Abstract

We report a compressive imaging method based on active illumination, which reconstructs a 3D scene at a frame rate beyond the acquisition speed limit of the camera. We have built an imaging prototype that projects temporally varying illumination pattern and demonstrated a joint reconstruction algorithm that iteratively retrieves both the range and high-temporal-frequency information from the 2D low-frame rate measurement. The reflectance and depth-map videos have been reconstructed at 1000 frames per second (fps) from the measurement captured at 200 fps. The range resolution is in agreement with the resolution calculated from the triangulation methods based on the same system geometry. We expect such an imaging method could become a simple solution to a wide range of applications, including industrial metrology, 3D printing, and vehicle navigations.

© 2016 Optical Society of America

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References

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  1. P. Treleaven and J. Wells, “3D body scanning and healthcare applications,” Computer 40(7), 28–34 (2007).
    [Crossref]
  2. A. Corns and R. Shaw, “High resolution 3-dimensional documentation of archaeological monuments & landscapes using airborne lidar,” J. Cult. Herit. 10, e72 (2009).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  5. M. Hebert, “Active and passive range sensing for robotics,” in Proc. 2000 ICRA. Millenn. Conf. IEEE Int. Conf. Robot. Autom. Symp. Proc. 1, 102–110 (2000).
    [Crossref]
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    [Crossref]
  7. F. Blais, “Review of 20 years of range sensor development,” J. Electron. Imaging 13(1), 231 (2004).
    [Crossref]
  8. P. Llull, X. Yuan, L. Carin, and D. J. Brady, “Image translation for single-shot focal tomography,” Optica 2(9), 822 (2015).
    [Crossref]
  9. S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express 16(5), 3484–3489 (2008).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  12. P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21(9), 10526–10545 (2013).
    [Crossref] [PubMed]
  13. X. Yuan, P. Llull, X. Liao, J. Yang, G. Sapiro, D. J. Brady, and L. Carin, “Low-cost compressive sensing for color video and depth,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2014), pp. 3318–3325.
    [Crossref]
  14. X. Yuan and S. Pang, “Structured illumination temporal compressive microscopy,” Biomed. Opt. Express 7(3), 746–758 (2016).
    [Crossref] [PubMed]
  15. T.-H. Tsai, P. Llull, X. Yuan, L. Carin, and D. J. Brady, “Spectral-temporal compressive imaging,” Opt. Lett. 40(17), 4054–4057 (2015).
    [Crossref] [PubMed]
  16. X. Yuan, “Generalized alternating projection based total variation minimization for compressive sensing,” in Proceedings of the IEEE Conference on Image Processing (2016), pp. 2539.
    [Crossref]
  17. J. Yang, X. Yuan, X. Liao, P. Llull, D. J. Brady, G. Sapiro, and L. Carin, “Video compressive sensing using Gaussian mixture models,” IEEE Trans. Image Process. 23(11), 4863–4878 (2014).
    [Crossref] [PubMed]
  18. X. Yuan, V. Rao, S. Han, and L. Carin, “Hierarchical infinite divisibility for multiscale shrinkage,” IEEE Trans. Signal Process. 62(17), 4363–4374 (2014).
    [Crossref]

2016 (1)

2015 (2)

2014 (2)

J. Yang, X. Yuan, X. Liao, P. Llull, D. J. Brady, G. Sapiro, and L. Carin, “Video compressive sensing using Gaussian mixture models,” IEEE Trans. Image Process. 23(11), 4863–4878 (2014).
[Crossref] [PubMed]

X. Yuan, V. Rao, S. Han, and L. Carin, “Hierarchical infinite divisibility for multiscale shrinkage,” IEEE Trans. Signal Process. 62(17), 4363–4374 (2014).
[Crossref]

2013 (1)

2010 (1)

F. Rengier, A. Mehndiratta, H. von Tengg-Kobligk, C. M. Zechmann, R. Unterhinninghofen, H. U. Kauczor, and F. L. Giesel, “3D printing based on imaging data: review of medical applications,” Int. J. CARS 5(4), 335–341 (2010).
[Crossref] [PubMed]

2009 (1)

A. Corns and R. Shaw, “High resolution 3-dimensional documentation of archaeological monuments & landscapes using airborne lidar,” J. Cult. Herit. 10, e72 (2009).
[Crossref]

2008 (1)

2007 (1)

P. Treleaven and J. Wells, “3D body scanning and healthcare applications,” Computer 40(7), 28–34 (2007).
[Crossref]

2006 (1)

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

2004 (2)

F. Blais, “Review of 20 years of range sensor development,” J. Electron. Imaging 13(1), 231 (2004).
[Crossref]

W. S. Wijesoma, K. R. S. Kodagoda, and A. P. Balasuriya, “Road-boundary detection and tracking using ladarladar sensing,” IEEE Trans. Robot. Autom. 20(3), 456–464 (2004).
[Crossref]

1988 (1)

P. J. Besl, “Active, optical range imaging sensors,” Mach. Vis. Appl. 1(2), 127–152 (1988).
[Crossref]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Balasuriya, A. P.

W. S. Wijesoma, K. R. S. Kodagoda, and A. P. Balasuriya, “Road-boundary detection and tracking using ladarladar sensing,” IEEE Trans. Robot. Autom. 20(3), 456–464 (2004).
[Crossref]

Besl, P. J.

P. J. Besl, “Active, optical range imaging sensors,” Mach. Vis. Appl. 1(2), 127–152 (1988).
[Crossref]

Blais, F.

F. Blais, “Review of 20 years of range sensor development,” J. Electron. Imaging 13(1), 231 (2004).
[Crossref]

Brady, D. J.

T.-H. Tsai, P. Llull, X. Yuan, L. Carin, and D. J. Brady, “Spectral-temporal compressive imaging,” Opt. Lett. 40(17), 4054–4057 (2015).
[Crossref] [PubMed]

P. Llull, X. Yuan, L. Carin, and D. J. Brady, “Image translation for single-shot focal tomography,” Optica 2(9), 822 (2015).
[Crossref]

J. Yang, X. Yuan, X. Liao, P. Llull, D. J. Brady, G. Sapiro, and L. Carin, “Video compressive sensing using Gaussian mixture models,” IEEE Trans. Image Process. 23(11), 4863–4878 (2014).
[Crossref] [PubMed]

P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21(9), 10526–10545 (2013).
[Crossref] [PubMed]

X. Yuan, P. Llull, X. Liao, J. Yang, G. Sapiro, D. J. Brady, and L. Carin, “Low-cost compressive sensing for color video and depth,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2014), pp. 3318–3325.
[Crossref]

Carin, L.

T.-H. Tsai, P. Llull, X. Yuan, L. Carin, and D. J. Brady, “Spectral-temporal compressive imaging,” Opt. Lett. 40(17), 4054–4057 (2015).
[Crossref] [PubMed]

P. Llull, X. Yuan, L. Carin, and D. J. Brady, “Image translation for single-shot focal tomography,” Optica 2(9), 822 (2015).
[Crossref]

X. Yuan, V. Rao, S. Han, and L. Carin, “Hierarchical infinite divisibility for multiscale shrinkage,” IEEE Trans. Signal Process. 62(17), 4363–4374 (2014).
[Crossref]

J. Yang, X. Yuan, X. Liao, P. Llull, D. J. Brady, G. Sapiro, and L. Carin, “Video compressive sensing using Gaussian mixture models,” IEEE Trans. Image Process. 23(11), 4863–4878 (2014).
[Crossref] [PubMed]

P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21(9), 10526–10545 (2013).
[Crossref] [PubMed]

X. Yuan, P. Llull, X. Liao, J. Yang, G. Sapiro, D. J. Brady, and L. Carin, “Low-cost compressive sensing for color video and depth,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2014), pp. 3318–3325.
[Crossref]

Corns, A.

A. Corns and R. Shaw, “High resolution 3-dimensional documentation of archaeological monuments & landscapes using airborne lidar,” J. Cult. Herit. 10, e72 (2009).
[Crossref]

Donoho, D. L.

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

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Giesel, F. L.

F. Rengier, A. Mehndiratta, H. von Tengg-Kobligk, C. M. Zechmann, R. Unterhinninghofen, H. U. Kauczor, and F. L. Giesel, “3D printing based on imaging data: review of medical applications,” Int. J. CARS 5(4), 335–341 (2010).
[Crossref] [PubMed]

Han, S.

X. Yuan, V. Rao, S. Han, and L. Carin, “Hierarchical infinite divisibility for multiscale shrinkage,” IEEE Trans. Signal Process. 62(17), 4363–4374 (2014).
[Crossref]

Kauczor, H. U.

F. Rengier, A. Mehndiratta, H. von Tengg-Kobligk, C. M. Zechmann, R. Unterhinninghofen, H. U. Kauczor, and F. L. Giesel, “3D printing based on imaging data: review of medical applications,” Int. J. CARS 5(4), 335–341 (2010).
[Crossref] [PubMed]

Kittle, D.

Kodagoda, K. R. S.

W. S. Wijesoma, K. R. S. Kodagoda, and A. P. Balasuriya, “Road-boundary detection and tracking using ladarladar sensing,” IEEE Trans. Robot. Autom. 20(3), 456–464 (2004).
[Crossref]

Liao, X.

J. Yang, X. Yuan, X. Liao, P. Llull, D. J. Brady, G. Sapiro, and L. Carin, “Video compressive sensing using Gaussian mixture models,” IEEE Trans. Image Process. 23(11), 4863–4878 (2014).
[Crossref] [PubMed]

P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21(9), 10526–10545 (2013).
[Crossref] [PubMed]

X. Yuan, P. Llull, X. Liao, J. Yang, G. Sapiro, D. J. Brady, and L. Carin, “Low-cost compressive sensing for color video and depth,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2014), pp. 3318–3325.
[Crossref]

Llull, P.

T.-H. Tsai, P. Llull, X. Yuan, L. Carin, and D. J. Brady, “Spectral-temporal compressive imaging,” Opt. Lett. 40(17), 4054–4057 (2015).
[Crossref] [PubMed]

P. Llull, X. Yuan, L. Carin, and D. J. Brady, “Image translation for single-shot focal tomography,” Optica 2(9), 822 (2015).
[Crossref]

J. Yang, X. Yuan, X. Liao, P. Llull, D. J. Brady, G. Sapiro, and L. Carin, “Video compressive sensing using Gaussian mixture models,” IEEE Trans. Image Process. 23(11), 4863–4878 (2014).
[Crossref] [PubMed]

P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21(9), 10526–10545 (2013).
[Crossref] [PubMed]

X. Yuan, P. Llull, X. Liao, J. Yang, G. Sapiro, D. J. Brady, and L. Carin, “Low-cost compressive sensing for color video and depth,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2014), pp. 3318–3325.
[Crossref]

Mehndiratta, A.

F. Rengier, A. Mehndiratta, H. von Tengg-Kobligk, C. M. Zechmann, R. Unterhinninghofen, H. U. Kauczor, and F. L. Giesel, “3D printing based on imaging data: review of medical applications,” Int. J. CARS 5(4), 335–341 (2010).
[Crossref] [PubMed]

Pang, S.

Pavani, S. R. P.

Piestun, R.

Rao, V.

X. Yuan, V. Rao, S. Han, and L. Carin, “Hierarchical infinite divisibility for multiscale shrinkage,” IEEE Trans. Signal Process. 62(17), 4363–4374 (2014).
[Crossref]

Rengier, F.

F. Rengier, A. Mehndiratta, H. von Tengg-Kobligk, C. M. Zechmann, R. Unterhinninghofen, H. U. Kauczor, and F. L. Giesel, “3D printing based on imaging data: review of medical applications,” Int. J. CARS 5(4), 335–341 (2010).
[Crossref] [PubMed]

Sapiro, G.

J. Yang, X. Yuan, X. Liao, P. Llull, D. J. Brady, G. Sapiro, and L. Carin, “Video compressive sensing using Gaussian mixture models,” IEEE Trans. Image Process. 23(11), 4863–4878 (2014).
[Crossref] [PubMed]

P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21(9), 10526–10545 (2013).
[Crossref] [PubMed]

X. Yuan, P. Llull, X. Liao, J. Yang, G. Sapiro, D. J. Brady, and L. Carin, “Low-cost compressive sensing for color video and depth,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2014), pp. 3318–3325.
[Crossref]

Shaw, R.

A. Corns and R. Shaw, “High resolution 3-dimensional documentation of archaeological monuments & landscapes using airborne lidar,” J. Cult. Herit. 10, e72 (2009).
[Crossref]

Treleaven, P.

P. Treleaven and J. Wells, “3D body scanning and healthcare applications,” Computer 40(7), 28–34 (2007).
[Crossref]

Tsai, T.-H.

Unterhinninghofen, R.

F. Rengier, A. Mehndiratta, H. von Tengg-Kobligk, C. M. Zechmann, R. Unterhinninghofen, H. U. Kauczor, and F. L. Giesel, “3D printing based on imaging data: review of medical applications,” Int. J. CARS 5(4), 335–341 (2010).
[Crossref] [PubMed]

von Tengg-Kobligk, H.

F. Rengier, A. Mehndiratta, H. von Tengg-Kobligk, C. M. Zechmann, R. Unterhinninghofen, H. U. Kauczor, and F. L. Giesel, “3D printing based on imaging data: review of medical applications,” Int. J. CARS 5(4), 335–341 (2010).
[Crossref] [PubMed]

Wells, J.

P. Treleaven and J. Wells, “3D body scanning and healthcare applications,” Computer 40(7), 28–34 (2007).
[Crossref]

Wijesoma, W. S.

W. S. Wijesoma, K. R. S. Kodagoda, and A. P. Balasuriya, “Road-boundary detection and tracking using ladarladar sensing,” IEEE Trans. Robot. Autom. 20(3), 456–464 (2004).
[Crossref]

Yang, J.

J. Yang, X. Yuan, X. Liao, P. Llull, D. J. Brady, G. Sapiro, and L. Carin, “Video compressive sensing using Gaussian mixture models,” IEEE Trans. Image Process. 23(11), 4863–4878 (2014).
[Crossref] [PubMed]

P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21(9), 10526–10545 (2013).
[Crossref] [PubMed]

X. Yuan, P. Llull, X. Liao, J. Yang, G. Sapiro, D. J. Brady, and L. Carin, “Low-cost compressive sensing for color video and depth,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2014), pp. 3318–3325.
[Crossref]

Yuan, X.

X. Yuan and S. Pang, “Structured illumination temporal compressive microscopy,” Biomed. Opt. Express 7(3), 746–758 (2016).
[Crossref] [PubMed]

P. Llull, X. Yuan, L. Carin, and D. J. Brady, “Image translation for single-shot focal tomography,” Optica 2(9), 822 (2015).
[Crossref]

T.-H. Tsai, P. Llull, X. Yuan, L. Carin, and D. J. Brady, “Spectral-temporal compressive imaging,” Opt. Lett. 40(17), 4054–4057 (2015).
[Crossref] [PubMed]

J. Yang, X. Yuan, X. Liao, P. Llull, D. J. Brady, G. Sapiro, and L. Carin, “Video compressive sensing using Gaussian mixture models,” IEEE Trans. Image Process. 23(11), 4863–4878 (2014).
[Crossref] [PubMed]

X. Yuan, V. Rao, S. Han, and L. Carin, “Hierarchical infinite divisibility for multiscale shrinkage,” IEEE Trans. Signal Process. 62(17), 4363–4374 (2014).
[Crossref]

P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21(9), 10526–10545 (2013).
[Crossref] [PubMed]

X. Yuan, P. Llull, X. Liao, J. Yang, G. Sapiro, D. J. Brady, and L. Carin, “Low-cost compressive sensing for color video and depth,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2014), pp. 3318–3325.
[Crossref]

X. Yuan, “Generalized alternating projection based total variation minimization for compressive sensing,” in Proceedings of the IEEE Conference on Image Processing (2016), pp. 2539.
[Crossref]

Zechmann, C. M.

F. Rengier, A. Mehndiratta, H. von Tengg-Kobligk, C. M. Zechmann, R. Unterhinninghofen, H. U. Kauczor, and F. L. Giesel, “3D printing based on imaging data: review of medical applications,” Int. J. CARS 5(4), 335–341 (2010).
[Crossref] [PubMed]

Biomed. Opt. Express (1)

Computer (1)

P. Treleaven and J. Wells, “3D body scanning and healthcare applications,” Computer 40(7), 28–34 (2007).
[Crossref]

IEEE Trans. Image Process. (1)

J. Yang, X. Yuan, X. Liao, P. Llull, D. J. Brady, G. Sapiro, and L. Carin, “Video compressive sensing using Gaussian mixture models,” IEEE Trans. Image Process. 23(11), 4863–4878 (2014).
[Crossref] [PubMed]

IEEE Trans. Inf. Theory (1)

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

IEEE Trans. Robot. Autom. (1)

W. S. Wijesoma, K. R. S. Kodagoda, and A. P. Balasuriya, “Road-boundary detection and tracking using ladarladar sensing,” IEEE Trans. Robot. Autom. 20(3), 456–464 (2004).
[Crossref]

IEEE Trans. Signal Process. (1)

X. Yuan, V. Rao, S. Han, and L. Carin, “Hierarchical infinite divisibility for multiscale shrinkage,” IEEE Trans. Signal Process. 62(17), 4363–4374 (2014).
[Crossref]

Int. J. CARS (1)

F. Rengier, A. Mehndiratta, H. von Tengg-Kobligk, C. M. Zechmann, R. Unterhinninghofen, H. U. Kauczor, and F. L. Giesel, “3D printing based on imaging data: review of medical applications,” Int. J. CARS 5(4), 335–341 (2010).
[Crossref] [PubMed]

J. Cult. Herit. (1)

A. Corns and R. Shaw, “High resolution 3-dimensional documentation of archaeological monuments & landscapes using airborne lidar,” J. Cult. Herit. 10, e72 (2009).
[Crossref]

J. Electron. Imaging (1)

F. Blais, “Review of 20 years of range sensor development,” J. Electron. Imaging 13(1), 231 (2004).
[Crossref]

Mach. Vis. Appl. (1)

P. J. Besl, “Active, optical range imaging sensors,” Mach. Vis. Appl. 1(2), 127–152 (1988).
[Crossref]

Nature (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Optica (1)

Other (3)

M. Hebert, “Active and passive range sensing for robotics,” in Proc. 2000 ICRA. Millenn. Conf. IEEE Int. Conf. Robot. Autom. Symp. Proc. 1, 102–110 (2000).
[Crossref]

X. Yuan, “Generalized alternating projection based total variation minimization for compressive sensing,” in Proceedings of the IEEE Conference on Image Processing (2016), pp. 2539.
[Crossref]

X. Yuan, P. Llull, X. Liao, J. Yang, G. Sapiro, D. J. Brady, and L. Carin, “Low-cost compressive sensing for color video and depth,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2014), pp. 3318–3325.
[Crossref]

Supplementary Material (1)

NameDescription
» Visualization 1: AVI (1993 KB)      Reconstructed high-speed 3D video

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Figures (6)

Fig. 1
Fig. 1

The forward model of our system. The projector projects high-speed masks onto the scene. The scene is modulated by the projected masks h*(x, y, t), the variants of the original mask at different range locations. The camera integrates the modulated high-speed frames into one measurement.

Fig. 2
Fig. 2

System geometry. The lateral offset of projector and camera is d = 135 mm. The distance between the projector and the camera along the z axis is l0 ≈1.3 m. θ denotes the projection angle. The focal lengths of the camera and the projector are both 50 mm.

Fig. 3
Fig. 3

Calibrated masks at different ranges. Same patterns of the 5th mask projected at different ranges are highlighted with the red boxes. The shift of the red boxes indicates one modulation on the range. The scales of the zoom in patterns is the other modulation on the range.

Fig. 4
Fig. 4

The measurement and the reconstructed high-speed frames of reflectance are shown in the top row. The corresponding depth maps are shown in the middle row. Bottom row shows the ground truth of the scene.

Fig. 5
Fig. 5

The top row presents a measurement at 200 fps and the reconstructed frames of reflectance at 1000 fps. The object is a disk rotating at 30 rps. The red box indicates the position of the black tape in the first frame. The bottom row shows a photo of the stationary object and the reconstructed depth map. A video of 1000-fps frames has been reconstructed from 200-fps measurement (see Visualization 1).

Fig. 6
Fig. 6

The simulated range resolution. The red curve is the normalized negative correlation of the masks. The blue curve is the normalized reconstruction error using the masks which are Δz away from the correct one. The inserts indicate the first frame of the reconstructed results using the masks with Δz = 0, 1 mm, 2 mm.

Tables (1)

Tables Icon

Table 1 Reconstruction algorithm

Equations (11)

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g(x',y')= t=0 T f( z+ l 0 f c x , z+ l 0 f c y ,t ) h ( z+ l 0 f c x , z+ l 0 f c y ,t )dt ,
h z (x,y,t)= h 0 ( f p z x+ f p z d, f p z y,t ),
g m,n = k=1 N T h m,n,k f m,n,k ,
g= H * [ f 1 f N T ].
( f ^ , H ^ * )= arg f, H * min g- H * f 2 2 +τR(f).
f ^ i block = arg f block min g block - H i f block 2 2 +τR( f block ),
z ^ block = z j ,j= arg i min g block - H i f ^ block i 2 2 ,i=1,, N z ,
j= arg i min g block - H i f ^ block i 2 2 +τR( f ^ block i ),i=1,, N z .
z= f c d l 0 x x + f c cotθ .
g( x )= t(θ)f(x,z(x))δ( z(x) f c d l 0 x x + f c cotθ ) dθ,
g H i f ^ i 2 2 = g H i f ^ i ,g H i f ^ i = g H i f ^ i ,g = g 2 2 H i f ^ i , H 1 f ^ 1 ,

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