Abstract

A hybrid system has been constructed to perform the complex Fourier transform of real 2-D data. The system is based on the Radon transform; i.e., operations are performed on 1-D projections of the data. The projections are derived optically from transmissive or reflective objects, and the complex Fourier transform is performed with SAW filters via the chirp transform algorithm. The real and imaginary parts of the 2-D transform are produced in two bipolar output channels.

© 1985 Optical Society of America

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References

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  1. R. H. Katyl, “Compensating Optical systems. 3: Achromatic Fourier Transformation,” Appl. Opt. 11, 1255 (1972).
    [CrossRef] [PubMed]
  2. L. Mertz, Transformation in Optics (Wiley, New York, 1965).
  3. J. M. Richardson, U.S. Patent3,669,528, “Device for Producing Identifiable Sine and Cosine (Fourier) Transforms of Input Signals by means of Noncoherent Optics” (June1972).
  4. I. Leifer, G. L. Rogers, N. W. F. Stephens, “Incoherent Fourier Transformation: a New Approach to Character Recognition,” Opt. Acta 16, 535 (1969).
    [CrossRef]
  5. M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent Optical Correlators,” Proc. IEEE 65, 121 (1977).
    [CrossRef]
  6. A. M. Tai, C. C. Aleksoff, “Grating-Based Interferometric Processor for Real-Time Optical Fourier Transformation,” Appl. Opt. 23, 2282 (1984).
    [CrossRef] [PubMed]
  7. K. Xu, R. Ji, Z. Zhang, “A New Method for Measuring the Phase of the Fourier Spectrum,” Opt. Commun. 50, 85 (1984).
    [CrossRef]
  8. N. George, S. Wang, “Cosinusoidal Transforms in White Light,” Appl. Opt. 23, 787 (1984).
    [CrossRef] [PubMed]
  9. I. Glaser, Y. Katzir, V. Toschi, “Incoherent Optical Two-Dimensional Fourier Transform Using the Chirp-z Algorithm,” Opt. Lett. 9, 199 (1984).
    [CrossRef] [PubMed]
  10. A. J. Ticknor, R. L. Easton, H. H. Barrett., “A Two-Dimensional Radon-Fourier Transformer,” Opt. Eng. 24, 082 (1985).
    [CrossRef]
  11. S. R. Deans, The Radon Transform and Some of its Applications (J Wiley, New York, 1983).
  12. G. Eichmann, B. Z. Dong, “Coherent Optical Production of the Hough Transform,” Appl. Opt. 22, 830 (1983).
    [CrossRef] [PubMed]
  13. G. R. Gindi, A. F. Gmitro, “Optical Feature Extraction via the Radon Transform,” Opt. Eng. 23, 499 (1984).
    [CrossRef]
  14. A. G. Gmitro, G. R. Gindi, H. H. Barrett, R. L. Easton, “Two-Dimensional Image Processing by One-Dimensional Filtering of Projection Data,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 132 (1983).
  15. H. H. Barrett, “Three-Dimensional Image Reconstruction from Planar Projections with Application to Optical Data Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 179 (1983).
  16. W. E. Smith, H. H. Barrett, “Radon Transform and Bandwidth Compression,” Opt. Lett. 8, 395 (1983).
    [CrossRef] [PubMed]
  17. D. Fraser, B. R. Hunt, J. C. Su, “Principles of Tomography in Image Data Compression,” Opt. Eng. 24, 298 (1985).
    [CrossRef]
  18. R. L. Easton, A. J. Ticknor, H. H. Barrett, “Application of the Radon Transform to Optical Production of the Wigner Distribution Function,” Opt. Eng. 23, 738 (1984).
  19. H. H. Barrett, “Optical Processing in Radon Space,” Opt. Lett. 7, 248 (1982).
    [CrossRef] [PubMed]
  20. R. L. Easton, H. H. Barrett, A. J. Ticknor, “Using SAW Filters to Process Two-Dimensional Data via the Radon Transform,” Proc. IEEE Ultrason. Symp. 185 (1983).
  21. N. H. Farhat, C. Y. Ho, L. Szu Chang, “Projection Theorems and Their Application in Multidimensional Signal Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 140 (1983).
  22. H. J. Whitehouse, “Role of Charge-Coupled Devices and Surface Acoustic Wave Devices in Optical Signal Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 118, 124 (1977).
  23. M. A. Jack, E. G. S. Paige, “Fourier Transformation Processors Based on Surface Acoustic Wave Chirp Filters,” Wave Electron. 3, 229 (1978).
  24. M. A. Jack, P. M. Grant, J. H. Collins, “The Theory, Design, and Applications of Surface Acoustic Wave Fourier-Transform Processors,” IEEE Proc. 68, 450 (1980).
    [CrossRef]
  25. H. M. Gerard, P. S. Yao, O. W. Otto, “Performance of a Programmable Radar Pulse Compression Filter Based on a Chirp Transformation with RAC Filters,” Proc. IEEE Ultrason. Symp. 947, (1977).
  26. K. A. Stetson, J. N. Elkins, “Optical System for Dynamic Analysis of Rotating Structures,” Tech. Rep. AFAPL-TR-77-51 (United Technologies Research Center, East Hartford, Conn., 1977).

1985 (2)

A. J. Ticknor, R. L. Easton, H. H. Barrett., “A Two-Dimensional Radon-Fourier Transformer,” Opt. Eng. 24, 082 (1985).
[CrossRef]

D. Fraser, B. R. Hunt, J. C. Su, “Principles of Tomography in Image Data Compression,” Opt. Eng. 24, 298 (1985).
[CrossRef]

1984 (6)

R. L. Easton, A. J. Ticknor, H. H. Barrett, “Application of the Radon Transform to Optical Production of the Wigner Distribution Function,” Opt. Eng. 23, 738 (1984).

G. R. Gindi, A. F. Gmitro, “Optical Feature Extraction via the Radon Transform,” Opt. Eng. 23, 499 (1984).
[CrossRef]

A. M. Tai, C. C. Aleksoff, “Grating-Based Interferometric Processor for Real-Time Optical Fourier Transformation,” Appl. Opt. 23, 2282 (1984).
[CrossRef] [PubMed]

K. Xu, R. Ji, Z. Zhang, “A New Method for Measuring the Phase of the Fourier Spectrum,” Opt. Commun. 50, 85 (1984).
[CrossRef]

N. George, S. Wang, “Cosinusoidal Transforms in White Light,” Appl. Opt. 23, 787 (1984).
[CrossRef] [PubMed]

I. Glaser, Y. Katzir, V. Toschi, “Incoherent Optical Two-Dimensional Fourier Transform Using the Chirp-z Algorithm,” Opt. Lett. 9, 199 (1984).
[CrossRef] [PubMed]

1983 (6)

G. Eichmann, B. Z. Dong, “Coherent Optical Production of the Hough Transform,” Appl. Opt. 22, 830 (1983).
[CrossRef] [PubMed]

A. G. Gmitro, G. R. Gindi, H. H. Barrett, R. L. Easton, “Two-Dimensional Image Processing by One-Dimensional Filtering of Projection Data,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 132 (1983).

H. H. Barrett, “Three-Dimensional Image Reconstruction from Planar Projections with Application to Optical Data Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 179 (1983).

W. E. Smith, H. H. Barrett, “Radon Transform and Bandwidth Compression,” Opt. Lett. 8, 395 (1983).
[CrossRef] [PubMed]

R. L. Easton, H. H. Barrett, A. J. Ticknor, “Using SAW Filters to Process Two-Dimensional Data via the Radon Transform,” Proc. IEEE Ultrason. Symp. 185 (1983).

N. H. Farhat, C. Y. Ho, L. Szu Chang, “Projection Theorems and Their Application in Multidimensional Signal Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 140 (1983).

1982 (1)

1980 (1)

M. A. Jack, P. M. Grant, J. H. Collins, “The Theory, Design, and Applications of Surface Acoustic Wave Fourier-Transform Processors,” IEEE Proc. 68, 450 (1980).
[CrossRef]

1978 (1)

M. A. Jack, E. G. S. Paige, “Fourier Transformation Processors Based on Surface Acoustic Wave Chirp Filters,” Wave Electron. 3, 229 (1978).

1977 (3)

M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent Optical Correlators,” Proc. IEEE 65, 121 (1977).
[CrossRef]

H. M. Gerard, P. S. Yao, O. W. Otto, “Performance of a Programmable Radar Pulse Compression Filter Based on a Chirp Transformation with RAC Filters,” Proc. IEEE Ultrason. Symp. 947, (1977).

H. J. Whitehouse, “Role of Charge-Coupled Devices and Surface Acoustic Wave Devices in Optical Signal Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 118, 124 (1977).

1972 (1)

1969 (1)

I. Leifer, G. L. Rogers, N. W. F. Stephens, “Incoherent Fourier Transformation: a New Approach to Character Recognition,” Opt. Acta 16, 535 (1969).
[CrossRef]

Aleksoff, C. C.

Barrett, H. H.

A. J. Ticknor, R. L. Easton, H. H. Barrett., “A Two-Dimensional Radon-Fourier Transformer,” Opt. Eng. 24, 082 (1985).
[CrossRef]

R. L. Easton, A. J. Ticknor, H. H. Barrett, “Application of the Radon Transform to Optical Production of the Wigner Distribution Function,” Opt. Eng. 23, 738 (1984).

R. L. Easton, H. H. Barrett, A. J. Ticknor, “Using SAW Filters to Process Two-Dimensional Data via the Radon Transform,” Proc. IEEE Ultrason. Symp. 185 (1983).

A. G. Gmitro, G. R. Gindi, H. H. Barrett, R. L. Easton, “Two-Dimensional Image Processing by One-Dimensional Filtering of Projection Data,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 132 (1983).

H. H. Barrett, “Three-Dimensional Image Reconstruction from Planar Projections with Application to Optical Data Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 179 (1983).

W. E. Smith, H. H. Barrett, “Radon Transform and Bandwidth Compression,” Opt. Lett. 8, 395 (1983).
[CrossRef] [PubMed]

H. H. Barrett, “Optical Processing in Radon Space,” Opt. Lett. 7, 248 (1982).
[CrossRef] [PubMed]

Bocker, R. P.

M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent Optical Correlators,” Proc. IEEE 65, 121 (1977).
[CrossRef]

Bromley, K.

M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent Optical Correlators,” Proc. IEEE 65, 121 (1977).
[CrossRef]

Collins, J. H.

M. A. Jack, P. M. Grant, J. H. Collins, “The Theory, Design, and Applications of Surface Acoustic Wave Fourier-Transform Processors,” IEEE Proc. 68, 450 (1980).
[CrossRef]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of its Applications (J Wiley, New York, 1983).

Dong, B. Z.

Easton, R. L.

A. J. Ticknor, R. L. Easton, H. H. Barrett., “A Two-Dimensional Radon-Fourier Transformer,” Opt. Eng. 24, 082 (1985).
[CrossRef]

R. L. Easton, A. J. Ticknor, H. H. Barrett, “Application of the Radon Transform to Optical Production of the Wigner Distribution Function,” Opt. Eng. 23, 738 (1984).

R. L. Easton, H. H. Barrett, A. J. Ticknor, “Using SAW Filters to Process Two-Dimensional Data via the Radon Transform,” Proc. IEEE Ultrason. Symp. 185 (1983).

A. G. Gmitro, G. R. Gindi, H. H. Barrett, R. L. Easton, “Two-Dimensional Image Processing by One-Dimensional Filtering of Projection Data,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 132 (1983).

Eichmann, G.

Elkins, J. N.

K. A. Stetson, J. N. Elkins, “Optical System for Dynamic Analysis of Rotating Structures,” Tech. Rep. AFAPL-TR-77-51 (United Technologies Research Center, East Hartford, Conn., 1977).

Farhat, N. H.

N. H. Farhat, C. Y. Ho, L. Szu Chang, “Projection Theorems and Their Application in Multidimensional Signal Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 140 (1983).

Fraser, D.

D. Fraser, B. R. Hunt, J. C. Su, “Principles of Tomography in Image Data Compression,” Opt. Eng. 24, 298 (1985).
[CrossRef]

George, N.

Gerard, H. M.

H. M. Gerard, P. S. Yao, O. W. Otto, “Performance of a Programmable Radar Pulse Compression Filter Based on a Chirp Transformation with RAC Filters,” Proc. IEEE Ultrason. Symp. 947, (1977).

Gindi, G. R.

G. R. Gindi, A. F. Gmitro, “Optical Feature Extraction via the Radon Transform,” Opt. Eng. 23, 499 (1984).
[CrossRef]

A. G. Gmitro, G. R. Gindi, H. H. Barrett, R. L. Easton, “Two-Dimensional Image Processing by One-Dimensional Filtering of Projection Data,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 132 (1983).

Glaser, I.

Gmitro, A. F.

G. R. Gindi, A. F. Gmitro, “Optical Feature Extraction via the Radon Transform,” Opt. Eng. 23, 499 (1984).
[CrossRef]

Gmitro, A. G.

A. G. Gmitro, G. R. Gindi, H. H. Barrett, R. L. Easton, “Two-Dimensional Image Processing by One-Dimensional Filtering of Projection Data,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 132 (1983).

Grant, P. M.

M. A. Jack, P. M. Grant, J. H. Collins, “The Theory, Design, and Applications of Surface Acoustic Wave Fourier-Transform Processors,” IEEE Proc. 68, 450 (1980).
[CrossRef]

Ho, C. Y.

N. H. Farhat, C. Y. Ho, L. Szu Chang, “Projection Theorems and Their Application in Multidimensional Signal Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 140 (1983).

Hunt, B. R.

D. Fraser, B. R. Hunt, J. C. Su, “Principles of Tomography in Image Data Compression,” Opt. Eng. 24, 298 (1985).
[CrossRef]

Jack, M. A.

M. A. Jack, P. M. Grant, J. H. Collins, “The Theory, Design, and Applications of Surface Acoustic Wave Fourier-Transform Processors,” IEEE Proc. 68, 450 (1980).
[CrossRef]

M. A. Jack, E. G. S. Paige, “Fourier Transformation Processors Based on Surface Acoustic Wave Chirp Filters,” Wave Electron. 3, 229 (1978).

Ji, R.

K. Xu, R. Ji, Z. Zhang, “A New Method for Measuring the Phase of the Fourier Spectrum,” Opt. Commun. 50, 85 (1984).
[CrossRef]

Katyl, R. H.

Katzir, Y.

Leifer, I.

I. Leifer, G. L. Rogers, N. W. F. Stephens, “Incoherent Fourier Transformation: a New Approach to Character Recognition,” Opt. Acta 16, 535 (1969).
[CrossRef]

Mertz, L.

L. Mertz, Transformation in Optics (Wiley, New York, 1965).

Monahan, M. A.

M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent Optical Correlators,” Proc. IEEE 65, 121 (1977).
[CrossRef]

Otto, O. W.

H. M. Gerard, P. S. Yao, O. W. Otto, “Performance of a Programmable Radar Pulse Compression Filter Based on a Chirp Transformation with RAC Filters,” Proc. IEEE Ultrason. Symp. 947, (1977).

Paige, E. G. S.

M. A. Jack, E. G. S. Paige, “Fourier Transformation Processors Based on Surface Acoustic Wave Chirp Filters,” Wave Electron. 3, 229 (1978).

Richardson, J. M.

J. M. Richardson, U.S. Patent3,669,528, “Device for Producing Identifiable Sine and Cosine (Fourier) Transforms of Input Signals by means of Noncoherent Optics” (June1972).

Rogers, G. L.

I. Leifer, G. L. Rogers, N. W. F. Stephens, “Incoherent Fourier Transformation: a New Approach to Character Recognition,” Opt. Acta 16, 535 (1969).
[CrossRef]

Smith, W. E.

Stephens, N. W. F.

I. Leifer, G. L. Rogers, N. W. F. Stephens, “Incoherent Fourier Transformation: a New Approach to Character Recognition,” Opt. Acta 16, 535 (1969).
[CrossRef]

Stetson, K. A.

K. A. Stetson, J. N. Elkins, “Optical System for Dynamic Analysis of Rotating Structures,” Tech. Rep. AFAPL-TR-77-51 (United Technologies Research Center, East Hartford, Conn., 1977).

Su, J. C.

D. Fraser, B. R. Hunt, J. C. Su, “Principles of Tomography in Image Data Compression,” Opt. Eng. 24, 298 (1985).
[CrossRef]

Szu Chang, L.

N. H. Farhat, C. Y. Ho, L. Szu Chang, “Projection Theorems and Their Application in Multidimensional Signal Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 140 (1983).

Tai, A. M.

Ticknor, A. J.

A. J. Ticknor, R. L. Easton, H. H. Barrett., “A Two-Dimensional Radon-Fourier Transformer,” Opt. Eng. 24, 082 (1985).
[CrossRef]

R. L. Easton, A. J. Ticknor, H. H. Barrett, “Application of the Radon Transform to Optical Production of the Wigner Distribution Function,” Opt. Eng. 23, 738 (1984).

R. L. Easton, H. H. Barrett, A. J. Ticknor, “Using SAW Filters to Process Two-Dimensional Data via the Radon Transform,” Proc. IEEE Ultrason. Symp. 185 (1983).

Toschi, V.

Wang, S.

Whitehouse, H. J.

H. J. Whitehouse, “Role of Charge-Coupled Devices and Surface Acoustic Wave Devices in Optical Signal Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 118, 124 (1977).

Xu, K.

K. Xu, R. Ji, Z. Zhang, “A New Method for Measuring the Phase of the Fourier Spectrum,” Opt. Commun. 50, 85 (1984).
[CrossRef]

Yao, P. S.

H. M. Gerard, P. S. Yao, O. W. Otto, “Performance of a Programmable Radar Pulse Compression Filter Based on a Chirp Transformation with RAC Filters,” Proc. IEEE Ultrason. Symp. 947, (1977).

Zhang, Z.

K. Xu, R. Ji, Z. Zhang, “A New Method for Measuring the Phase of the Fourier Spectrum,” Opt. Commun. 50, 85 (1984).
[CrossRef]

Appl. Opt. (4)

IEEE Proc. (1)

M. A. Jack, P. M. Grant, J. H. Collins, “The Theory, Design, and Applications of Surface Acoustic Wave Fourier-Transform Processors,” IEEE Proc. 68, 450 (1980).
[CrossRef]

Opt. Acta (1)

I. Leifer, G. L. Rogers, N. W. F. Stephens, “Incoherent Fourier Transformation: a New Approach to Character Recognition,” Opt. Acta 16, 535 (1969).
[CrossRef]

Opt. Commun. (1)

K. Xu, R. Ji, Z. Zhang, “A New Method for Measuring the Phase of the Fourier Spectrum,” Opt. Commun. 50, 85 (1984).
[CrossRef]

Opt. Eng. (4)

G. R. Gindi, A. F. Gmitro, “Optical Feature Extraction via the Radon Transform,” Opt. Eng. 23, 499 (1984).
[CrossRef]

D. Fraser, B. R. Hunt, J. C. Su, “Principles of Tomography in Image Data Compression,” Opt. Eng. 24, 298 (1985).
[CrossRef]

R. L. Easton, A. J. Ticknor, H. H. Barrett, “Application of the Radon Transform to Optical Production of the Wigner Distribution Function,” Opt. Eng. 23, 738 (1984).

A. J. Ticknor, R. L. Easton, H. H. Barrett., “A Two-Dimensional Radon-Fourier Transformer,” Opt. Eng. 24, 082 (1985).
[CrossRef]

Opt. Lett. (3)

Proc. IEEE (1)

M. A. Monahan, K. Bromley, R. P. Bocker, “Incoherent Optical Correlators,” Proc. IEEE 65, 121 (1977).
[CrossRef]

Proc. IEEE Ultrason. Symp. (2)

R. L. Easton, H. H. Barrett, A. J. Ticknor, “Using SAW Filters to Process Two-Dimensional Data via the Radon Transform,” Proc. IEEE Ultrason. Symp. 185 (1983).

H. M. Gerard, P. S. Yao, O. W. Otto, “Performance of a Programmable Radar Pulse Compression Filter Based on a Chirp Transformation with RAC Filters,” Proc. IEEE Ultrason. Symp. 947, (1977).

Proc. Soc. Photo-Opt. Instrum. Eng. (4)

N. H. Farhat, C. Y. Ho, L. Szu Chang, “Projection Theorems and Their Application in Multidimensional Signal Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 140 (1983).

H. J. Whitehouse, “Role of Charge-Coupled Devices and Surface Acoustic Wave Devices in Optical Signal Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 118, 124 (1977).

A. G. Gmitro, G. R. Gindi, H. H. Barrett, R. L. Easton, “Two-Dimensional Image Processing by One-Dimensional Filtering of Projection Data,” Proc. Soc. Photo-Opt. Instrum. Eng. 388, 132 (1983).

H. H. Barrett, “Three-Dimensional Image Reconstruction from Planar Projections with Application to Optical Data Processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 373, 179 (1983).

Wave Electron. (1)

M. A. Jack, E. G. S. Paige, “Fourier Transformation Processors Based on Surface Acoustic Wave Chirp Filters,” Wave Electron. 3, 229 (1978).

Other (4)

L. Mertz, Transformation in Optics (Wiley, New York, 1965).

J. M. Richardson, U.S. Patent3,669,528, “Device for Producing Identifiable Sine and Cosine (Fourier) Transforms of Input Signals by means of Noncoherent Optics” (June1972).

K. A. Stetson, J. N. Elkins, “Optical System for Dynamic Analysis of Rotating Structures,” Tech. Rep. AFAPL-TR-77-51 (United Technologies Research Center, East Hartford, Conn., 1977).

S. R. Deans, The Radon Transform and Some of its Applications (J Wiley, New York, 1983).

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Figures (6)

Fig. 1
Fig. 1

Schematic of the 1-D SAW complex Fourier transformer. The temporal signal f(t) from the photomultiplier in the flying-line scanner is proportional to one projection. The impulse response of the SAW filters is h±(t). The microcomputer controller sends a trigger signal to the digital delay generator, which in turn produces a. 1-nsec pulse that is applied to the downchirp SAW filter. The resulting impulse-response signal h(t) is multiplied by the incoming projection signal in a rf mixer. The product signal is applied to the upchirp SAW filter, and the output goes to the signal-input port of the rf phase comparator. After a delay of 14 μsec (1-nsec resolution), the digital delay generator outputs a second 1-nsec pulse, which is applied to the postmultiplication SAW filter. An rf phase shifter at the SAW filter output allows fine adjustment of the post-multiplication timing. This signal is applied to the reference port of the phase comparator. After low pass filtering, the in-phase I output of the phase comparator is proportional to the real part of the Fourier transform F(ν) (i.e., cosine transform) of the input signal f(t). Similarly, the output of the quadrature port Q of the phase comparator is proportional to the imginary part of F(ν), (i.e., sine transform).

Fig. 2
Fig. 2

Performance of the SAW chirp complex Fourier transformer. In each of the four cases shown, the top trace is the signal from the flying-line scanner, i.e., a single projection of the 2-D input. The second and third traces are the cosine transform and sine transform, respectively, produced by the SAW chirp transformer. The traces on the right-hand side are a computer simulation of the same signal. The object was a grating of 25% duty cycle in a circular aperture. In the first case, the grating was centered in the aperture creating a symmetric signal whose Fourier transform is purely real. In the other three cases, the grating was translated relative to the circular aperture giving an asymmetric signal with a complex transform. Each horizontal division in the oscilloscope traces represents 5 μsec, indicating that the complete transform is computed within 30 μsec.

Fig. 3
Fig. 3

Two-dimensional complex Fourier transforms of a circular aperture. The object was a single circular aperture of 1.0-mm diameter, as shown at top. The letters denote the origin of coordinates (i.e., the optical axis) for each case. The display was biased up, so that zero amplitude is the brightness level shown in the imaginary part of (A). The brightest areas represent the most positive amplitude of the transform, and the darkest areas represent the most negative amplitude. (A) With the aperture centered at the origin, the transform is purely real. (B), (C) The aperture was translated from the optic axis by ~1.4 and 2.4 diameters, respectively, producing fringes due to the constant phase term.

Fig. 4
Fig. 4

Two-dimensional complex Fourier transforms of a grating in a circular aperture. The spatial frequency of the grating was 1.5 cycles/mm, with a duty cycle of 80% and the aperture diameter was 6 mm. (A) Cosine transform with the object centered on the optical axis as shown. The transform is even, and the Airy patterns of the circular aperture at the ±1 orders of the grating are clearly seen. (B) Cosine transform after translating the object by one-half of a grating cycle. The linear phase term resulting from the translation has inverted the phase of the Airy patterns. (C), (D) Sine transform of the object after translation by ±one-fourth of a grating cycle, respectively, relative to (A). The transforms are odd, and the Airy patterns at the ±1 orders are out of phase. (E), (F) The aperture diameter was reduced to 2.5 mm, and the center was translated relative the optic axis by a sufficient distance (2 mm) so that several cycles of the linear phase are visible within the central disk of the Airy pattern. (E) is the real part of the transform and is even. (F) is the imaginary part and is odd. Note that the translation was in different directions in the two cases, so that the fringe direction differs.

Fig. 5
Fig. 5

Two-dimensional complex Fourier transforms of two circular apertures. The diameter of the apertures was 1 mm with their centers separated by 5 mm, as shown at top. Again the letters denote the position of the optical axis in each case. (A) Cosine transform with the optic axis centered on the object’s axis of symmetry. Note the phase change as a fringe passes from the central lobe of the Airy disk to the first ring. The faint fringes in the imaginary part of the transform are due to wobble in the image rotating prism. (B) The optic axis was located 1 mm above the symmetry axis producing fringes perpendicular to those from the double aperture. The cosine transform is even, and the sine transform is odd. (C) The optic axis was located on the symmetry axis but displaced from the center of symmetry by 1 mm multiplying the fringes by a linear phase term of lower frequency.

Fig. 6
Fig. 6

Two-dimensional complex transforms of a reflective object. A beam splitter was introduced into the flying-line scanner to direct the reflected line onto the detector. Fourier transformation and display were performed as before. The object was a grating in a circular aperture of 6-mm diameter, as in Fig. 4. The main features of the transform are easily seen, i.e., the location and phase of the Airy patterns on the first orders of the grating spectrum. The signal-to-noise is less than in the transmissive case due to the lower reflectance and lower modulation in reflectance. The real and imaginary parts of the Fourier transform of the object centered on the optical axis are shown in (A). Since the object is symmetric in this case, the imaginary part of the transform vanishes. (B) The object was translated by one-half of a grating cycle.

Equations (11)

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

exp ( 2 π i ν t ) = { exp [ i π ( ν β ) 2 } × { exp [ i π ( β t ) 2 } × { exp [ i π ( ν β β t ) 2 ] } .
F ( ν ) = + f ( t ) exp ( 2 π i ν t ) d t = exp [ i π ( ν β ) 2 ] × + { f ( t ) exp [ i π ( β t ) 2 ] } × exp [ i π ( ν β β t ) 2 ] d t = [ exp [ i π ( β t ) 2 ] × ( { F ( t ) exp [ i π ( β t ) 2 ] } * exp [ i π ( β t ) 2 ] ) ] | ( ν = β 2 t ) ,
1 2 π × d ϕ d t | ( t = t n ) = + β 2 t n 2 ,
h ( t ) = cos [ ( ω 0 t ± α t 2 2 ) ] .
ν n = 1 2 π × d d t ( ω 0 t ± α t 2 2 ) | ( t ¯ t n ) = ω 0 ± α t n 2 π = ν 0 ± α t n 2 π .
Rect ( t τ 1 2 ) × cos ( ω t a t 2 2 ) ,
h ( t ) = Rect [ t τ + 1 2 ] × cos ( ω + t + α t 2 2 ) ,
( a ) Rect ( t t τ + 1 2 ) × cos ( ω + t + α t 2 2 ) ( b ) Rect ( t t τ + 1 2 ) × cos ( ω + t + α t 2 2 π 2 ) = Rect ( t t τ + 1 2 ) × sin ( ω + t + α t 2 2 ) ,
ν = ( ω + ω + α t ) / 2 π .
A ( t ) cos ( ω a t ) × B ( t ) cos ( ω b t ) = [ A ( t ) B ( t ) ] 2 × { cos [ ( ω a + ω b ) t ] + cos [ ( ω a ω b ) t ] } .
( | ν | α τ + 8 π ) .

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