Abstract

Human task performance using a passive interferometric millimeter wave imaging sensor is modeled using a task performance modeling approach developed by the U.S. Army Night Vision and Electronic Sensors Directorate. The techniques used are illustrated for an imaging system composed of an interferometric antenna array, optical upconversion, and image formation using a shortwave infrared focal plane array. Two tasks, target identification and pilotage, are modeled. The effects of sparse antenna arrays on task performance are considered. Applications of this model include system trade studies for concealed weapon identification, navigation in fog, and brownout conditions.

© 2010 Optical Society of America

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References

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  1. R. H. Vollmerhausen, E. Jacobs, and R. G. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43, 2806–2818 (2004).
    [CrossRef]
  2. R. H. Vollmerhausen and T. Bui, “Using a targeting metric to predict the utility of an EO imager as a pilotage aid,” Proc. SPIE 6207, 62070C (2006).
    [CrossRef]
  3. S. R. Murrill, E. L. Jacobs, S. K. Moyer, C. E. Halford, S. T. Griffin, F. C. D. Lucia, D. T. Petkie, and C. C. Franck, “Terahertz imaging system performance model for concealed-weapon identification,” Appl. Opt. 47, 1286–1297 (2008).
    [CrossRef] [PubMed]
  4. R. L. Espinola, E. L. Jacobs, C. E. Halford, R. Vollmerhausen, and D. H. Tofsted, “Modeling the target acquisition performance of active imaging systems,” Opt. Express 15, 3816–3832 (2007).
    [CrossRef] [PubMed]
  5. P. M. Blanchard, A. H. Greenaway, A. R. Harvey, and K. Webster, “Coherent optical beam forming with passive millimeter-wave arrays,” J. Lightwave Technol. 17, 418–425(1999).
    [CrossRef]
  6. C. A. Schuetz, J. Murakowski, G. J. Schneider, and D. W. Prather, “Radiometric millimeter-wave detection via optical upconversion and carrier suppression,” IEEE Trans. Microwave Theory Tech. 53, 1732–1738 (2005).
  7. D. Prather, I. Biswas, C. Schuetz, R. Martin, and M. Mirotznik, “Multiple aperture imaging of millimeter sources via image-plane interferometry,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS, 2007). pp. 2967–2970.
    [CrossRef]
  8. E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).
  9. S. S. Young, R. G. Driggers, and E. L. Jacobs, Signal Processing and Performance Analysis for Imaging Systems, Artech House Optoelectronics Series (Artech House, 2008).
  10. R. H. Vollmerhausen, E. L. Jacobs, J. Hixson, and M. Friedman, “The targeting task performance (TTP) metric.” Technical Report AMSEL-NV-TR-230 (U.S. Army Night Vision and Electronics Sensor Directorate, 2006), p. 140.
  11. C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W.Prather, “Optical techniques for sparse-aperture millimeter-wave imaging,” Proc. SPIE 6211, 62110G (2006).
    [CrossRef]
  12. N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46, 5933–5943 (2007).
    [CrossRef] [PubMed]
  13. C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. (Wiley, 1997).
  14. C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Applications of optical upconversion to sparse aperture millimeter-wave imaging,” Proc. SPIE 5989, 59891C (2005)
    [CrossRef]

2008 (1)

2007 (2)

2006 (2)

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W.Prather, “Optical techniques for sparse-aperture millimeter-wave imaging,” Proc. SPIE 6211, 62110G (2006).
[CrossRef]

R. H. Vollmerhausen and T. Bui, “Using a targeting metric to predict the utility of an EO imager as a pilotage aid,” Proc. SPIE 6207, 62070C (2006).
[CrossRef]

2005 (2)

C. A. Schuetz, J. Murakowski, G. J. Schneider, and D. W. Prather, “Radiometric millimeter-wave detection via optical upconversion and carrier suppression,” IEEE Trans. Microwave Theory Tech. 53, 1732–1738 (2005).

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Applications of optical upconversion to sparse aperture millimeter-wave imaging,” Proc. SPIE 5989, 59891C (2005)
[CrossRef]

2004 (1)

R. H. Vollmerhausen, E. Jacobs, and R. G. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43, 2806–2818 (2004).
[CrossRef]

1999 (1)

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. (Wiley, 1997).

Biswas, I.

D. Prather, I. Biswas, C. Schuetz, R. Martin, and M. Mirotznik, “Multiple aperture imaging of millimeter sources via image-plane interferometry,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS, 2007). pp. 2967–2970.
[CrossRef]

Blanchard, P. M.

Boettcher, E.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Bui, T.

R. H. Vollmerhausen and T. Bui, “Using a targeting metric to predict the utility of an EO imager as a pilotage aid,” Proc. SPIE 6207, 62070C (2006).
[CrossRef]

Dierking, M. P.

Driggers, R. G.

S. S. Young, R. G. Driggers, and E. L. Jacobs, Signal Processing and Performance Analysis for Imaging Systems, Artech House Optoelectronics Series (Artech House, 2008).

Driggers, R.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Driggers, R. G.

R. H. Vollmerhausen, E. Jacobs, and R. G. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43, 2806–2818 (2004).
[CrossRef]

Duncan, B. D.

Espinola, R. L.

Franck, C. C.

Friedman, M.

R. H. Vollmerhausen, E. L. Jacobs, J. Hixson, and M. Friedman, “The targeting task performance (TTP) metric.” Technical Report AMSEL-NV-TR-230 (U.S. Army Night Vision and Electronics Sensor Directorate, 2006), p. 140.

Garcia, J.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Grata, J.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Greenaway, A. H.

Griffin, S. T.

Halford, C. E.

Harris, R.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Harvey, A. R.

Hixson, J.

R. H. Vollmerhausen, E. L. Jacobs, J. Hixson, and M. Friedman, “The targeting task performance (TTP) metric.” Technical Report AMSEL-NV-TR-230 (U.S. Army Night Vision and Electronics Sensor Directorate, 2006), p. 140.

Jacobs, E. L.

R. L. Espinola, E. L. Jacobs, C. E. Halford, R. Vollmerhausen, and D. H. Tofsted, “Modeling the target acquisition performance of active imaging systems,” Opt. Express 15, 3816–3832 (2007).
[CrossRef] [PubMed]

R. H. Vollmerhausen, E. L. Jacobs, J. Hixson, and M. Friedman, “The targeting task performance (TTP) metric.” Technical Report AMSEL-NV-TR-230 (U.S. Army Night Vision and Electronics Sensor Directorate, 2006), p. 140.

S. S. Young, R. G. Driggers, and E. L. Jacobs, Signal Processing and Performance Analysis for Imaging Systems, Artech House Optoelectronics Series (Artech House, 2008).

Jacobs, E.

R. H. Vollmerhausen, E. Jacobs, and R. G. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43, 2806–2818 (2004).
[CrossRef]

Jacobs, E. L.

Kiser, W.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Krapels, K.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Lucia, F. C. D.

Martin, R.

D. Prather, I. Biswas, C. Schuetz, R. Martin, and M. Mirotznik, “Multiple aperture imaging of millimeter sources via image-plane interferometry,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS, 2007). pp. 2967–2970.
[CrossRef]

Miller, N. J.

Mirotznik, M.

D. Prather, I. Biswas, C. Schuetz, R. Martin, and M. Mirotznik, “Multiple aperture imaging of millimeter sources via image-plane interferometry,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS, 2007). pp. 2967–2970.
[CrossRef]

Mirotznik, M. S.

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W.Prather, “Optical techniques for sparse-aperture millimeter-wave imaging,” Proc. SPIE 6211, 62110G (2006).
[CrossRef]

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Applications of optical upconversion to sparse aperture millimeter-wave imaging,” Proc. SPIE 5989, 59891C (2005)
[CrossRef]

Moyer, S. K.

Murakowski, J.

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W.Prather, “Optical techniques for sparse-aperture millimeter-wave imaging,” Proc. SPIE 6211, 62110G (2006).
[CrossRef]

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Applications of optical upconversion to sparse aperture millimeter-wave imaging,” Proc. SPIE 5989, 59891C (2005)
[CrossRef]

Murakowski, J.

C. A. Schuetz, J. Murakowski, G. J. Schneider, and D. W. Prather, “Radiometric millimeter-wave detection via optical upconversion and carrier suppression,” IEEE Trans. Microwave Theory Tech. 53, 1732–1738 (2005).

Murrill, S. R.

Petkie, D. T.

Prather, D. W.

C. A. Schuetz, J. Murakowski, G. J. Schneider, and D. W. Prather, “Radiometric millimeter-wave detection via optical upconversion and carrier suppression,” IEEE Trans. Microwave Theory Tech. 53, 1732–1738 (2005).

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Applications of optical upconversion to sparse aperture millimeter-wave imaging,” Proc. SPIE 5989, 59891C (2005)
[CrossRef]

Prather, D.

D. Prather, I. Biswas, C. Schuetz, R. Martin, and M. Mirotznik, “Multiple aperture imaging of millimeter sources via image-plane interferometry,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS, 2007). pp. 2967–2970.
[CrossRef]

Prather, D. W.

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W.Prather, “Optical techniques for sparse-aperture millimeter-wave imaging,” Proc. SPIE 6211, 62110G (2006).
[CrossRef]

Samluk, J.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Schneider, G.

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W.Prather, “Optical techniques for sparse-aperture millimeter-wave imaging,” Proc. SPIE 6211, 62110G (2006).
[CrossRef]

Schneider, G. J.

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Applications of optical upconversion to sparse aperture millimeter-wave imaging,” Proc. SPIE 5989, 59891C (2005)
[CrossRef]

Schneider, G. J.

C. A. Schuetz, J. Murakowski, G. J. Schneider, and D. W. Prather, “Radiometric millimeter-wave detection via optical upconversion and carrier suppression,” IEEE Trans. Microwave Theory Tech. 53, 1732–1738 (2005).

Schuetz, C.

D. Prather, I. Biswas, C. Schuetz, R. Martin, and M. Mirotznik, “Multiple aperture imaging of millimeter sources via image-plane interferometry,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS, 2007). pp. 2967–2970.
[CrossRef]

Schuetz, C.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Schuetz, C. A.

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W.Prather, “Optical techniques for sparse-aperture millimeter-wave imaging,” Proc. SPIE 6211, 62110G (2006).
[CrossRef]

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Applications of optical upconversion to sparse aperture millimeter-wave imaging,” Proc. SPIE 5989, 59891C (2005)
[CrossRef]

C. A. Schuetz, J. Murakowski, G. J. Schneider, and D. W. Prather, “Radiometric millimeter-wave detection via optical upconversion and carrier suppression,” IEEE Trans. Microwave Theory Tech. 53, 1732–1738 (2005).

Shi, S.

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W.Prather, “Optical techniques for sparse-aperture millimeter-wave imaging,” Proc. SPIE 6211, 62110G (2006).
[CrossRef]

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Applications of optical upconversion to sparse aperture millimeter-wave imaging,” Proc. SPIE 5989, 59891C (2005)
[CrossRef]

Stein, L.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Tofsted, D. H.

Visnansky, A.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Vollmerhausen, R.

Vollmerhausen, R. H.

R. H. Vollmerhausen and T. Bui, “Using a targeting metric to predict the utility of an EO imager as a pilotage aid,” Proc. SPIE 6207, 62070C (2006).
[CrossRef]

R. H. Vollmerhausen, E. Jacobs, and R. G. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43, 2806–2818 (2004).
[CrossRef]

R. H. Vollmerhausen, E. L. Jacobs, J. Hixson, and M. Friedman, “The targeting task performance (TTP) metric.” Technical Report AMSEL-NV-TR-230 (U.S. Army Night Vision and Electronics Sensor Directorate, 2006), p. 140.

Webster, K.

Wikner, D.

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

Young, S. S.

S. S. Young, R. G. Driggers, and E. L. Jacobs, Signal Processing and Performance Analysis for Imaging Systems, Artech House Optoelectronics Series (Artech House, 2008).

Opt. Eng. (1)

R. H. Vollmerhausen, E. Jacobs, and R. G. Driggers, “New metric for predicting target acquisition performance,” Opt. Eng. 43, 2806–2818 (2004).
[CrossRef]

Appl. Opt. (2)

J. Lightwave Technol. (1)

Opt. Express (1)

Proc. SPIE (3)

R. H. Vollmerhausen and T. Bui, “Using a targeting metric to predict the utility of an EO imager as a pilotage aid,” Proc. SPIE 6207, 62070C (2006).
[CrossRef]

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W.Prather, “Optical techniques for sparse-aperture millimeter-wave imaging,” Proc. SPIE 6211, 62110G (2006).
[CrossRef]

C. A. Schuetz, M. S. Mirotznik, S. Shi, G. J. Schneider, J. Murakowski, and D. W. Prather, “Applications of optical upconversion to sparse aperture millimeter-wave imaging,” Proc. SPIE 5989, 59891C (2005)
[CrossRef]

Other (6)

C. A. Balanis, Antenna Theory: Analysis and Design, 2nd ed. (Wiley, 1997).

C. A. Schuetz, J. Murakowski, G. J. Schneider, and D. W. Prather, “Radiometric millimeter-wave detection via optical upconversion and carrier suppression,” IEEE Trans. Microwave Theory Tech. 53, 1732–1738 (2005).

D. Prather, I. Biswas, C. Schuetz, R. Martin, and M. Mirotznik, “Multiple aperture imaging of millimeter sources via image-plane interferometry,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS, 2007). pp. 2967–2970.
[CrossRef]

E. Boettcher, K. Krapels, R. Driggers, J. Garcia, C. Schuetz, J. Samluk, L. Stein, W. Kiser, A. Visnansky, J. Grata, D. Wikner, and R. Harris, “Modeling field performance of passive millimeter wave imagery for discriminating small watercraft,” in Proceedings of the Military Sensing Symposium (SENSIAC, 2009).

S. S. Young, R. G. Driggers, and E. L. Jacobs, Signal Processing and Performance Analysis for Imaging Systems, Artech House Optoelectronics Series (Artech House, 2008).

R. H. Vollmerhausen, E. L. Jacobs, J. Hixson, and M. Friedman, “The targeting task performance (TTP) metric.” Technical Report AMSEL-NV-TR-230 (U.S. Army Night Vision and Electronics Sensor Directorate, 2006), p. 140.

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Figures (12)

Fig. 1
Fig. 1

Schematic of a PMMW interferometric sensor.

Fig. 2
Fig. 2

Antenna and source plane coordinates.

Fig. 3
Fig. 3

Fiber bundle configuration.

Fig. 4
Fig. 4

Approximate MTF and central horizontal plane cut.

Fig. 5
Fig. 5

Images of two coherent point sources at 200 m; the sources are separated by 3 m.

Fig. 6
Fig. 6

MTFs, CTFs, and identification range performance for a 17 × 17 element array.

Fig. 7
Fig. 7

MTFs, CTFs, and identification range performance for a 33 × 33 element array.

Fig. 8
Fig. 8

MTFs, CTFs, and identification range performance for a 17 × 17 element array with element spacing of 4.9622 cm .

Fig. 9
Fig. 9

Fiber bundle layout for a circular array with 17 elements per radial arm.

Fig. 10
Fig. 10

MTFs, CTFs, and identification performance of the circular array shown in Fig. 9.

Fig. 11
Fig. 11

Fiber layout for a six-legged polar array with 17 elements per leg.

Fig. 12
Fig. 12

MTFs, CTFs, and identification range performance for the polar array in Fig. 11.

Tables (2)

Tables Icon

Table 1 Simulation Parameters

Tables Icon

Table 2 Performance Calculation Parameters

Equations (31)

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

M TTP = f 1 f 2 C T CTF sys ( f ) d f .
CTF sys ( f ) = CTF eye ( f M ) MTF sys ( f ) [ 1 + k 2 σ N 2 ( f ) ] 1 2 .
P ( R ) = χ ( R ) 1 + χ ( R ) , χ ( R ) = ( V ( R ) V 50 ) α ( V ( R ) V 50 ) + β , V ( R ) = A T R M TTP ,
V oc ( x ) = e j k z j λ z [ U ( x ) * J ˜ n ( x λ z ) e j k 2 z | x | 2 ] * S array ( x ) ,
E mod ( x ) = j m 2 | E opt | | V oc ( x ) | e j ( ω 0 + ω ¯ m ) t + j ( ϕ 0 + ϕ m ) ,
I f pa ( x f pa ) = 1 ( λ f ) 2 | + [ E mod ( p ̲ ̲ 1 · x ) as * Φ ( x f ) ] e - j 2 π λ f ( x f pa · x f ) d x f | 2 ,
I disp ( x f pa ) = { [ I f pa ( x f pa ) * h det ( x f pa ) ] * S f pa ( x f pa ) } * h disp ( x f pa ) ,
I disp ( x f pa ) = { [ I target ( x f pa ) * h int ( x f pa ) * h det ( x f pa ) ] * S f pa ( x f pa ) } * h disp ( x f pa ) .
MTF sys ( u ) = MTF int ( u ) MTF det ( u ) MTF disp ( u ) ,
MTF det ( u ) = sinc ( u x θ x ) sinc ( u y θ y ) ,
θ x = Δ x f l M , θ y = Δ y f l N ,
θ d = Δ d d v ,
α x = θ d θ x , α y = θ d θ y .
MTF disp ( u ̲ ) = e π [ ( u x α x ) 2 + ( u y α y ) 2 ] θ d 2 .
V oc = E i · l e
E r = j k η e - j k r 4 π r I in l e .
E r = j k η e - j k r 4 π r J ˜ ( r ^ λ ) ,
l e = J ˜ ( r ^ λ ) I in = J ˜ n ( r ^ λ ) .
E i ( x , x 0 ) = e j k z j λ z U ( x 0 ) e j k 2 z | x - x 0 | 2 ,
V oc ( x ) = e j k z j λ z U ( x 0 ) · J ˜ n ( x - x 0 λ z ) e j k 2 z | x - x 0 | 2 d x 0 = e j k z j λ z U ( x ) * J ˜ n ( x λ z ) e j k 2 z | x ̲ | 2 ,
V oc ( x ) = e j k z j λ z [ U ( x ) * J ˜ n ( x λ z ) e j k 2 z | x | 2 ] * S array ( x ) .
V MMW ( x , t ) = V oc ( x ) e j ω ¯ m t ,
E mod ( x ) = j m 2 E opt V oc ( x ) e j ( ω 0 + ω ¯ m ) t = j m 2 | E opt | | V oc ( x ) | e j ( ω 0 + ω ¯ m ) t + j ( ϕ 0 + ϕ m ) ,
x f = M x x ^ + N y y ^
x = p ̲ ̲ · x f = [ 1 M x ^ x ^ + 1 N y ^ y ^ ] · x f ,
E bundle ( x f ) = E mod ( p ̲ ̲ 1 · x ) * Φ ( x f ) .
E lens ( x f ) = E bundle ( x f ) e j π λ f | x ̲ f | 2 ,
E f pa ( x f pa ) = j λ f e j π λ f | x f pa | 2 + E bundle ( x f ) e j 2 π λ f ( x f pa · x f ) d x f = j λ f e j π λ f | x f pa | 2 E ˜ bundle ( x f ) ,
I f pa ( x f pa ) = | E f pa ( x f pa ) | 2 = 1 ( λ f ) 2 | E ˜ bundle ( x f pa λ f ) | 2 .
I det ( x f pa ) = [ I f pa ( x f pa ) * h det ( x f pa ) ] * S f pa ( x f pa ) ,
I disp ( x f pa ) = I det ( x f pa ) * h disp ( x f pa ) ,

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