Abstract

Many infrared optical systems in wide-ranging applications such as surveillance and security frequently require large fields of view (FOVs). Often this necessitates a focal plane array (FPA) with a large number of pixels, which, in general, is very expensive. In a previous paper, we proposed a method for increasing the FOV without increasing the pixel resolution of the FPA by superimposing multiple sub-images within a static scene and disambiguating the observed data to reconstruct the original scene. This technique, in effect, allows each sub-image of the scene to share a single FPA, thereby increasing the FOV without compromising resolution. In this paper, we demonstrate the increase of FOVs in a realistic setting by physically generating a superimposed video from a single scene using an optical system employing a beamsplitter and a movable mirror. Without prior knowledge of the contents of the scene, we are able to disambiguate the two sub-images, successfully capturing both large-scale features and fine details in each sub-image. We improve upon our previous reconstruction approach by allowing each sub-image to have slowly changing components, carefully exploiting correlations between sequential video frames to achieve small mean errors and to reduce run times. We show the effectiveness of this improved approach by reconstructing the constituent images of a surveillance camera video.

© 2008 Optical Society of America

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2008 (2)

E. Be'ery and A. Yeredor, "Blind separation of superimposed shifted images using parameterized joint diagonalization," IEEE Trans. Image Process. 17, 340-353 (2008).
[CrossRef] [PubMed]

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, "Single-pixel imaging via compressive sampling, " IEEE Signal Process. Mag. 25, 83 - 91, March 2008.
[CrossRef]

2007 (1)

2006 (1)

J. Bobin, J.-L. Starck, J. Fadili, and Y. Moudden, "Morphological Diversity and Source Separation," IEEE Trans. Signal Process. 13, 409-412 (2006).
[CrossRef]

2005 (4)

P. D. O'Grady, B. A. Pearlmutter, and S. T. Rickard, "Survey of sparse and non-sparse methods in source separation," Int. J. Imag. Syst. Tech. 15, 18-33 (2005).
[CrossRef]

A. M. Bronstein, M. M. Bronstein, M. Zibulevsky, and Y. Y. Zeevi, "Sparse ICA for blind separation of transmitted and reflected images," Int. J. Imag. Syst. Tech. 15, 84-91 (2005).
[CrossRef]

S. D. Gunapala, S. V. Bandara, J. K. Liu, C. J. Hill, S. B. Rafol, J. M. Mumolo, J. T. Trinh, M. Z. Tidrow, and P. D. Le Van, "1024 x 1024 pixel mid-wavelength and long-wavelength infrared QWIP focal plane arrays for imaging applications," Semicond. Sci. Technol. 20, 473-480 (2005).
[CrossRef]

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
[CrossRef]

2003 (1)

S. C. Park, M. K. Park, and M. G. Kang, "Super-resolution image reconstruction: A technical overview," IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

1998 (3)

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

H. S. P. Wong, R. T. Chang, E. Crabbe, and P. D. Agnello, "CMOS active pixel image sensors fabricated using a 1.8-V, 0.25-mu m CMOS technology," IEEE Trans. Electron Devices 45, 889-894 (1998).
[CrossRef]

S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM J. Sci. Comput.  20, 33-61 (electronic) (1998).

1996 (2)

R. Tibshirani, "Regression shrinkage and selection via the lasso," J. Roy. Statist. Soc. Ser. B 58, 267-288 (1996).

Y. Hagiwara, "High-density and high-quality frame transfer CCD imager with very low smear, low dark current, and very high blue sensitivity," IEEE Trans. Electron Devices 43, 2122-2130 (1996).
[CrossRef]

1995 (1)

J. C. Gillett, T. M. Stadtmiller, and R. C. Hardie, "Aliasing reduction in staring infrared imagers utilizing subpixel techniques," Opt. Eng. 34, 3130-3137 (1995).
[CrossRef]

1991 (1)

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

Agnello, P. D.

H. S. P. Wong, R. T. Chang, E. Crabbe, and P. D. Agnello, "CMOS active pixel image sensors fabricated using a 1.8-V, 0.25-mu m CMOS technology," IEEE Trans. Electron Devices 45, 889-894 (1998).
[CrossRef]

Annamalai, S.

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
[CrossRef]

Armstrong, E. E.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Bandara, S. V.

S. D. Gunapala, S. V. Bandara, J. K. Liu, C. J. Hill, S. B. Rafol, J. M. Mumolo, J. T. Trinh, M. Z. Tidrow, and P. D. Le Van, "1024 x 1024 pixel mid-wavelength and long-wavelength infrared QWIP focal plane arrays for imaging applications," Semicond. Sci. Technol. 20, 473-480 (2005).
[CrossRef]

Baraniuk, R.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, "Single-pixel imaging via compressive sampling, " IEEE Signal Process. Mag. 25, 83 - 91, March 2008.
[CrossRef]

Barnard, K. J.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Be'ery, E.

E. Be'ery and A. Yeredor, "Blind separation of superimposed shifted images using parameterized joint diagonalization," IEEE Trans. Image Process. 17, 340-353 (2008).
[CrossRef] [PubMed]

Bobin, J.

J. Bobin, J.-L. Starck, J. Fadili, and Y. Moudden, "Morphological Diversity and Source Separation," IEEE Trans. Signal Process. 13, 409-412 (2006).
[CrossRef]

Bognar, J. G.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Bronstein, A. M.

A. M. Bronstein, M. M. Bronstein, M. Zibulevsky, and Y. Y. Zeevi, "Sparse ICA for blind separation of transmitted and reflected images," Int. J. Imag. Syst. Tech. 15, 84-91 (2005).
[CrossRef]

Bronstein, M. M.

A. M. Bronstein, M. M. Bronstein, M. Zibulevsky, and Y. Y. Zeevi, "Sparse ICA for blind separation of transmitted and reflected images," Int. J. Imag. Syst. Tech. 15, 84-91 (2005).
[CrossRef]

Campbell, J.

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
[CrossRef]

Carothers, D.

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
[CrossRef]

Chang, R. T.

H. S. P. Wong, R. T. Chang, E. Crabbe, and P. D. Agnello, "CMOS active pixel image sensors fabricated using a 1.8-V, 0.25-mu m CMOS technology," IEEE Trans. Electron Devices 45, 889-894 (1998).
[CrossRef]

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM J. Sci. Comput.  20, 33-61 (electronic) (1998).

Crabbe, E.

H. S. P. Wong, R. T. Chang, E. Crabbe, and P. D. Agnello, "CMOS active pixel image sensors fabricated using a 1.8-V, 0.25-mu m CMOS technology," IEEE Trans. Electron Devices 45, 889-894 (1998).
[CrossRef]

Davenport, M.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, "Single-pixel imaging via compressive sampling, " IEEE Signal Process. Mag. 25, 83 - 91, March 2008.
[CrossRef]

Donoho, D. L.

S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM J. Sci. Comput.  20, 33-61 (electronic) (1998).

Dowd, P.

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
[CrossRef]

Duarte, M.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, "Single-pixel imaging via compressive sampling, " IEEE Signal Process. Mag. 25, 83 - 91, March 2008.
[CrossRef]

Fadili, J.

J. Bobin, J.-L. Starck, J. Fadili, and Y. Moudden, "Morphological Diversity and Source Separation," IEEE Trans. Signal Process. 13, 409-412 (2006).
[CrossRef]

Figueiredo, M. A. T.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, "Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems," IEEE J. Sel. Topics in Signal Processing: Special Issue on Convex Optimization Methods for Signal Processing (To appear).
[PubMed]

Forman, D.

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
[CrossRef]

Gillett, J. C.

J. C. Gillett, T. M. Stadtmiller, and R. C. Hardie, "Aliasing reduction in staring infrared imagers utilizing subpixel techniques," Opt. Eng. 34, 3130-3137 (1995).
[CrossRef]

Gray, A.

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
[CrossRef]

Gunapala, S. D.

S. D. Gunapala, S. V. Bandara, J. K. Liu, C. J. Hill, S. B. Rafol, J. M. Mumolo, J. T. Trinh, M. Z. Tidrow, and P. D. Le Van, "1024 x 1024 pixel mid-wavelength and long-wavelength infrared QWIP focal plane arrays for imaging applications," Semicond. Sci. Technol. 20, 473-480 (2005).
[CrossRef]

Hagiwara, Y.

Y. Hagiwara, "High-density and high-quality frame transfer CCD imager with very low smear, low dark current, and very high blue sensitivity," IEEE Trans. Electron Devices 43, 2122-2130 (1996).
[CrossRef]

Hardie, R. C.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

J. C. Gillett, T. M. Stadtmiller, and R. C. Hardie, "Aliasing reduction in staring infrared imagers utilizing subpixel techniques," Opt. Eng. 34, 3130-3137 (1995).
[CrossRef]

Hicks, R. A.

Hill, C. J.

S. D. Gunapala, S. V. Bandara, J. K. Liu, C. J. Hill, S. B. Rafol, J. M. Mumolo, J. T. Trinh, M. Z. Tidrow, and P. D. Le Van, "1024 x 1024 pixel mid-wavelength and long-wavelength infrared QWIP focal plane arrays for imaging applications," Semicond. Sci. Technol. 20, 473-480 (2005).
[CrossRef]

Irani, M.

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

Kang, M. G.

S. C. Park, M. K. Park, and M. G. Kang, "Super-resolution image reconstruction: A technical overview," IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Kelly, K.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, "Single-pixel imaging via compressive sampling, " IEEE Signal Process. Mag. 25, 83 - 91, March 2008.
[CrossRef]

Krishna, S.

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
[CrossRef]

Kurzweg, T. P.

Laska, J.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, "Single-pixel imaging via compressive sampling, " IEEE Signal Process. Mag. 25, 83 - 91, March 2008.
[CrossRef]

Le Van, P. D.

S. D. Gunapala, S. V. Bandara, J. K. Liu, C. J. Hill, S. B. Rafol, J. M. Mumolo, J. T. Trinh, M. Z. Tidrow, and P. D. Le Van, "1024 x 1024 pixel mid-wavelength and long-wavelength infrared QWIP focal plane arrays for imaging applications," Semicond. Sci. Technol. 20, 473-480 (2005).
[CrossRef]

Liu, J. K.

S. D. Gunapala, S. V. Bandara, J. K. Liu, C. J. Hill, S. B. Rafol, J. M. Mumolo, J. T. Trinh, M. Z. Tidrow, and P. D. Le Van, "1024 x 1024 pixel mid-wavelength and long-wavelength infrared QWIP focal plane arrays for imaging applications," Semicond. Sci. Technol. 20, 473-480 (2005).
[CrossRef]

Liu, M. G.

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
[CrossRef]

Moudden, Y.

J. Bobin, J.-L. Starck, J. Fadili, and Y. Moudden, "Morphological Diversity and Source Separation," IEEE Trans. Signal Process. 13, 409-412 (2006).
[CrossRef]

Mumolo, J. M.

S. D. Gunapala, S. V. Bandara, J. K. Liu, C. J. Hill, S. B. Rafol, J. M. Mumolo, J. T. Trinh, M. Z. Tidrow, and P. D. Le Van, "1024 x 1024 pixel mid-wavelength and long-wavelength infrared QWIP focal plane arrays for imaging applications," Semicond. Sci. Technol. 20, 473-480 (2005).
[CrossRef]

Nasis, V. T.

Nowak, R. D.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, "Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems," IEEE J. Sel. Topics in Signal Processing: Special Issue on Convex Optimization Methods for Signal Processing (To appear).
[PubMed]

O'Grady, P. D.

P. D. O'Grady, B. A. Pearlmutter, and S. T. Rickard, "Survey of sparse and non-sparse methods in source separation," Int. J. Imag. Syst. Tech. 15, 18-33 (2005).
[CrossRef]

Park, M. K.

S. C. Park, M. K. Park, and M. G. Kang, "Super-resolution image reconstruction: A technical overview," IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Park, S. C.

S. C. Park, M. K. Park, and M. G. Kang, "Super-resolution image reconstruction: A technical overview," IEEE Signal Process. Mag. 20, 21-36 (2003).
[CrossRef]

Pearlmutter, B. A.

P. D. O'Grady, B. A. Pearlmutter, and S. T. Rickard, "Survey of sparse and non-sparse methods in source separation," Int. J. Imag. Syst. Tech. 15, 18-33 (2005).
[CrossRef]

Peleg, S.

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

Rafol, S. B.

S. D. Gunapala, S. V. Bandara, J. K. Liu, C. J. Hill, S. B. Rafol, J. M. Mumolo, J. T. Trinh, M. Z. Tidrow, and P. D. Le Van, "1024 x 1024 pixel mid-wavelength and long-wavelength infrared QWIP focal plane arrays for imaging applications," Semicond. Sci. Technol. 20, 473-480 (2005).
[CrossRef]

Rickard, S. T.

P. D. O'Grady, B. A. Pearlmutter, and S. T. Rickard, "Survey of sparse and non-sparse methods in source separation," Int. J. Imag. Syst. Tech. 15, 18-33 (2005).
[CrossRef]

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM J. Sci. Comput.  20, 33-61 (electronic) (1998).

Stadtmiller, T. M.

J. C. Gillett, T. M. Stadtmiller, and R. C. Hardie, "Aliasing reduction in staring infrared imagers utilizing subpixel techniques," Opt. Eng. 34, 3130-3137 (1995).
[CrossRef]

Starck, J.-L.

J. Bobin, J.-L. Starck, J. Fadili, and Y. Moudden, "Morphological Diversity and Source Separation," IEEE Trans. Signal Process. 13, 409-412 (2006).
[CrossRef]

Sun, K.

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
[CrossRef]

Sun, T.

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Takhar, D.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly, and R. Baraniuk, "Single-pixel imaging via compressive sampling, " IEEE Signal Process. Mag. 25, 83 - 91, March 2008.
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S. D. Gunapala, S. V. Bandara, J. K. Liu, C. J. Hill, S. B. Rafol, J. M. Mumolo, J. T. Trinh, M. Z. Tidrow, and P. D. Le Van, "1024 x 1024 pixel mid-wavelength and long-wavelength infrared QWIP focal plane arrays for imaging applications," Semicond. Sci. Technol. 20, 473-480 (2005).
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Appl. Phys. Lett. (1)

S. Krishna, D. Forman, S. Annamalai, P. Dowd, P. Varangis, T. Tumolillo, A. Gray, J. Zilko, K. Sun, M. G. Liu, J. Campbell, and D. Carothers, "Demonstration of a 320x256 two-color focal plane array using InAs/InGaAs quantum dots in well detectors," Appl. Phys. Lett. 86, 193,501 (2005).
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E. Be'ery and A. Yeredor, "Blind separation of superimposed shifted images using parameterized joint diagonalization," IEEE Trans. Image Process. 17, 340-353 (2008).
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R. Tibshirani, "Regression shrinkage and selection via the lasso," J. Roy. Statist. Soc. Ser. B 58, 267-288 (1996).

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R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
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R. F. Marcia and R. M. Willett, "Compressive coded aperture video reconstruction," Accepted to "Proc. Sixteenth European Signal Processing Conference (EUSIPCO 2008)".

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Figures (4)

Fig. 1.
Fig. 1.

(a) Basic concept of superimposition and disambiguation. (b) Proposed camera architecture for superimposing two sub-images from the top view. The scene is split into two halves, x (1) t and x (2) t . The optical field from the left half propagates directly through the beamsplitter to hit the FPA in the camera. The optical field from the right half hits a movable mirror before propagating to the beamsplitter and being reflected to the FPA in the camera.

Fig. 2.
Fig. 2.

(a) Original “Duke Earth Day” scene used in the experiment. The box with a solid red border represents the extent of x (1), which is stationary during the superimposition process. As the mirror moves in a circular motion in the x-z plane shown in Fig. 1(b), the blue box with a solid border, which represents the moving boundary of x (2) at object plane, oscillates between the left and right turning points, represented by the blue boxes with dashed and dotted borders, respectively. (b) Superimposed image (left panel) and re-constructed scene (right panel) when the moving boundary is near the mid-point of the oscillation in the superimposed video. (c) Superimposed image (left panel) and reconstructed scene (right panel) when the moving boundary is near the left turning point, where sub-images are not completely disambiguated: the man in the hat and the banner (circled in yellow) partly appear in the left half of the disambiguated image.

Fig. 3.
Fig. 3.

(a) Original surveillance video with two moving components (circled in yellow), (b) the observed superimposed sub-images of the left and right half of the scene, and (c) the reconstruction using the 8-Frame Method allowing 20 seconds of GPSR iterations.

Fig. 4.
Fig. 4.

MSE values for the 90 frames allowing 5 seconds (a) and 20 seconds (b) to solve the optimization problems for the different n-Frame Methods for the surveillance video.

Equations (5)

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

z t = A t x t + ε t ,
z t = [ I S t ] [ x t ( 1 ) x t ( 2 ) ] + ε t = S ˜ t W ˜ θ t + ε t ,
θ ̂ t = arg min θ t z t S ˜ t W ˜ θ t 2 2 + τ θ t 1 .
θ ̂ t [ 2 ] [ θ ̂ t Δ θ ̂ t ] = arg min θ t , Δ θ t [ z t z t + 1 ] [ S ˜ t 0 0 S ˜ t + 1 ] [ W ˜ 0 W ˜ W ˜ ] [ θ t Δ θ t ] 2 2 + τ [ θ t Δ θ t ] 1 ,
θ ̂ t [ 4 ] = arg min θ ̄ t z ̄ t S ̄ t W ̄ θ ̄ t 2 2 + τ θ t 1 + ρ j = 0 2 Δ θ t + j 1 ,

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