresponding polar distributions, FB(r,θ). FB(r,θ) and FA(r,θ), are the same in the θ direction, but differ by the linear scaling factor (2 in this case) in the r direction. (f) Corresponding polar-logarithmic distribution, GB(ρ,θ). GA(ρ,θ) and GB(ρ,θ) are identical in shape, except for a shift in the ρ direction [equaling the logarithm of the scale factor: log(2)0.3]. (g) Final output signal |Sf|2 when GA(ρ,θ) and GB(ρ,θ) are applied to the correlator. (h) Final signal after thresholding for σ=2.

Fig. 10.
Fig. 10.

(a) VC(x,y) is smaller in area than VA(x,y) by a factor of 1.44. (b) Corresponding polar distributions, FC(r,θ). (c) Corresponding polar-logarithmic distribution is GC(ρ,θ). (d) Final output signals GA(ρ,θ) and GC(ρ,θ) are applied to HOC. Final output signal has one peak at the center implying that a match is found, but no scale information is revealed due to the fact that the displacement between GA(ρ,θ) and GC(ρ,θ) is very small. (d) Final signal after thresholding for σ=1.2.

Fig. 11.
Fig. 11.

(a) VA(x,y) is the amplitude of the FT of an arbitrary image and VB(x,y) represents the amplitude of the FT of the same image, but rotated by an angle of θ0=45°. (b) Corresponding polar distributions, FB(r,θ) and FA(r,θ), now differ in the θ direction. (c) Dotted part of GB(ρ,θ) is now shifted from the solid part of GA(ρ,θ) in the θ direction by an amount of θ0=45°; the dotted part of GB(ρ,θ) is now shifted from the solid part of GA(ρ,θ) in the θ-direction by an amount of (2πθ0)=315°.

Fig. 12.
Fig. 12.

(a) Cross-correlation and anti-cross-correlation (T3+T4) signal when GA(ρ,θ) and GB(ρ,θ) are applied to the HOC. (b) Cross-sectional view of the T3+T4 signal showing two peaks shifted in the θ direction by an amount of θ0 and θ0, which correspond to the cross-correlation terms GA1(ρ,θ)GB1(ρ,θ) and GB1(ρ,θ)GA1(ρ,θ). (c) Cross-correlation signal T3, which shows the two peaks corresponding to the cross-correlation GA1(ρ,θ)GB1(ρ,θ) and GA2(ρ,θ)GB2(ρ,θ). (d) Anti-cross-correlation signal T4, which shows the two peaks corresponding to the cross-correlations GB1(ρ,θ)GA1(ρ,θ) and GB2(ρ,θ)GA2(ρ,θ). (e) Final output signal |Sf|2. (f) Final output signal after thresholding.

Fig. 13.
Fig. 13.

Normalized amplitude of T3 for different angles. See text for details.

Fig. 14.
Fig. 14.

(a) VC(x,y) is smaller in area by a factor of 4 than VA(x,y) and is rotated by 45°. (b) Corresponding polar distributions, FC(r,θ) and FA(r,θ), now differ in the r direction and the θ direction. (c) GC(ρ,θ) is now shifted from GA(ρ,θ) in the ρ direction by an amount log(2)=0.3 and the θ direction by an amount of θ0=45°.

Fig. 15.
Fig. 15.

(a) Cross-correlation and anti-cross-correlation (T3+T4) signal when GA(ρ,θ) and GC(ρ,θ) are applied to the HOC, showing two peaks at (ρ=0.3; θ=45°) and (ρ=0.3; θ=45°). These correspond to the cross-correlation terms GA1(ρ,θ)GC1(ρ,θ) and GC1(ρ,θ)GA1(ρ,θ), respectively. (b) Cross-sectional view shows the peaks 3a and 4a. (c) Cross-correlation signal T3, which shows the two peaks corresponding to the cross-correlation GA1(ρ,θ)GC1(ρ,θ) and GA2(ρ,θ)GC2(ρ,θ). (d) Anti-cross-correlation signal T4, which shows the two peaks corresponding to the cross-correlations GC1(ρ,θ)GA1(ρ,θ) and GC2(ρ,θ)GA2(ρ,θ) (e) Final output signal |Sf|2. (f) Final output signal after thresholding.

Fig. 16.
Fig. 16.

(a) We consider two images, U1(x,y) and U2(x,y), where U2(x,y) is smaller in area than U1(x,y) by a factor of 4. (b) The corresponding magnitudes of FTs are denoted as V1(x,y) and V2(x,y), which are similar in shape, except that V2(x,y) has a larger area than that of V1(x,y) by a factor of 4. (c) A hole of radius r0 is created in the center of V1(x,y) and V2(x,y), and the resulting functions are denoted as V1H(x,y) and V2H(x,y), respectively. (d) The corresponding polar distributions, F1(r,θ) and F2(r,θ), are the same in the θ direction, but differ by a linear scaling factor of 2 in the r direction. (e) The polar-logarithmic distributions, G1(ρ,θ) and G2(ρ,θ), are identical in shape, except for a shift in the ρ direction [equaling the logarithm of the scale factor: log(2)0.3]. (f) The final output signal, |Sf|2 of the HOC when U1(x,y) and U2(x,y), is converted to the log-polar domain and then acts as an input to the HOC. (g) After thresholding, we get two peaks corresponding to the two cross-correlation terms (T3 and T4).

Fig. 17.
Fig. 17.

Illustration of the fact that cutting a hole of certain radius in the center of the FT of the image does not change the image significantly. See text for details. [Note that in Figs. 18(b), 18(c), 18(f), and 18(g) the color has been inverted for clear visualization.]

Fig. 18.
Fig. 18.

(a) We consider two images, U1(x,y) and U3(x,y), where U3(x,y) is smaller in area than U1(x,y) by a factor of 4 and also rotated by an angle of θ0=30°. (b) The corresponding FTs are denoted as V1(x,y) and V3(x,y), which are similar in shape, except V3(x,y) has a larger area than that of V1(x,y) by a factor of 4 and also rotated by an angle of θ0=30°. (c) A hole of radius r0 is created in the center of V1(x,y) and V3(x,y), which are denoted as V1H(x,y) and V3H(x,y), respectively. (d) The corresponding polar distributions, F1(r,θ) and F3(r,θ), are shifted in the θ direction by an amount of θ0=30° and also shifted in the r direction by a linear scaling factor of 2. (e) Polar-logarithmic distributions, G1(ρ,θ) and G3(ρ,θ), are identical in shapes, except for a shift in the ρ direction [equaling the logarithm of the scale factor: log(2)0.3] and also a shift in the θ-direction by an amount of θ0=30°. (e) Final output signal, |Sf|2 of the HOC when U1(x,y) and U3(x,y) are converted to G1(ρ,θ) and G3(ρ,θ) and act as inputs to the HOC. (f) After thresholding, we get two peaks corresponding to the two cross-correlation terms (T3 and T4).

Fig. 19.
Fig. 19.

Illustration of multiple object detection without rotation and scale change using the HOC architecture. See text for details.

Fig. 20.
Fig. 20.

Illustration of the process of mapping multiple query objects, each with potentially a different position, a different scale factor, and a different angular orientation, to the right half of the query image plane, which has a grid size of N×N. See text for details.

Fig. 21.
Fig. 21.

Process of Fourier transforming a query image plane with multiple images using an SLM and a lens. See text for details.

Fig. 22.
Fig. 22.

(a) FT of the reference image. (b) Array of FTs of multiple objects, where one side is intentionally left blank. (c) PMT of the reference image. (d) PMT of the array of query images. (e) Final results from the HOC. We have to consider only the right half of this plane. See text for details.

Tables (1)

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Table 1. Summary of Definitions of Various Transform

Equations (10)

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

Ar=|Mrejϕr+Crejψr|2=|Mr|2+|Cr|2+|Mr||Cr|ej(ϕrψr)+|Mr||Cr|ej(ϕrψr),
Br=|Mr|2.
Sr=ArBr|Cr|2=MrCr*+Mr*Cr=|Mr||Cr|ej(ϕrψr)+|Mr||Cr|ej(ϕrψr).
Aq=|Mq+Cq|2=|Mq|2+|Cq|2+|Mq||Cq|ej(ϕqψq)+|Mq||Cq|ej(ϕqψq),
Bq=|Mq|2,
Sq=AqBq|Cq|2=MqCq*+Mq*Cq=|Mq||Cq|ej(ϕqψq)+|Mq||Cq|ej(ϕqψq).
S=Sr·Sq=(MrCr*+Mr*Cr)·(MqCq*+Mq*Cq)=[α*MrMq+αMr*Mq*+β*MrMq*+βMr*Mq],
Sf=α*F(MrMq)+αF(Mr*Mq*)+β*F(MrMq*)+βF(Mr*Mq).
Sf=α*T1+αT2+β*T3+βT4,T1=Rj(x,y)Qj(x,y),T2=Rj*(x,y)Qj*(x,y),T3=Rj(x,y)Qj(x,y),T4=Qj(x,y)Rj(x,y),
SfD=|β*T3+βT4|2=|β*F(MrMq*)+βF(Mr*Mq)|2.

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