Abstract

A principal difficulty of long dwell coherent imaging ladar is its extreme sensitivity to target or platform motion. This paper describes a motion compensated frequency modulated continuous wave 3D coherent imaging ladar method that overcomes this motion sensitivity, making it possible to work with nonstatic targets such as human faces, as well as imaging of targets through refractive turbulence. Key features of this method include scannerless imaging and high range resolution. The reduced motion sensitivity is shown with mathematical analysis and demonstration 3D images. Images of static and dynamic targets are provided demonstrating up to 600×800pixel imaging with millimeter range resolution.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Acharekar, P. Gatt, and L. Mizerka, “Laser vibration sensor,” Proc. SPIE 2472, 2–11 (1995).
    [CrossRef]
  2. S. M. Beck, J. R. Buck, W. F. Buell, R. P. Dickinson, D. A. Kozlowski, N. J. Marechal, and T. J. Wright, “Synthetic-aperture imaging laser radar: laboratory demonstration and signal processing,” Appl. Opt. 44, 7621–7629 (2005).
    [CrossRef]
  3. J. Buck, B. W. Krause, A. I. Malm, and C. M. Ryan, “Synthetic aperture imaging at optical wavelengths,” in Conference on Lasers and Electro-Optics/International Quantum Electronics (Optical Society of America, 2009), paper PThB3.
  4. B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic aperture ladar flight demonstration,” in CLEO:2011—Laser Applications to Photonic Applications (Optical Society of America, 2011), paper PDPB7.
  5. B. L. Stann, W. C. Ruff, and Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
    [CrossRef]
  6. C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, Spotlight-mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer, 1999).
  7. C. E. Cook and M. Bernfeld, Radar Signals: An Introduction to Theory and Application (Artech House, 1993).
  8. P. Gatt, J. A. Thomson, and S. W. Henderson, “Coherent laser radar range precision for range resolved and unresolved targets,” Proceedings of the 11th Coherent Laser Radar Conference, 2001 (Defense Evaluation & Research Agency, 2001).
  9. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. 2nd ed. (Springer, 1984).
  10. G. W. Deley, “Waveform design,” in Radar Handbook, M. I. Skolnik, ed. (McGraw-Hill, 1970).
  11. M. C. Teich and R. Y. Yen, “Three-frequency nonlinear heterodyne detection. 1: cw radar and analog communications,” Appl. Opt. 14, 666–679 (1975).
    [CrossRef]
  12. D. Fink, “Coherent detection signal-to-noise,” Appl. Opt. 14, 689–690 (1975).
    [CrossRef]
  13. B. W. Krause, P. Gatt, C. Embry, and J. R. Buck, “High resolution 3D coherent laser radar imaging,” Proc. SPIE 6214, 62140V (2006).
    [CrossRef]

2006

B. W. Krause, P. Gatt, C. Embry, and J. R. Buck, “High resolution 3D coherent laser radar imaging,” Proc. SPIE 6214, 62140V (2006).
[CrossRef]

2005

1996

B. L. Stann, W. C. Ruff, and Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
[CrossRef]

1995

M. Acharekar, P. Gatt, and L. Mizerka, “Laser vibration sensor,” Proc. SPIE 2472, 2–11 (1995).
[CrossRef]

1975

Acharekar, M.

M. Acharekar, P. Gatt, and L. Mizerka, “Laser vibration sensor,” Proc. SPIE 2472, 2–11 (1995).
[CrossRef]

Ashby, S.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic aperture ladar flight demonstration,” in CLEO:2011—Laser Applications to Photonic Applications (Optical Society of America, 2011), paper PDPB7.

Beck, S. M.

Bernfeld, M.

C. E. Cook and M. Bernfeld, Radar Signals: An Introduction to Theory and Application (Artech House, 1993).

Buck, J.

J. Buck, B. W. Krause, A. I. Malm, and C. M. Ryan, “Synthetic aperture imaging at optical wavelengths,” in Conference on Lasers and Electro-Optics/International Quantum Electronics (Optical Society of America, 2009), paper PThB3.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic aperture ladar flight demonstration,” in CLEO:2011—Laser Applications to Photonic Applications (Optical Society of America, 2011), paper PDPB7.

Buck, J. R.

Buell, W. F.

Cook, C. E.

C. E. Cook and M. Bernfeld, Radar Signals: An Introduction to Theory and Application (Artech House, 1993).

Deley, G. W.

G. W. Deley, “Waveform design,” in Radar Handbook, M. I. Skolnik, ed. (McGraw-Hill, 1970).

Dickinson, R. P.

Eichel, P. H.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, Spotlight-mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer, 1999).

Embry, C.

B. W. Krause, P. Gatt, C. Embry, and J. R. Buck, “High resolution 3D coherent laser radar imaging,” Proc. SPIE 6214, 62140V (2006).
[CrossRef]

Fink, D.

Gatt, P.

B. W. Krause, P. Gatt, C. Embry, and J. R. Buck, “High resolution 3D coherent laser radar imaging,” Proc. SPIE 6214, 62140V (2006).
[CrossRef]

M. Acharekar, P. Gatt, and L. Mizerka, “Laser vibration sensor,” Proc. SPIE 2472, 2–11 (1995).
[CrossRef]

P. Gatt, J. A. Thomson, and S. W. Henderson, “Coherent laser radar range precision for range resolved and unresolved targets,” Proceedings of the 11th Coherent Laser Radar Conference, 2001 (Defense Evaluation & Research Agency, 2001).

Ghiglia, D. C.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, Spotlight-mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer, 1999).

Gleason, A.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic aperture ladar flight demonstration,” in CLEO:2011—Laser Applications to Photonic Applications (Optical Society of America, 2011), paper PDPB7.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. 2nd ed. (Springer, 1984).

Henderson, S. W.

P. Gatt, J. A. Thomson, and S. W. Henderson, “Coherent laser radar range precision for range resolved and unresolved targets,” Proceedings of the 11th Coherent Laser Radar Conference, 2001 (Defense Evaluation & Research Agency, 2001).

Hwang, D.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic aperture ladar flight demonstration,” in CLEO:2011—Laser Applications to Photonic Applications (Optical Society of America, 2011), paper PDPB7.

Jakowatz, C. V.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, Spotlight-mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer, 1999).

Kondratko, P.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic aperture ladar flight demonstration,” in CLEO:2011—Laser Applications to Photonic Applications (Optical Society of America, 2011), paper PDPB7.

Kozlowski, D. A.

Krause, B.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic aperture ladar flight demonstration,” in CLEO:2011—Laser Applications to Photonic Applications (Optical Society of America, 2011), paper PDPB7.

Krause, B. W.

B. W. Krause, P. Gatt, C. Embry, and J. R. Buck, “High resolution 3D coherent laser radar imaging,” Proc. SPIE 6214, 62140V (2006).
[CrossRef]

J. Buck, B. W. Krause, A. I. Malm, and C. M. Ryan, “Synthetic aperture imaging at optical wavelengths,” in Conference on Lasers and Electro-Optics/International Quantum Electronics (Optical Society of America, 2009), paper PThB3.

Malm, A.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic aperture ladar flight demonstration,” in CLEO:2011—Laser Applications to Photonic Applications (Optical Society of America, 2011), paper PDPB7.

Malm, A. I.

J. Buck, B. W. Krause, A. I. Malm, and C. M. Ryan, “Synthetic aperture imaging at optical wavelengths,” in Conference on Lasers and Electro-Optics/International Quantum Electronics (Optical Society of America, 2009), paper PThB3.

Marechal, N. J.

Mizerka, L.

M. Acharekar, P. Gatt, and L. Mizerka, “Laser vibration sensor,” Proc. SPIE 2472, 2–11 (1995).
[CrossRef]

Ruff, W. C.

B. L. Stann, W. C. Ruff, and Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
[CrossRef]

Ryan, C.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic aperture ladar flight demonstration,” in CLEO:2011—Laser Applications to Photonic Applications (Optical Society of America, 2011), paper PDPB7.

Ryan, C. M.

J. Buck, B. W. Krause, A. I. Malm, and C. M. Ryan, “Synthetic aperture imaging at optical wavelengths,” in Conference on Lasers and Electro-Optics/International Quantum Electronics (Optical Society of America, 2009), paper PThB3.

Stann, B. L.

B. L. Stann, W. C. Ruff, and Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
[CrossRef]

Sztankay, Z. G.

B. L. Stann, W. C. Ruff, and Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
[CrossRef]

Teich, M. C.

Thompson, P. A.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, Spotlight-mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer, 1999).

Thomson, J. A.

P. Gatt, J. A. Thomson, and S. W. Henderson, “Coherent laser radar range precision for range resolved and unresolved targets,” Proceedings of the 11th Coherent Laser Radar Conference, 2001 (Defense Evaluation & Research Agency, 2001).

Wahl, D. E.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, Spotlight-mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer, 1999).

Wright, T. J.

Yen, R. Y.

Appl. Opt.

Opt. Eng.

B. L. Stann, W. C. Ruff, and Z. G. Sztankay, “Intensity-modulated diode laser radar using frequency-modulation/continuous-wave ranging techniques,” Opt. Eng. 35, 3270–3278 (1996).
[CrossRef]

Proc. SPIE

B. W. Krause, P. Gatt, C. Embry, and J. R. Buck, “High resolution 3D coherent laser radar imaging,” Proc. SPIE 6214, 62140V (2006).
[CrossRef]

M. Acharekar, P. Gatt, and L. Mizerka, “Laser vibration sensor,” Proc. SPIE 2472, 2–11 (1995).
[CrossRef]

Other

J. Buck, B. W. Krause, A. I. Malm, and C. M. Ryan, “Synthetic aperture imaging at optical wavelengths,” in Conference on Lasers and Electro-Optics/International Quantum Electronics (Optical Society of America, 2009), paper PThB3.

B. Krause, J. Buck, C. Ryan, D. Hwang, P. Kondratko, A. Malm, A. Gleason, and S. Ashby, “Synthetic aperture ladar flight demonstration,” in CLEO:2011—Laser Applications to Photonic Applications (Optical Society of America, 2011), paper PDPB7.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, Spotlight-mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer, 1999).

C. E. Cook and M. Bernfeld, Radar Signals: An Introduction to Theory and Application (Artech House, 1993).

P. Gatt, J. A. Thomson, and S. W. Henderson, “Coherent laser radar range precision for range resolved and unresolved targets,” Proceedings of the 11th Coherent Laser Radar Conference, 2001 (Defense Evaluation & Research Agency, 2001).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. 2nd ed. (Springer, 1984).

G. W. Deley, “Waveform design,” in Radar Handbook, M. I. Skolnik, ed. (McGraw-Hill, 1970).

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 (16)

Fig. 1.
Fig. 1.

Simplified block diagram of the FMCW ladar system showing major components and the assembly configuration. T / R = transmit receive switch .

Fig. 2.
Fig. 2.

FMCW concept: waveform carrier frequency as a function of time showing the constant IF resulting from mixing the target echo with the LO. The IF is linearly related to target range.

Fig. 3.
Fig. 3.

Magnitude of the ambiguity function of the LFM waveform showing the relationship between velocity resolution and range resolution. The ellipse depicts an approximation of the main lobe. Also shown is the range-Doppler coupling.

Fig. 4.
Fig. 4.

Total waveform: carrier frequency ( f ) of the LFM and pilot waveforms as a function of time ( t ).

Fig. 5.
Fig. 5.

Illustration of negative pilot IF. The asterisks denote aliasing. The dashed arrow is the aliased f 1 . The difference frequency (tall arrow) is the sum of the aliased negative f 1 with positive f 2 .

Fig. 6.
Fig. 6.

Illustration of aliased operation in which f 2 aliases. Asterisks denote aliasing. The difference frequency (tall dashed arrow) is the aliased version of the sum of f 1 and aliased f 2 .

Fig. 7.
Fig. 7.

(a) Photo of the model Humvee target taken from approximately the same pose as the 3D image. (b) A rendered surface of the 3D data with simulated lighting to reveal the surface details. Range is represented by surface color. Waveform bandwidth = 34 GHz . Range resolution = 4.4 mm . Range precision = 1 mm . Cross range pixels = 540 × 612 . Ramp duration = 2 s . Target range = 1 m . Wavelength = 1550 nm . Transmitted power = 1 W . Number of ramps averaged = 7 . Median filtering applied to 5 × 5 windows. Static target: motion compensation not used.

Fig. 8.
Fig. 8.

Photograph of the mannequin showing the 3D fine features.

Fig. 9.
Fig. 9.

3D image of the mannequin’s face. (a) A gray surface with simulated illumination. The creases around the mouth and eyes are clearly visible. The warts on the forehead and nose are also visible. (b) Dot cloud of the 3D image of the mannequin’s face viewed from the side. Two warts on the nose and the on the side of the forehead are clearly visible in this view. Waveform bandwidth = 34 GHz . Range resolution = 4.4 mm . Range precision = 1 mm . Cross range pixels = 512 × 640 . Cross range resolution = 280 μm . Ramp duration = 2 s . Target range = 1 m . Wavelength = 1550 nm . Number of ramps averaged = 7 . Median filtering applied to 5 × 5 windows. Static target: motion compensation not used.

Fig. 10.
Fig. 10.

3D image of chess pieces on a table. (a) Rendered surface with simulated illumination. (b) Angled view of 3D dot cloud. Waveform bandwidth = 20 GHz . Range resolution = 7.5 mm . Cross range pixels = 256 × 256 . Ramp duration = 381 ms . Wavelength = 1550 nm . Transmitted power = 2 W . No ramp averaging. Range precision = 2 mm . Median filtering applied to 5 × 5 windows. Motion compensation used. Aliasing not enabled.

Fig. 11.
Fig. 11.

3D image of a mannequin seated in a chair 1.5 m away from the sensor. Comparison with and without motion compensation. Waveform bandwidth = 41 GHz . Range resolution = 3.7 mm . Ramp duration = 0.96 s . Cross range pixels = 256 × 256 . No ramp averaging. No median filtering. Wavelength = 1550 nm . Transmitted power = 500 mW . All dimensions above are in cm. (a) Image without motion compensation. (b) Image with motion compensation.

Fig. 12.
Fig. 12.

3D image of an outdoor mannequin face, imaged through refractive turbulence. (a) Range image where gray scale indicates relative range. (b) Angled view of 3D dot cloud. Range to the target is 17.5 m. Waveform bandwidth = 6.3 GHz . Range resolution = 4.8 cm . Cross range pixels = 256 × 256 . Cross range resolution = 520 μm . Ramp duration = 1 s . Wavelength = 1550 nm . No ramp averaging. Median filtering applied to 7 × 7 windows. LO downshifted by 400 Hz. Motion compensation enabled. Integration time = 725 μs allowing 3 × aliasing. Receiver aperture diameter = 10 cm . Refractive index structure constant C n 2 = 1.04 × 10 13 m 2 / 3 .

Fig. 13.
Fig. 13.

3D dot cloud of live human face. Waveform bandwidth = 100 GHz . Range resolution = 1.5 mm . Ramp duration = 1.2 s . Range precision < 1 mm . Cross range pixels = 288 × 288 . Range to target = 10 m . Wavelength = 780 nm . Transmitted power = 1 W . No ramp averaging. 3 × 3 median filtering.

Fig. 14.
Fig. 14.

Time domain camera signal for a single pixel. The observable beating is due to the roughly fixed frequency difference between the pilot and LFM IF signals. The beat frequency is constant despite the changing carrier frequency.

Fig. 15.
Fig. 15.

Power spectrum of the camera signal before and after squaring. The sharp peak at 572 Hz in the lower plot, labeled “Range Report” and its absence in the upper plot illustrates the effect of the motion compensation.

Fig. 16.
Fig. 16.

3D dot cloud of live human face. Bounding box indicates perspective. Waveform bandwidth = 143 GHz . Range resolution = 1 mm . Ramp duration = 170 ms . Range precision < 1 mm . Range to target = 2 m . Wavelength = 780 nm . Transmitted power = 1 W . No ramp averaging. No median filtering. 3 × aliasing enabled.

Equations (56)

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

f if τ t 0 = B t r ,
f if = 2 B ( R R 0 ) c t r ,
R = c t r f if 2 B + R 0 .
Δ R = c t r Δ f if 2 B ,
t d = t r ( τ t 0 ) .
Δ f if = 1 / t r ,
Δ R = c 2 B .
σ τ 1 / β 2 N avg CNR ,
β = ( 2 π ) 2 f 2 | U ( f ) | 2 d f | U ( f ) | 2 d f ,
σ R 6 2 π Δ R N avg CNR .
f r = γ N pix ,
f max = γ 2 N pix .
R max R 0 = c t r 2 B f max .
N pix RSI Δ R = γ t r 2 ,
f d = 2 v λ = 2 v f c c ,
Δ v = λ 2 t r = c 2 t r f c ,
Δ f d = 1 / t r .
| v | λ B a / 2 ,
x ( t ) = cos ( 2 π f 1 t ) + cos ( 2 π f 2 t ) ,
x ( m T ) = cos ( 2 π f 1 m T ) + cos ( 2 π f 2 m T ) ,
x 2 ( m T ) = 1 + 1 2 cos ( 4 π f 1 m T ) + 1 2 cos ( 4 π f 2 m T ) + cos [ 2 π ( f 2 + f 1 ) m T ] + cos [ 2 π ( f 2 f 1 ) m T ] .
x ( m T ) = cos ( 2 π f 1 m T ) + cos [ 2 π ( 1 2 T + f a ) m T ] = cos ( 2 π f 1 m T ) + cos [ 2 π ( 1 2 T f a ) m T ] .
x 2 ( m T ) = 1 + 1 2 cos ( 4 π f 1 m T ) + 1 2 cos [ 4 π ( 1 2 T f a ) m T ] + cos [ 2 π ( 1 2 T f a f 1 ) m T ] + cos [ 2 π ( Δ f ) m T ] ,
Δ f = 1 2 T + f a f 1
| v | λ 4 t i .
U x a ( 1 ) ( x , y , t ) = 2 z m I x a ( 1 ) ( x , y , t ) exp [ 2 π i ( f 1 t + B 2 t r t 2 ) ] ,
u x a ( 1 ) ( x , y , t ) = 1 2 z m U x a ( 1 ) ( x , y , t ) = I x a ( 1 ) ( x , y , t ) exp [ 2 π i ( f 1 t + B 2 t r t 2 ) ] .
| u x a ( x , y , t ) | 2 = I x a ( x , y , t ) .
f ( t ) = 1 2 π d ϕ d t ,
f ( t ) = f 1 + B t r t ,
P x ( 1 ) ( t ) = aperture I x a ( 1 ) ( x , y , t ) d x d y .
u x a ( 2 ) ( x , y , t ) = I x a ( 2 ) ( x , y , t ) exp [ 2 π i f p t ] ,
P x ( 2 ) ( t ) = aperture I x a ( 2 ) ( x , y , t ) d x d y .
τ ( t ) = 2 c R ( t τ ( t ) 2 ) .
T = t τ ( t ) 2 .
u s d ( 1 ) ( x , y , t ) = I s d ( 1 ) ( x , y , t ) exp [ 2 π i ( f 1 [ t 2 R ( T ) c ] + B 2 t r [ t 2 R ( T ) c ] 2 ) ] ,
P s ( 1 ) ( t ) = pixel I s d ( 1 ) ( x , y , t ) d x d y
P s ( 2 ) ( t ) = pixel I s d ( 2 ) ( x , y , t ) d x d y .
u s d ( 2 ) ( x , y , t ) = I s d ( 2 ) ( x , y , t ) exp [ 2 π i f p ( t 2 R ( T ) c ) ] .
u o d ( 1 ) ( x , y , t ) = I o d ( 1 ) ( x , y , t ) exp [ 2 π i ( f 1 t + B 2 t r t 2 ) ] .
u o d ( 2 ) ( x , y , t ) = I o d ( 2 ) ( x , y , t ) exp [ 2 π i f p t ] .
I ( x , y , t ) = | u s d ( 1 ) ( x , y , t ) + u o d ( 1 ) ( x , y , t ) + u s d ( 2 ) ( x , y , t ) + u o d ( 2 ) ( x , y , t ) | 2 .
I ( x , y , t ) = I s d ( 1 ) ( x , y , t ) + I o d ( 1 ) ( x , y , t ) + I s d ( 2 ) ( x , y , t ) + I o d ( 2 ) ( x , y , t ) + 2 I o d ( 1 ) ( x , y , t ) I s d ( 1 ) ( x , y , t ) cos [ 2 π ( 2 f 1 R ( T ) c + 2 B R ( T ) t c t r 2 B R 2 ( T ) c 2 t r ) ] + 2 I o d ( 2 ) ( x , y , t ) I s d ( 2 ) ( x , y , t ) cos [ 2 π ( 2 f p R ( T ) c ) ] + 2 I o d ( 1 ) ( x , y , t ) I o d ( 2 ) ( x , y , t ) cos [ 2 π ( f p t f 1 t B t 2 2 t r ) ] + 2 I o d ( 1 ) ( x , y , t ) I s d ( 2 ) ( x , y , t ) cos [ 2 π ( 2 f p R ( T ) c + t ( f p f 1 ) B t 2 2 t r ) ] + 2 I s d ( 1 ) ( x , y , t ) I o d ( 2 ) ( x , y , t ) cos [ 2 π ( 2 f 1 R ( T ) c + t ( f p f 1 ) 2 B R 2 ( T ) c 2 t r + 2 B R ( T ) t c t r B t 2 2 t r ) ] + 2 I s d ( 1 ) ( x , y , t ) I s d ( 2 ) ( x , y , t ) cos [ 2 π ( 2 ( f 1 f p ) R ( T ) c + t ( f p f 1 ) 2 B R 2 ( T ) c 2 t r + 2 B R t c t r B t 2 2 t r ) ] .
I ( x , y , t ) I s d ( 1 ) ( x , y , t ) + I o d ( 1 ) ( x , y , t ) + I s d ( 2 ) ( x , y , t ) + I o d ( 2 ) ( x , y , t ) + 2 I o d ( 1 ) ( x , y , t ) I s d ( 1 ) ( x , y , t ) cos [ 2 π ( 2 f 1 R ( T ) c + 2 B R ( T ) t c t r 2 B R 2 ( T ) c 2 t r ) ] + 2 I o d ( 2 ) ( x , y , t ) I s d ( 2 ) ( x , y , t ) cos [ 2 π ( 2 f p R ( T ) c ) ] .
x ( t ) = G pixel R ( x , y ) I ( x , y , t ) d x d y ,
x ( t ) = G R [ P s ( 1 ) ( t ) + P o ( 1 ) ( t ) + P s ( 2 ) ( t ) + P o ( 2 ) ( t ) ] + 2 G R η 1 P o ( 1 ) ( t ) P s ( 1 ) ( t ) cos [ 2 π ( 2 f 1 R ( T ) c + 2 B R ( T ) t c t r + 2 B R 2 ( T ) c 2 t r ) ] + 2 G R η 2 P o ( 2 ) ( t ) P s ( 2 ) ( t ) cos [ 2 π ( 2 f p R ( T ) c ) ] ,
P 0 ( 1 ) ( t ) = pixel I o d ( 1 ) ( x , y , t ) d x d y
P 0 ( 2 ) ( t ) = pixel I o d ( 2 ) ( x , y , t ) d x d y .
x 2 ( t ) = y DC + 4 G 2 R 2 η 1 η 2 P o ( 1 ) P o ( 2 ) P s ( 1 ) P s ( 2 ) cos [ 2 π ( 2 ( f 1 f p ) R ( T ) c + 2 R ( T ) B t c t r 2 B R 2 ( T ) c 2 t r ) ] + 4 G 2 R 2 η 1 η 2 P o ( 1 ) P o ( 2 ) P s ( 1 ) P s ( 2 ) cos [ 2 π ( 2 ( f 1 + f p ) R ( T ) c + 2 R ( T ) B t c t r 2 B R 2 ( T ) c 2 t r ) ] + 4 G 2 R 2 P DC η 1 P o ( 1 ) P s ( 1 ) cos [ 2 π ( 2 f 1 R ( T ) c + 2 R ( T ) B t c t r 2 B R 2 ( T ) c 2 t r ) ] + 4 G 2 R 2 P DC η 1 P o ( 2 ) P s ( 2 ) cos [ 2 π ( 2 f p R ( T ) c ) ] + 2 G 2 R 2 P o ( 1 ) P s ( 1 ) η 1 cos [ 2 π ( 4 f 1 R ( T ) c + 4 R ( T ) B t c t r 4 B R 2 ( T ) c 2 t r ) ] + 2 G 2 R 2 P o ( 2 ) P s ( 2 ) η 2 cos [ 2 π ( 4 f p R ( T ) c ) ] ,
y DC = ( P DC ) 2 G 2 R 2 + 2 η 1 P o ( 1 ) P s ( 1 ) G 2 R 2 + 2 η 2 P o ( 2 ) P s ( 2 ) G 2 R 2
P DC = P s ( 1 ) + P o ( 1 ) + P s ( 2 ) + P o ( 2 )
θ ( t ) = 2 π ( 2 R ( T ) c ( f 1 f p + B t t r ) 2 B R 2 ( T ) c 2 t r ) .
T t R 1 c .
f ( t ) = 1 2 π d θ d t = 2 B R ( T ) c t r + 2 R ( T ) c [ f 1 f p + B t t r ] 4 B R ( T ) R ( T ) c 2 t r .
Δ v c 2 t r B .
1 420 Hz 2 3.6 = 1.32 ms ,

Metrics