Abstract

We exploit recent advances in active high-resolution imaging through scattering media with ballistic photons. We derive the fundamental limits on the accuracy of the estimated parameters of a mathematical model that describes such an imaging scenario and compare the performance of ballistic and conventional imaging systems. This model is later used to derive optimal single-pixel statistical tests for detecting objects hidden in turbid media. To improve the detection rate of the aforementioned single-pixel detectors, we develop a multiscale algorithm based on the generalized likelihood ratio test framework. Moreover, considering the effect of diffraction, we derive a lower bound on the achievable spatial resolution of the proposed imaging systems. Furthermore, we present the first experimental ballistic scanner that directly takes advantage of novel adaptive sampling and reconstruction techniques.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. C. Dunsby and P. M. W. French, "Techniques for depth-resolved imaging through turbid media including coherence-gated imaging," J. Phys. D 36, 207-227 (2003).
    [CrossRef]
  2. K. Yoo and R. R. Alfano, "Time-resolved coherent and incoherent components of forward light scattering in random media," Opt. Lett. 15, 320-322 (1990).
    [CrossRef] [PubMed]
  3. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography --principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
    [CrossRef]
  4. M. E. Zevallos, S. K. Gayen, M. Alrubaiee, and R. R. Alfano, "Time-gated backscattered ballistic light imaging of objects in turbid water," Appl. Phys. Lett. 86, 0111151-0111153 (2005).
    [CrossRef]
  5. M. Paciaroni and M. Linne, "Single-shot, two-dimensional ballistic imaging through scattering media," Appl. Opt. 43, 5100-5109 (2004).
    [CrossRef] [PubMed]
  6. D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory," Appl. Opt. 36, 4587-4599 (1997).
    [CrossRef] [PubMed]
  7. I. Delfino, M. Lepore, and P. L. Indovina, "Experimental tests of different solutions to the diffusion equation for optical characterization of scattering media by time-resolved transmittance," Appl. Opt. 38, 4228-4236 (1999).
    [CrossRef]
  8. W. Cai, S. K. Gayen, M. Xu, M. Zevallos, M. Alrubaiee, M. Lax, and R. R. Alfano, "Optical tomographic image reconstruction from ultrafast time-sliced transmission measurements," Appl. Opt. 38, 4237-4246 (1999).
    [CrossRef]
  9. B. B. Das, F. Liu, and R. R. Alfano, "Time-resolved fluorescence and photon migration studies in biomedical and model random media," Rep. Prog. Phys. 60, 227-292 (1997).
    [CrossRef]
  10. A. Gibson, J. Hebden, and S. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
    [CrossRef] [PubMed]
  11. H. Lischke, T. J. Loeffler, and A. Fischlin, "Aggregation of individual trees and patches in forest succession models: capturing variability with height structured, random, spatial distributions," Theor. Popul. Biol. 54, 213-236 (1998).
    [CrossRef]
  12. S. V. Aert, D. V. Dyck, and A. J. den Dekker, "Resolution of coherent and incoherent imaging systems reconsidered--classical criteria and a statistical alternative," Opt. Express 14, 3830-3839 (2006).
    [CrossRef] [PubMed]
  13. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993), Vol. 1.
  14. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).
  15. G. W. Sutton, "Fog hole boring with pulsed high-energy lasers--an exact solution including scattering and absorption," Appl. Opt. 17, 3424-3430 (1978).
    [CrossRef] [PubMed]
  16. S. M. Kay, Fundamentals of Statistical Signal Processing Detection Theory (Prentice-Hall, 1998), Vol. 2.
  17. M. Shahram and P. Milanfar, "Imaging below the diffraction limit: a statistical analysis," IEEE Trans. Image Processing 13, 677-689 (2004).
    [CrossRef]
  18. A. Sommerfeld, Optics Lectures on Theortical Physics (Academic, 1954), Vol. 4.
  19. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).
  20. G. B. Parrent, Jr. and B. J. Thompson, "On the Fraunhofer (far field) diffraction patterns of opaque and transparent objects with coherent background," J. Mod. Opt. 11, 183-193 (1964).
    [CrossRef]
  21. D. N. Nikogosyan, "Beta barium borate (BBO)," Appl. Phys. A 52, 359-368 (1991).
    [CrossRef]
  22. M. Ghotbi and M. Ebrahim-Zadeh, "Optical second harmonic generation properties of BiB3O6," Opt. Express 12, 6002-6019 (2004).
    [CrossRef] [PubMed]
  23. E. Candès, J. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. 59, 1207-1223 (2006).
    [CrossRef]
  24. J. Haupt and R. Nowak, "Signal reconstruction from noisy random projections," IEEE Trans. Inf. Theory 52, 4036-4048 (2006).
    [CrossRef]
  25. R. Castro, J. Haupt, and R. Nowak, "Compressed sensing vs. active learning," in 2006 International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 820-823.
  26. D. Donoho, M. Elad, and V. Temlyakov, "Stable recovery of sparse overcomplete representations in the presence of noise," IEEE Trans. Inf. Theory 52, 6-18 (2006).
    [CrossRef]
  27. M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, "An architecture for compressive imaging," in 2006 International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.
    [CrossRef]
  28. R. Castro, R. Willett, and R. Nowak, "Faster rates in regression via active learning," in 2005 Advances in Neural Information Processing Systems 18 (MIT Press, 2005), pp. 179-186.
  29. S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multi-frame super-resolution," IEEE Trans. Image Processing 13, 1327-1344 (2004).
    [CrossRef]
  30. B. Eriksson and R. Nowak, "Maximum likelihood methods for time-resolved imaging through turbid media," in 2006 International Conference on Image Processing (IEEE, 2006), pp. 641-644.
    [CrossRef]
  31. S. Fantini, S. Walker, M. Franceschini, M. Kaschke, P. Schlag, and K. Moesta, "Assessment of the size, position, and optical properties of breast tumors in vivoo by noninvasive optical methods," Appl. Opt. 37, 1982-1989 (1998).
    [CrossRef]
  32. F. Anscombe, "The transformation of Poisson, binomial and negative-binomial data," Biometrika 35, 246-254 (1948).
  33. R. Miller, Simultaneous Statistical Inference (Springer, 1991).
  34. A. Gandjbakhche, G. Weiss, R. Bonner, and R. Nossal, "Photon path-length distributions for transmission through optically turbid slabs," Phys. Rev. E 48, 810-818 (1993).
    [CrossRef]
  35. A. Gandjbakhche, R. Nossal, and R. Bonner, "Resolution limits for optical transillumination of abnormalities deeply embedded in tissues," Med. Phys. 21, 185-191 (1994).
    [CrossRef] [PubMed]
  36. V. Chernomordik, R. Nossal, and A. Gandjbakhche, "Point spread functions of photons in time-resolved transillumination experiments using simple scaling arguments," Med. Phys. 23, 1857-1861 (1996).
    [CrossRef] [PubMed]

2006 (4)

E. Candès, J. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

J. Haupt and R. Nowak, "Signal reconstruction from noisy random projections," IEEE Trans. Inf. Theory 52, 4036-4048 (2006).
[CrossRef]

D. Donoho, M. Elad, and V. Temlyakov, "Stable recovery of sparse overcomplete representations in the presence of noise," IEEE Trans. Inf. Theory 52, 6-18 (2006).
[CrossRef]

S. V. Aert, D. V. Dyck, and A. J. den Dekker, "Resolution of coherent and incoherent imaging systems reconsidered--classical criteria and a statistical alternative," Opt. Express 14, 3830-3839 (2006).
[CrossRef] [PubMed]

2005 (2)

M. E. Zevallos, S. K. Gayen, M. Alrubaiee, and R. R. Alfano, "Time-gated backscattered ballistic light imaging of objects in turbid water," Appl. Phys. Lett. 86, 0111151-0111153 (2005).
[CrossRef]

A. Gibson, J. Hebden, and S. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

2004 (4)

M. Shahram and P. Milanfar, "Imaging below the diffraction limit: a statistical analysis," IEEE Trans. Image Processing 13, 677-689 (2004).
[CrossRef]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multi-frame super-resolution," IEEE Trans. Image Processing 13, 1327-1344 (2004).
[CrossRef]

M. Paciaroni and M. Linne, "Single-shot, two-dimensional ballistic imaging through scattering media," Appl. Opt. 43, 5100-5109 (2004).
[CrossRef] [PubMed]

M. Ghotbi and M. Ebrahim-Zadeh, "Optical second harmonic generation properties of BiB3O6," Opt. Express 12, 6002-6019 (2004).
[CrossRef] [PubMed]

2003 (2)

C. Dunsby and P. M. W. French, "Techniques for depth-resolved imaging through turbid media including coherence-gated imaging," J. Phys. D 36, 207-227 (2003).
[CrossRef]

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography --principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

1999 (2)

1998 (2)

S. Fantini, S. Walker, M. Franceschini, M. Kaschke, P. Schlag, and K. Moesta, "Assessment of the size, position, and optical properties of breast tumors in vivoo by noninvasive optical methods," Appl. Opt. 37, 1982-1989 (1998).
[CrossRef]

H. Lischke, T. J. Loeffler, and A. Fischlin, "Aggregation of individual trees and patches in forest succession models: capturing variability with height structured, random, spatial distributions," Theor. Popul. Biol. 54, 213-236 (1998).
[CrossRef]

1997 (2)

B. B. Das, F. Liu, and R. R. Alfano, "Time-resolved fluorescence and photon migration studies in biomedical and model random media," Rep. Prog. Phys. 60, 227-292 (1997).
[CrossRef]

D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory," Appl. Opt. 36, 4587-4599 (1997).
[CrossRef] [PubMed]

1996 (1)

V. Chernomordik, R. Nossal, and A. Gandjbakhche, "Point spread functions of photons in time-resolved transillumination experiments using simple scaling arguments," Med. Phys. 23, 1857-1861 (1996).
[CrossRef] [PubMed]

1994 (1)

A. Gandjbakhche, R. Nossal, and R. Bonner, "Resolution limits for optical transillumination of abnormalities deeply embedded in tissues," Med. Phys. 21, 185-191 (1994).
[CrossRef] [PubMed]

1993 (1)

A. Gandjbakhche, G. Weiss, R. Bonner, and R. Nossal, "Photon path-length distributions for transmission through optically turbid slabs," Phys. Rev. E 48, 810-818 (1993).
[CrossRef]

1991 (1)

D. N. Nikogosyan, "Beta barium borate (BBO)," Appl. Phys. A 52, 359-368 (1991).
[CrossRef]

1990 (1)

1978 (1)

1964 (1)

G. B. Parrent, Jr. and B. J. Thompson, "On the Fraunhofer (far field) diffraction patterns of opaque and transparent objects with coherent background," J. Mod. Opt. 11, 183-193 (1964).
[CrossRef]

1948 (1)

F. Anscombe, "The transformation of Poisson, binomial and negative-binomial data," Biometrika 35, 246-254 (1948).

Appl. Opt. (6)

Appl. Phys. A (1)

D. N. Nikogosyan, "Beta barium borate (BBO)," Appl. Phys. A 52, 359-368 (1991).
[CrossRef]

Appl. Phys. Lett. (1)

M. E. Zevallos, S. K. Gayen, M. Alrubaiee, and R. R. Alfano, "Time-gated backscattered ballistic light imaging of objects in turbid water," Appl. Phys. Lett. 86, 0111151-0111153 (2005).
[CrossRef]

Biometrika (1)

F. Anscombe, "The transformation of Poisson, binomial and negative-binomial data," Biometrika 35, 246-254 (1948).

Commun. Pure Appl. Math. (1)

E. Candès, J. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccurate measurements," Commun. Pure Appl. Math. 59, 1207-1223 (2006).
[CrossRef]

IEEE Trans. Image Processing (2)

M. Shahram and P. Milanfar, "Imaging below the diffraction limit: a statistical analysis," IEEE Trans. Image Processing 13, 677-689 (2004).
[CrossRef]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multi-frame super-resolution," IEEE Trans. Image Processing 13, 1327-1344 (2004).
[CrossRef]

IEEE Trans. Inf. Theory (2)

D. Donoho, M. Elad, and V. Temlyakov, "Stable recovery of sparse overcomplete representations in the presence of noise," IEEE Trans. Inf. Theory 52, 6-18 (2006).
[CrossRef]

J. Haupt and R. Nowak, "Signal reconstruction from noisy random projections," IEEE Trans. Inf. Theory 52, 4036-4048 (2006).
[CrossRef]

J. Mod. Opt. (1)

G. B. Parrent, Jr. and B. J. Thompson, "On the Fraunhofer (far field) diffraction patterns of opaque and transparent objects with coherent background," J. Mod. Opt. 11, 183-193 (1964).
[CrossRef]

J. Phys. D (1)

C. Dunsby and P. M. W. French, "Techniques for depth-resolved imaging through turbid media including coherence-gated imaging," J. Phys. D 36, 207-227 (2003).
[CrossRef]

Med. Phys. (2)

A. Gandjbakhche, R. Nossal, and R. Bonner, "Resolution limits for optical transillumination of abnormalities deeply embedded in tissues," Med. Phys. 21, 185-191 (1994).
[CrossRef] [PubMed]

V. Chernomordik, R. Nossal, and A. Gandjbakhche, "Point spread functions of photons in time-resolved transillumination experiments using simple scaling arguments," Med. Phys. 23, 1857-1861 (1996).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (1)

A. Gibson, J. Hebden, and S. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Phys. Rev. E (1)

A. Gandjbakhche, G. Weiss, R. Bonner, and R. Nossal, "Photon path-length distributions for transmission through optically turbid slabs," Phys. Rev. E 48, 810-818 (1993).
[CrossRef]

Rep. Prog. Phys. (2)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography --principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

B. B. Das, F. Liu, and R. R. Alfano, "Time-resolved fluorescence and photon migration studies in biomedical and model random media," Rep. Prog. Phys. 60, 227-292 (1997).
[CrossRef]

Theor. Popul. Biol. (1)

H. Lischke, T. J. Loeffler, and A. Fischlin, "Aggregation of individual trees and patches in forest succession models: capturing variability with height structured, random, spatial distributions," Theor. Popul. Biol. 54, 213-236 (1998).
[CrossRef]

Other (10)

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993), Vol. 1.

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).

S. M. Kay, Fundamentals of Statistical Signal Processing Detection Theory (Prentice-Hall, 1998), Vol. 2.

R. Castro, J. Haupt, and R. Nowak, "Compressed sensing vs. active learning," in 2006 International Conference on Acoustics, Speech and Signal Processing (IEEE, 2006), pp. 820-823.

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, "An architecture for compressive imaging," in 2006 International Conference on Image Processing (IEEE, 2006), pp. 1273-1276.
[CrossRef]

R. Castro, R. Willett, and R. Nowak, "Faster rates in regression via active learning," in 2005 Advances in Neural Information Processing Systems 18 (MIT Press, 2005), pp. 179-186.

A. Sommerfeld, Optics Lectures on Theortical Physics (Academic, 1954), Vol. 4.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

B. Eriksson and R. Nowak, "Maximum likelihood methods for time-resolved imaging through turbid media," in 2006 International Conference on Image Processing (IEEE, 2006), pp. 641-644.
[CrossRef]

R. Miller, Simultaneous Statistical Inference (Springer, 1991).

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 (17)

Fig. 1
Fig. 1

(Color online) Optimal distance for calibrating the medium extinction factor for heavy fog, light fog, and haze. (a) Experimental setup, where the detector is moved to different locations [marked by lighter (red) dots] inside the turbid medium. (b) Bias of estimation that is calculated over 60,000 Monte Carlo simulations. (c) Summation of squared bias and variance (solid curves) that is dominated by the variance component and perfectly fits the predicted results from CRB formulation (dotted curves) in short distances.

Fig. 2
Fig. 2

(Color online) Experimental setup for characterizing the optical properties of the medium ( μ t ) and a semitransparent object ( μ t i n c ) .

Fig. 3
Fig. 3

(Color online) Comparison of the bias and variance from 5000 Monte Carlo simulations (numerically experimented) and the estimated CRB values of the medium and the semitransparent target's optical properties versus distance. The bias, variance, and CRB of the medium extinction factor are compared in (a), (b), and (c). The bias, variance, and CRB of the target's extinction factor are compared in (d), (e), and (f).

Fig. 4
Fig. 4

(Color online) (a) ROC plots at different distances for detecting opaque objects in heavy fog ( μ t = 12.5 1 m 1 ) and X e = 20 . (b) By fixing the P F A at different values, the detection rate ( P D ) is plotted versus the distance.

Fig. 5
Fig. 5

(Color online) (a) ROC plots for detecting transilluminative objects at 300   m distance in heavy fog ( μ t = 12.5 1   m ) and X e = 20 . (b) By fixing the P F A at different values, detection rate ( P D ) is plotted versus the object's transparency ( μ t i n c ) .

Fig. 6
Fig. 6

(Color online) Illustrative example showing the outline of the proposed multiscale GLRT algorithm. The check-marked second scale gives the highest confidence value for the central pixel.

Fig. 7
Fig. 7

(a) Ideal deterministic and noise-free image of four objects of different sizes and shapes. (b) Corresponding image as a Poissonian noise-free stochastic signal, with X s = 1 .

Fig. 8
Fig. 8

(Color online) Application of the proposed multiscale GLRT technique for improving the detection rate. (a) Result of adding Poisson noise ( X e = 20 ) to Fig. 7(a). (b) Result of the single-pixel detection. (c) Result of the proposed multiscale detection technique. (d) Image that corresponds to the selected scales for the image shown in (c). (e) Corresponding confidence values. (f) Misclassification probability in different scales.

Fig. 9
Fig. 9

(Color online) Application of the proposed multiscale GLRT technique for improving the detection rate. (a) Result of adding Poisson noise ( X e = 40 ) to Fig. 7(a). (b) Result of the single-pixel detection. (c) Result of the proposed multiscale detection technique. (d) Image that corresponds to the selected scales for the image shown in (c). (e) Corresponding confidence values. (f) Misclassification probability in different scales.

Fig. 10
Fig. 10

(Color online) Application of the proposed multiscale GLRT technique for enhancing the detection rate. ROC plots for the proposed multiscale detection scenario in the imaging scenarios of Figs. 8 and 9 (with 25 Monte Carlo experiments) are shown in (a) and (b), respectively. The numerical labels “1, 4, … , 23” correspond to the scale at which detection tests are performed, and the plot labeled “Final” represents the performance of the proposed multiscale (fused) technique.

Fig. 11
Fig. 11

(Color online) Shadow of an opaque object illuminated by a homogeneous widespread light beam. The dashed (red) curve represents the intensity of the measured light ignoring diffraction. The solid (green) curve represents the diffraction-induced PSF.

Fig. 12
Fig. 12

(Color online) (a) 1-D slice of the diffraction pattern of a circular object of radius 3 mm at 100 m distance and 800 nm wavelength. (b) 1-D slice of the imaging scenario, where z is the distance between the opaque circular object (radius ρ) and the detector. d is the distance between the laser and the detector. The pass length L 1 L 2 when z is very long.

Fig. 13
Fig. 13

(Color online) Detection rate versus the (unknown) opaque circular target's radius and the distance between the laser and the detector considering the diffraction limit with P F A = 0.1 and X e = 20 in heavy fog ( μ t = 12.5 1 m 1 ) .

Fig. 14
Fig. 14

(Color online) Laser setup at the Ballistic Imaging Laboratory at the University of California, Santa Cruz.

Fig. 15
Fig. 15

(Color online) Comparison of diffusive and ballistic imaging. (a), (b) Two diffusive (no time gating) scans. (c), (d) Two corresponding ballistic (time-gated) scans through five solid ground glass diffusers.

Fig. 16
Fig. 16

Comparison of passive and active imaging. (a) Result of scanning the turbid medium on a regular 128 × 128 grid (16,384 dense passive sampling). (b) Result of scanning the turbid medium on a regular 32 × 32 grid (1024 sparse passive sampling) followed by interpolation via bicubic interpolation to reconstruct the image on a 256 × 256 grid. (c) Result of scanning the turbid medium on an irregular grid (984 sparse adaptive sampling) followed by interpolation via adaptive interpolation to reconstruct the image on a 256 × 256 grid. (d) Distribution of the 984 irregular samples.

Fig. 17
Fig. 17

Simulation experiments with FOV = 50   m × 50   m , d i n c = 0.3   m , and μ t = 12.5 1 m 1 . (a)–(c) Ballistic, Bonferroni, and conventional observations at d = 350   m , respectively. (d)–(f) Ballistic, Bonferroni, and conventional observations at d = 400   m , respectively. (g)–(i) Ballistic, Bonferroni, and conventional observations at the critical distane d = d c r i t i c a l = 417   m , respectively.

Equations (43)

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

e pulse = 60 × 10 3 × 1   s 1000 = 6 × 10 5   J ,
e = h f = h c Λ = 2.4830 × 10 1 9   J ,
I 0 = 6 × 10 5 2.4830 × 1 0 1 9 = 2.4164 × 10 14   photons .
I b = I 0   exp ( d L ) .
I = I 0   exp ( μ t d ) + X e = X s + X e ,
f ( y ¯ X s ¯ + X e ¯ ) = k = 1 N e ( X e k + X s k ) ( X e k + X s k ) y k y k ! ,
  log [ f ( y ¯ | X s ¯ + X e ¯ ) ] μ t = 0 μ ^ t = ln ( N I 0 N X e k = 1 N y k ) d .
I i , j = E 2   log [ f ( y ¯ | X s ¯ + X e ¯ ) ] Θ i Θ j = k = 1 N ( 1 X e + X s k X s k Θ i X s k Θ j ) ,
I ( μ t ) = N I 0 2 d 2 e 2 μ t d X e + N I 0 e μ t d .
  log [ f ( y ¯ | I 0 e μ t d + X e ) ] μ t = ? I ( μ t ) ( μ ^ t μ t ) .
  log [ f ( y ¯ | I 0 e μ t d + X e ) ] μ t = I ( μ t ) [ I 0 d e μ t d ( X e + I 0 e μ t d ) I 0 d e μ t d N k = 1 N y k ] ,
V a r ( μ t ) > X e + I 0 e μ t d N I 0 2 d 2 ( e μ t d ) 2 .
μ ^ t = ln ( N 1 I 0 N 1 X e k = 1 N 1 y k ) d ,
μ ^ t i n c = d   ln ( N 2 I 0 N 2 X e k = 1 N 2 y k ) ( d d i n c ) ln ( N 1 I 0 N 1 X e k = 1 N 1 y k ) d d i n c ,
V a r ( μ t ) > X e + I 0 e μ t d N 1 I 0 2 d 2 ( e μ t d ) 2 ,
V a r ( μ t i n c ) > Ω e 2 μ t d 2 μ t d i n c + 2 μ t i n c d i n c I 0 2 d 2 N 1 d i n c 2 N 2
Ω = ( N 1 d 2 X e + N 1 d 2 e μ t d + μ t d i n c μ t i n c d i n c I 0 + N 2 e 2 μ t d i n c 2 μ t i n c d i n c d 2 X e + N 2 e μ t d + 2 μ t d i n c 2 μ t i n c d i n c d 2 I 0 2 N 2 e 2 μ t d i n c 2 μ t i n c d i n c d d i n c X e 2 N 2 e μ t d + 2 μ t d i n c 2 μ t i n c d i n c d d i n c I 0 + N 2 e 2 μ t d i n c 2 μ t i n c d i n c d i n c 2 X e + N 2 e μ t d + 2 μ t d i n c 2 μ t i n c d i n c d i n c 2 I 0 ) .
0 : f ( y ¯ | X e ) = k = 1 N e ( X e ) ( X e ) y k y k ! ,
1 : f ( y ¯ | X s + X e ) = k = 1 N e ( X e + X s ) ( X e + X s ) y k y k ! ,
log   k = 1 N [ e ( X e + X s ) ( X e + X s ) y k y k ! e X e ( X e ) y k y k ! ] 0 1     γ k = 1 N y k 0 1    log ( γ ) + N X s log ( X e + X s X e ) = γ .
P F A = P { k = 1 N y k > γ | 0 } = k = γ + 1 e N X e ( N X e ) k k ! = 1 k = 0 γ e N X e ( N X e ) k k ! = 1 CDF ( N X e ) ,
P D = P { k = 1 N y k > γ | 1 } = k = γ + 1 e N X e N X S ( N X e + N X S ) k k ! = 1 k = 0 γ e N X e N X S ( N X e + N X S ) k k ! = 1 CDF ( N X e + N X S ) ,
X s i n c = I 0 e μ t i n c d i n c μ t ( d d i n c ) ,
k = 1 N y k 0 1 log ( γ ) + N ( X s i n c X s ) log ( X e + X s i n c X e + X s ) = γ .
P F A = P { k = 1 N y k > γ | 0 } = k = γ + 1 e N X e N X s ( N X e + N X s ) k k ! = 1 k = 0 γ e N X e N X s ( N X e + N X s ) k k ! = 1 CDF ( N X e + N X S ) ,
P D = P { k = 1 N y k > γ | 1 } = k = γ + 1 e N X e N X s i n c ( N X e + N X s i n c ) k k ! = 1 k = 0 γ e N X e N X s i n c ( N X e + N X s i n c ) k k ! = 1 CDF ( N X e + N X s i n c ) .
y m , l scale 0 1 log ( γ s c a l e ) + N s c a l e X s log ( X e + X s X e ) ,
Confidence m , l s c a l e = | y m , l s c a l e log ( γ s c a l e ) + N s c a l e X s log ( X e + X s X e ) | .
H ( r ) = 1 2 k ρ 2 z  sin   k r 2 2 z J 1 ( k ρ r z ) k ρ r z + k 2 ρ 4 z 2 [ J 1 ( k ρ r z ) k ρ r z ] 2 ,
I = I   exp ( μ t z ) H ( r ) ,
I ( k , 0 ) = I 0 e μ t d + X e .
I ( k , ρ ^ ) = I 0 e μ t d H ( r k ) + X e ,
ρ ^ = arg   max ρ k = 1 N e I ( k , ρ ) [ I ( k , ρ ) ] y k y k ! .
ϱ [ g ] = k = 1 N y k log { I ( k , ρ [ g ] ) } I ( k , ρ [ g ] ) .
k = 1 N y k log { I ( k , ρ [ g max ] ) I ( k , 0 ) } γ .
X ^ ¯ ( t ) = arg   min X ¯ ( t ) [ A ( X ¯ Z ^ ¯ ) 2 2 + ψ l , m = P P α | m | + | l | X ¯ S x l S y m X ¯ 1 ] ,
X P ( χ ) 2 X + 3 8 N ( 2 χ , 1 ) ,
0 : X N ( 2 N X e + N X s , 1 ) ,
1 : X N ( 2 N X e + N X s i n c , 1 ) .
0 : X N ( 2 N X e + N X s , 1 W ) ,
1 : X N ( 2 N X e + N X s i n c , 1 W ) .
W [ 1 2 Φ 1 ( 1 P F A K ) Φ 1 ( P F A K ) N X e + N X s N X e + N X s i n c ] 2 .
w c o n v ballistic conventional w b 0.94 ( Δ t c μ s ) 1 / 2 ballistic conventional FOV × W K .

Metrics