Abstract

In order to improve speed and efficiency over traditional scanning methods, a Bayesian compressive sensing algorithm using adaptive spatial sampling is developed for single detector millimeter wave synthetic aperture imaging. The application of this algorithm is compared to random sampling to demonstrate that the adaptive algorithm converges faster for simple targets and generates more reliable reconstructions for complex targets.

© 2017 Optical Society of America

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References

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2015 (1)

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Top. Signal Process. 9(6), 964–976 (2015).
[Crossref]

2014 (4)

2013 (1)

2012 (3)

M. S. Heimbeck, D. L. Marks, D. Brady, and H. O. Everitt, “Terahertz interferometric synthetic aperture tomography for confocal imaging systems,” Opt. Lett. 37(8), 1316–1318 (2012).
[Crossref] [PubMed]

L. Xiao, K. Liu, D. Han, and J. Liu, “A compressed sensing approach for enhancing infrared imaging resolution,” Opt. Laser Technol. 44(8), 2354–2360 (2012).
[Crossref]

L. Yu and Y. Zhang, “Random step frequency CSAR imaging based on compressive sensing,” Prog. Electromagn. Res. C 32, 81–94 (2012).
[Crossref]

2010 (1)

J. H. G. Ender, “On compressive sensing applied to radar,” Signal Process. 90(5), 1402–1414 (2010).
[Crossref]

2008 (3)

S. Ji, Y. Xue, and L. Carin, “Bayesian Compressive Sensing,” IEEE Trans. Signal Process. 56(6), 2346–2356 (2008).
[Crossref]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

2007 (2)

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70(8), 1325–1379 (2007).
[Crossref]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

2006 (3)

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proced. Comput. Imaging, SPIE 6065, Computational Imaging IV, 606509 (2006).

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

D. L. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

2005 (1)

L. P. Song, C. Yu, and Q. H. Liu, “Through-wall imaging (TWI) by radar: 2-D tomographic results and analyses,” IEEE Trans. Geosci. Remote Sens. 43(12), 2793–2798 (2005).
[Crossref]

2004 (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

2001 (2)

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech. 49(9), 1581–1592 (2001).
[Crossref]

M. Tipping, “Sparse Bayesian Learning and the Relevance Vector Machine,” J. Mach. Learn. Res. 1, 211–214 (2001).

1998 (1)

G. Lauritsch and W. H. Haerer, “Theoretical framework for filtered back projection in tomosynthesis,” Proc. SPIE 3338, 1127–1137 (1998).

1990 (1)

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36(5), 961–1005 (1990).

Baraniuk, R.

R. Baraniuk and P. Steeghs, “Compressive radar imaging,” in IEEE National Radar Conference Proceedings, 128–133 (2007)

Baraniuk, R. G.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proced. Comput. Imaging, SPIE 6065, Computational Imaging IV, 606509 (2006).

Baron, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proced. Comput. Imaging, SPIE 6065, Computational Imaging IV, 606509 (2006).

Boppart, S. A.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Brady, D.

Brady, D. J.

Bryllert, T.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Candès, E. J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Carin, L.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Top. Signal Process. 9(6), 964–976 (2015).
[Crossref]

S. Ji, Y. Xue, and L. Carin, “Bayesian Compressive Sensing,” IEEE Trans. Signal Process. 56(6), 2346–2356 (2008).
[Crossref]

Carney, P. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Chan, W. L.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70(8), 1325–1379 (2007).
[Crossref]

Charan, K.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

Chattopadhyay, G.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Cooper, K. B.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Daubechies, I.

I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36(5), 961–1005 (1990).

Deibel, J.

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70(8), 1325–1379 (2007).
[Crossref]

Dengler, R. J.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Donoho, D. L. L.

D. L. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[Crossref]

Duarte, M. F.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proced. Comput. Imaging, SPIE 6065, Computational Imaging IV, 606509 (2006).

Ender, J. H. G.

J. H. G. Ender, “On compressive sensing applied to radar,” Signal Process. 90(5), 1402–1414 (2010).
[Crossref]

Everitt, H. O.

Furxhi, O.

Gill, J.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Greenberg, J.

Haerer, W. H.

G. Lauritsch and W. H. Haerer, “Theoretical framework for filtered back projection in tomosynthesis,” Proc. SPIE 3338, 1127–1137 (1998).

Hall, T. E.

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech. 49(9), 1581–1592 (2001).
[Crossref]

Han, D.

L. Xiao, K. Liu, D. Han, and J. Liu, “A compressed sensing approach for enhancing infrared imaging resolution,” Opt. Laser Technol. 44(8), 2354–2360 (2012).
[Crossref]

Heimbeck, M.

Heimbeck, M. S.

Ji, S.

S. Ji, Y. Xue, and L. Carin, “Bayesian Compressive Sensing,” IEEE Trans. Signal Process. 56(6), 2346–2356 (2008).
[Crossref]

Kelly, K. F.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proced. Comput. Imaging, SPIE 6065, Computational Imaging IV, 606509 (2006).

Krishnamurthy, K.

Laska, J. N.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proced. Comput. Imaging, SPIE 6065, Computational Imaging IV, 606509 (2006).

Lauritsch, G.

G. Lauritsch and W. H. Haerer, “Theoretical framework for filtered back projection in tomosynthesis,” Proc. SPIE 3338, 1127–1137 (1998).

Lee, C.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Liu, J.

L. Xiao, K. Liu, D. Han, and J. Liu, “A compressed sensing approach for enhancing infrared imaging resolution,” Opt. Laser Technol. 44(8), 2354–2360 (2012).
[Crossref]

Liu, K.

L. Xiao, K. Liu, D. Han, and J. Liu, “A compressed sensing approach for enhancing infrared imaging resolution,” Opt. Laser Technol. 44(8), 2354–2360 (2012).
[Crossref]

Liu, Q. H.

L. P. Song, C. Yu, and Q. H. Liu, “Through-wall imaging (TWI) by radar: 2-D tomographic results and analyses,” IEEE Trans. Geosci. Remote Sens. 43(12), 2793–2798 (2005).
[Crossref]

Llombart, N.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Llull, P.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Top. Signal Process. 9(6), 964–976 (2015).
[Crossref]

MacCabe, K. P.

Marks, D. L.

McMakin, D. L.

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech. 49(9), 1581–1592 (2001).
[Crossref]

Mehdi, I.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Mittleman, D. M.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

W. L. Chan, J. Deibel, and D. M. Mittleman, “Imaging with terahertz radiation,” Rep. Prog. Phys. 70(8), 1325–1379 (2007).
[Crossref]

Mrozack, A.

O’Sullivan, J. A.

Ralston, T. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Richard, J.

Romberg, J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Sarvotham, S.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proced. Comput. Imaging, SPIE 6065, Computational Imaging IV, 606509 (2006).

Schlecht, E.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Sheen, D. M.

D. M. Sheen, D. L. McMakin, and T. E. Hall, “Three-dimensional millimeter-wave imaging for concealed weapon detection,” IEEE Trans. Microw. Theory Tech. 49(9), 1581–1592 (2001).
[Crossref]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Siegel, P. H.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: From error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Skalare, A.

K. B. Cooper, R. J. Dengler, N. Llombart, T. Bryllert, G. Chattopadhyay, E. Schlecht, J. Gill, C. Lee, A. Skalare, I. Mehdi, and P. H. Siegel, “Penetrating 3-D imaging at 4- and 25-m range using a submillimeter-wave radar,” IEEE Trans. Microw. Theory Tech. 56(12), 2771–2778 (2008).
[Crossref]

Song, L. P.

L. P. Song, C. Yu, and Q. H. Liu, “Through-wall imaging (TWI) by radar: 2-D tomographic results and analyses,” IEEE Trans. Geosci. Remote Sens. 43(12), 2793–2798 (2005).
[Crossref]

Steeghs, P.

R. Baraniuk and P. Steeghs, “Compressive radar imaging,” in IEEE National Radar Conference Proceedings, 128–133 (2007)

Takhar, D.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93(12), 121105 (2008).
[Crossref]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proced. Comput. Imaging, SPIE 6065, Computational Imaging IV, 606509 (2006).

Tao, T.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[Crossref]

Tipping, M.

M. Tipping, “Sparse Bayesian Learning and the Relevance Vector Machine,” J. Mach. Learn. Res. 1, 211–214 (2001).

Tsai, T.

X. Yuan, T. Tsai, R. Zhu, P. Llull, D. Brady, and L. Carin, “Compressive hyperspectral imaging with side information,” IEEE J. Sel. Top. Signal Process. 9(6), 964–976 (2015).
[Crossref]

Wakin, M. B.

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Figures (7)

Fig. 1
Fig. 1 a) Imaging configuration for traditional ISAR imaging. b) Example of a measurement at a single angle in the frequency domain F( θ i ,ν) and the range data in the spatial domain obtained by Fourier transform. c) Complete range data of all angles f(θ,r) . d) Reconstruction of the reflectivity function ρ(x,y) by filtered back projection.
Fig. 2
Fig. 2 a) Visualization of the next measurement under a hypothetical traditional BCS methodology. Colored points indicate the points sampled in step k (orange) and k + 1 (green). b) Visualization of the next measurement for ISAR imaging. c) Vectorized form of h K+1 for ideal BCS, and d) h K+1 for ISAR imaging.
Fig. 3
Fig. 3 (a) A schematic drawing of the experiment. (b) Drawing of the target half-cylinder nautilus. (c) The range signal measured at 300 angles separated at 0.1 degrees per measurement. The fully-sampled reconstruction is treated as the ground truth for the simulated experiments. (d) The ISAR reconstruction via the filtered backpropagation method with all 3000 measurements.
Fig. 4
Fig. 4 (a) The reconstruction of the wavelet coefficient of the signal after 25 measurements. (b) The ISAR reconstruction via filtered back projection on the data with 25 measurements. (c) The average MSE comparison of adaptive spatial sampling and random spatial sampling for 60 simulated experiments. Error bars indicate the standard deviation of the MSE. (d) Histogram of how many measurements the adaptive and random sampling has taken to reach 15% MSE.
Fig. 5
Fig. 5 (a)-(d). Results from the adaptive algorithm after 4, 8, 12, 16 measurements respectively. Lines with different colors indicate the measurements made during each stage, i.e., white lines are the first 4 measurements; blue lines are the 5th-8th measurements. These figures demonstrate the process of adaptive selection.
Fig. 6
Fig. 6 Image of the target metal posts a) from the side and b) from above. c) The range signal measured at 300 angles separated at 0.1 degrees per measurement. d) The ISAR reconstruction via back propagation.
Fig. 7
Fig. 7 a) The reconstruction of the wavelet coefficient of the signal after 35 measurements. b) The ISAR reconstruction via filtered back projection on the data with 35 measurements. c) The average MSE comparison of adaptive spatial sampling comparing to random spatial sampling for 60 simulated experiments. Error bars indicate the standard deviation of the MSE. d) Histogram of how many measurements the adaptive and random sampling has taken to reach 15% MSE.

Equations (23)

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f(θ,r)= F(θ,ν) e j2πνr dν.
f(θ,r)= ρ(x,y)δ(xcosθ+ysinθr)dxdy .
ρ(x,y)= 0 π f'(θ,xcosθ+ysinθ)dθ .
g(θ,r)= h(θθ',rr')f(θ',r')dθ'dr' .
g Δ (θ,r)= n=1 N m=1 M h Δ (θmΔθ',rnΔr')f(mΔ θ ,nΔr') .
g=Hf.
ω ^ =arg min ω { gΦω 2 2 +λ ω 1 }.
g=Φω+ n m =Φ ω s +Φ ω e + n m . =Φ ω s +n
p(g| ω s , σ 2 )= (2π σ 2 ) K/2 exp( 1 2 σ 2 gΦ ω s 2 ).
p(ω|α)= i N N( ω i |0, α i 1 ) .
p(α|a,b)= i N Γ( α i |a,b ) .
p(ω|a,b)= i N 0 N( ω i |0, α i 1 )Γ( α i |a,b ) d α i .
p(n|a,b)= i K 0 N( n k |0, α 0 1 )Γ( α 0 |c,d ) d α i .
Σ= ( α 0 Φ T Φ+A) 1 μ= α 0 Σ Φ T g
(α, α 0 )=logp(g|α, α 0 ) =log p(g|ω, α 0 ) p(ω|α)dω. = 1 2 [Klog2π+log|C|]+ g T C 1 g
α i new = γ i μ i 2 α 0 new = K Σ i γ i ||g-Φμ| | 2 2 .
E(f)=Bμ Cov(f)=BΣ B T
S(f)= p(f)logp(f)df = 1 2 log|BΣ B T |+const. = 1 2 log|Σ|+const = 1 2 log|A+ α 0 Φ T Φ|+const
S new (f)=h(f) 1 2 log|A+ α 0 r K+1 T Σ r K+1 |.
r K+1 T Σ r K+1 = r K+1 T Cov(ω) r K+1 . =Var( g K+1 )
h k+1 = i j δ(θiΔθ,rjΔr) .
h ISAR k+1 = [ h k+1,1 , h k+1,2 ,..., h k+1,n ] T
h k+1,n =δ(θ θ k+1 ,rnΔr).

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