Abstract

This paper addresses the problem of reconstructing an image from 1-bit-quantized measurements, considering a simple but non-conventional optical acquisition model. Following a compressed-sensing design, a known pseudo-random phase-shifting mask is introduced at the aperture of the optical system. The associated reconstruction algorithm is tailored to this mask. Our results demonstrate the feasibility of the whole approach for reconstructing grayscale images.

© 2010 Optical Society of America

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References

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  1. A. Stern, Y. Rivenson, and B. Javidi, “Optically compressed image sensing using random aperture coding,” in “Proceedings of the SPIE - The International Society for Optical Engineering,” (2008), pp. 69750D–1–10.
  2. J. Romberg, “Sensing by random convolution,” in “2nd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing,” (2007), pp. 137–140.
  3. M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras,” IEEE Signal Process. Mag. 25,83–91 (2008).
    [CrossRef]
  4. R. F. Marcia and R. M. Willett, “Compressive coded aperture superresolution image reconstruction,” in “IEEE International Conference on Acoustic, Speech and Signal Processes,” (2008), pp. 833–836.
  5. F. Sebert, Y. M. Zou, and L. Ying, “Toeplitz block matrices in compressed sensing and their applications in imaging,” in “Proceedings of the 5th International Conference on Information Technology and Application in Biomedicine,” (2008), pp. 47–50.
  6. P. T. Boufounos and R. G. Baraniuk, “1-bit compressive sensing,” in “42nd Annual Conference on Information Sciences and Systems,” (2008), pp. 16–21.
  7. A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
    [CrossRef]
  8. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
    [CrossRef]
  9. W. U. Bajwa, J. D. Haupt, G. M. Raz, S. J. Wright, and R. D. Nowak, “Toeplitz-structured compressed sensing matrices,” in “IEEE Workshop on Statistical Signal Processing Proceedings,” (2007), pp. 294–298.
  10. H. Rauhut, “Circulant and toeplitz matrices in compressed sensing,” in “Proceedings of SPARS’09,” (2009).
  11. W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Tech. rep., CAAM, Rice University (2010).
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw Hill Higher Education, 1996), 2nd ed.
  13. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1959), 7th ed.
  14. M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
    [CrossRef]
  15. R. T. Rockafellar, Convex Analysis (Princeton Mathematical Series) (Princeton University Press, 1970).
  16. S. P. Lloyd, “Least squares quantization in PCM,” IEEE Trans. Inf. Theory 28, 129–137 (1982).
    [CrossRef]
  17. J. Max, “Quantizing for minimum distortion,” IRE Trans. Inf. Theory IT-6, 7–12 (1960).
    [CrossRef]
  18. L. Sbaiz, F. Yang, E. Charbon, S. Susstrunk, and M. Vetterli, “The gigavision camera,” in “2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (2009), pp. 1093–1096.

2009 (2)

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

L. Sbaiz, F. Yang, E. Charbon, S. Susstrunk, and M. Vetterli, “The gigavision camera,” in “2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (2009), pp. 1093–1096.

2008 (4)

M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras,” IEEE Signal Process. Mag. 25,83–91 (2008).
[CrossRef]

R. F. Marcia and R. M. Willett, “Compressive coded aperture superresolution image reconstruction,” in “IEEE International Conference on Acoustic, Speech and Signal Processes,” (2008), pp. 833–836.

F. Sebert, Y. M. Zou, and L. Ying, “Toeplitz block matrices in compressed sensing and their applications in imaging,” in “Proceedings of the 5th International Conference on Information Technology and Application in Biomedicine,” (2008), pp. 47–50.

P. T. Boufounos and R. G. Baraniuk, “1-bit compressive sensing,” in “42nd Annual Conference on Information Sciences and Systems,” (2008), pp. 16–21.

2007 (2)

W. U. Bajwa, J. D. Haupt, G. M. Raz, S. J. Wright, and R. D. Nowak, “Toeplitz-structured compressed sensing matrices,” in “IEEE Workshop on Statistical Signal Processing Proceedings,” (2007), pp. 294–298.

J. Romberg, “Sensing by random convolution,” in “2nd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing,” (2007), pp. 137–140.

1999 (1)

M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
[CrossRef]

1992 (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

1982 (1)

S. P. Lloyd, “Least squares quantization in PCM,” IEEE Trans. Inf. Theory 28, 129–137 (1982).
[CrossRef]

1960 (1)

J. Max, “Quantizing for minimum distortion,” IRE Trans. Inf. Theory IT-6, 7–12 (1960).
[CrossRef]

Bajwa, W. U.

W. U. Bajwa, J. D. Haupt, G. M. Raz, S. J. Wright, and R. D. Nowak, “Toeplitz-structured compressed sensing matrices,” in “IEEE Workshop on Statistical Signal Processing Proceedings,” (2007), pp. 294–298.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras,” IEEE Signal Process. Mag. 25,83–91 (2008).
[CrossRef]

P. T. Boufounos and R. G. Baraniuk, “1-bit compressive sensing,” in “42nd Annual Conference on Information Sciences and Systems,” (2008), pp. 16–21.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1959), 7th ed.

Boufounos, P. T.

P. T. Boufounos and R. G. Baraniuk, “1-bit compressive sensing,” in “42nd Annual Conference on Information Sciences and Systems,” (2008), pp. 16–21.

Bruckstein, A. M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Charbon, E.

L. Sbaiz, F. Yang, E. Charbon, S. Susstrunk, and M. Vetterli, “The gigavision camera,” in “2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (2009), pp. 1093–1096.

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras,” IEEE Signal Process. Mag. 25,83–91 (2008).
[CrossRef]

Donoho, D. L.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras,” IEEE Signal Process. Mag. 25,83–91 (2008).
[CrossRef]

Elad, M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill Higher Education, 1996), 2nd ed.

Haupt, J. D.

W. U. Bajwa, J. D. Haupt, G. M. Raz, S. J. Wright, and R. D. Nowak, “Toeplitz-structured compressed sensing matrices,” in “IEEE Workshop on Statistical Signal Processing Proceedings,” (2007), pp. 294–298.

Javidi, B.

A. Stern, Y. Rivenson, and B. Javidi, “Optically compressed image sensing using random aperture coding,” in “Proceedings of the SPIE - The International Society for Optical Engineering,” (2008), pp. 69750D–1–10.

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras,” IEEE Signal Process. Mag. 25,83–91 (2008).
[CrossRef]

Laska, J.

M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras,” IEEE Signal Process. Mag. 25,83–91 (2008).
[CrossRef]

Lloyd, S. P.

S. P. Lloyd, “Least squares quantization in PCM,” IEEE Trans. Inf. Theory 28, 129–137 (1982).
[CrossRef]

Marcia, R. F.

R. F. Marcia and R. M. Willett, “Compressive coded aperture superresolution image reconstruction,” in “IEEE International Conference on Acoustic, Speech and Signal Processes,” (2008), pp. 833–836.

Max, J.

J. Max, “Quantizing for minimum distortion,” IRE Trans. Inf. Theory IT-6, 7–12 (1960).
[CrossRef]

Morgan, S.

W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Tech. rep., CAAM, Rice University (2010).

Nowak, R. D.

W. U. Bajwa, J. D. Haupt, G. M. Raz, S. J. Wright, and R. D. Nowak, “Toeplitz-structured compressed sensing matrices,” in “IEEE Workshop on Statistical Signal Processing Proceedings,” (2007), pp. 294–298.

Osher, S.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Rauhut, H.

H. Rauhut, “Circulant and toeplitz matrices in compressed sensing,” in “Proceedings of SPARS’09,” (2009).

Raz, G. M.

W. U. Bajwa, J. D. Haupt, G. M. Raz, S. J. Wright, and R. D. Nowak, “Toeplitz-structured compressed sensing matrices,” in “IEEE Workshop on Statistical Signal Processing Proceedings,” (2007), pp. 294–298.

Rivenson, Y.

A. Stern, Y. Rivenson, and B. Javidi, “Optically compressed image sensing using random aperture coding,” in “Proceedings of the SPIE - The International Society for Optical Engineering,” (2008), pp. 69750D–1–10.

Rockafellar, R. T.

R. T. Rockafellar, Convex Analysis (Princeton Mathematical Series) (Princeton University Press, 1970).

Romberg, J.

J. Romberg, “Sensing by random convolution,” in “2nd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing,” (2007), pp. 137–140.

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Sbaiz, L.

L. Sbaiz, F. Yang, E. Charbon, S. Susstrunk, and M. Vetterli, “The gigavision camera,” in “2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (2009), pp. 1093–1096.

Sebert, F.

F. Sebert, Y. M. Zou, and L. Ying, “Toeplitz block matrices in compressed sensing and their applications in imaging,” in “Proceedings of the 5th International Conference on Information Technology and Application in Biomedicine,” (2008), pp. 47–50.

Stern, A.

A. Stern, Y. Rivenson, and B. Javidi, “Optically compressed image sensing using random aperture coding,” in “Proceedings of the SPIE - The International Society for Optical Engineering,” (2008), pp. 69750D–1–10.

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras,” IEEE Signal Process. Mag. 25,83–91 (2008).
[CrossRef]

Susstrunk, S.

L. Sbaiz, F. Yang, E. Charbon, S. Susstrunk, and M. Vetterli, “The gigavision camera,” in “2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (2009), pp. 1093–1096.

Takbar, D.

M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras,” IEEE Signal Process. Mag. 25,83–91 (2008).
[CrossRef]

Unser, M.

M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
[CrossRef]

Vetterli, M.

L. Sbaiz, F. Yang, E. Charbon, S. Susstrunk, and M. Vetterli, “The gigavision camera,” in “2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (2009), pp. 1093–1096.

Willett, R. M.

R. F. Marcia and R. M. Willett, “Compressive coded aperture superresolution image reconstruction,” in “IEEE International Conference on Acoustic, Speech and Signal Processes,” (2008), pp. 833–836.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1959), 7th ed.

Wright, S. J.

W. U. Bajwa, J. D. Haupt, G. M. Raz, S. J. Wright, and R. D. Nowak, “Toeplitz-structured compressed sensing matrices,” in “IEEE Workshop on Statistical Signal Processing Proceedings,” (2007), pp. 294–298.

Yang, F.

L. Sbaiz, F. Yang, E. Charbon, S. Susstrunk, and M. Vetterli, “The gigavision camera,” in “2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (2009), pp. 1093–1096.

Yang, J.

W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Tech. rep., CAAM, Rice University (2010).

Yin, W.

W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Tech. rep., CAAM, Rice University (2010).

Ying, L.

F. Sebert, Y. M. Zou, and L. Ying, “Toeplitz block matrices in compressed sensing and their applications in imaging,” in “Proceedings of the 5th International Conference on Information Technology and Application in Biomedicine,” (2008), pp. 47–50.

Zhang, Y.

W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Tech. rep., CAAM, Rice University (2010).

Zou, Y. M.

F. Sebert, Y. M. Zou, and L. Ying, “Toeplitz block matrices in compressed sensing and their applications in imaging,” in “Proceedings of the 5th International Conference on Information Technology and Application in Biomedicine,” (2008), pp. 47–50.

IEEE Signal Process. Mag. (2)

M. F. Duarte, M. A. Davenport, D. Takbar, J. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling: Building simpler, smaller, and less-expensive digital cameras,” IEEE Signal Process. Mag. 25,83–91 (2008).
[CrossRef]

M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22–38 (1999).
[CrossRef]

IEEE Trans. Inf. Theory (1)

S. P. Lloyd, “Least squares quantization in PCM,” IEEE Trans. Inf. Theory 28, 129–137 (1982).
[CrossRef]

IRE Trans. Inf. Theory (1)

J. Max, “Quantizing for minimum distortion,” IRE Trans. Inf. Theory IT-6, 7–12 (1960).
[CrossRef]

Physica D (1)

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

SIAM Rev. (1)

A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Rev. 51, 34–81 (2009).
[CrossRef]

Other (12)

L. Sbaiz, F. Yang, E. Charbon, S. Susstrunk, and M. Vetterli, “The gigavision camera,” in “2009 IEEE International Conference on Acoustics, Speech and Signal Processing,” (2009), pp. 1093–1096.

R. T. Rockafellar, Convex Analysis (Princeton Mathematical Series) (Princeton University Press, 1970).

A. Stern, Y. Rivenson, and B. Javidi, “Optically compressed image sensing using random aperture coding,” in “Proceedings of the SPIE - The International Society for Optical Engineering,” (2008), pp. 69750D–1–10.

J. Romberg, “Sensing by random convolution,” in “2nd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing,” (2007), pp. 137–140.

W. U. Bajwa, J. D. Haupt, G. M. Raz, S. J. Wright, and R. D. Nowak, “Toeplitz-structured compressed sensing matrices,” in “IEEE Workshop on Statistical Signal Processing Proceedings,” (2007), pp. 294–298.

H. Rauhut, “Circulant and toeplitz matrices in compressed sensing,” in “Proceedings of SPARS’09,” (2009).

W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Tech. rep., CAAM, Rice University (2010).

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill Higher Education, 1996), 2nd ed.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1959), 7th ed.

R. F. Marcia and R. M. Willett, “Compressive coded aperture superresolution image reconstruction,” in “IEEE International Conference on Acoustic, Speech and Signal Processes,” (2008), pp. 833–836.

F. Sebert, Y. M. Zou, and L. Ying, “Toeplitz block matrices in compressed sensing and their applications in imaging,” in “Proceedings of the 5th International Conference on Information Technology and Application in Biomedicine,” (2008), pp. 47–50.

P. T. Boufounos and R. G. Baraniuk, “1-bit compressive sensing,” in “42nd Annual Conference on Information Sciences and Systems,” (2008), pp. 16–21.

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Figures (6)

Fig. 1.
Fig. 1.

Given an object fo(x) and a pseudo-random-phase mask, the optical system produces an (intermediate) image fi(x). A 1-bit sensor array samples and binarizes the latter to the measurements γ[k]. The sampling process is uniform but non-ideal, meaning that each sensing area has some pre-integration effect. Note that the problem non-linearity (and hence complexity) is due to quantization. The binary and discrete data γ[k] are submitted to our compressed-sensing reconstruction algorithm, which is the last stage of our hybrid system. Assuming that the PSF is known, this algorithm reconstructs an image of the object. The result f͂o(x) can then be directly observed or evaluated.

Fig. 2.
Fig. 2.

The acquisition device is diffraction-limited; its circular exit pupil is phase-masked, which corresponds to a generalized pupil function. Geometric light propagation between the entrance and exit pupils [12] and unit magnification are assumed. Thus, the same coordinate system holds for both object fo(x) and image fi(x). All device components are centered and placed perpendicularly to the optical axis. The numerical aperture is then defined as NA = n sin α.

Fig. 3.
Fig. 3.

Shape of our penalty function. The transition between the linear and arctan regimes of ψ(t) is C2-continuous, and takes place at t = 0. When t goes to infinity, the applied penalty tends to zero. The convexity of ψ(t) clearly appears in this graph.

Fig. 4.
Fig. 4.

Comparison between standard and preconditioned gradient descent for our algorithm. The reconstruction SNR is shown as a function of the number of iterations (logarithmic scale). In the preconditioned case, the corresponding SNR (solid line) reaches a plateau after relatively few iterations. Comparatively, the same result without preconditioning (dashed line) shows an extremely slow SNR progression.

Fig. 5.
Fig. 5.

House image. (a) Original object (b) Conventional solution with minimum-error threshold (Lloyd-Max): SNR = 16.59 dB (c) Binary acquisition using our method (d) Reconstruction using our method: SNR = 23.21 dB

Fig. 6.
Fig. 6.

Peppers image. (a) Original object (b) Conventional solution with minimum-error threshold (Lloyd-Max): SNR = 13.72 dB (c) Binary acquisition using our method (d) Reconstruction using our method: SNR = 19.10 dB

Tables (1)

Tables Icon

Table 1. Reconstruction quality [dB] for different images. The results of our method with several sensor resolutions are reported on the left, while the standard-acquisition performance is shown in the middle. The TV norm of each reference image is given on the right.

Equations (24)

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

𝒫 : min c Φ T c 0 subject to g Ac 2 ε .
𝒫 : min c g Ac 2 2 + λ Φ T c 1 ,
f i ( x ) = ( f o * h ) ( x ) ,
circ ( ξ ) = { 1 , ξ 1 0 , therwise .
p ( ξ ) = k∊ 2 ν [ k ] rect ( ρ 2 ξ k ) .
q ( ξ ) = cicr ( ξ ) exp [ j ( k 0 W 20 ξ 2 + p ( ξ ) ) ] ,
h ( x ) = 𝒦 h 𝓕 { q ( ξ ) } ( NA λ 0 x ) 2 ,
𝓕 { f } ( ω ) = 2 f ( x ) exp { j 2 π ω T x } d x .
ϕ ( x ) = s ( x Δ s ) ,
g [ k ] = ( f i * ϕ ) ( x ) x = s .
γ [ k ] = 𝓑 ( g [ k ] , τ ) = { + 1 , g [ k ] τ 1 , g [ k ] < τ ,
f o ( x ) = k 2 c [ k ] φ ( x Δ c k ) ,
g = D ( 𝓝 ) A χ U ( 𝓜 ) c = Ac ,
χ [ k ] = I Δ s 2 𝓝 2 ( ( h * ϕ ) ( Δ s · 𝓝 ) * φ ( · 𝓜 ) ) ( x ) x = k .
c ˜ = arg min c 𝓓 ( c ) + λ𝓡 ( c ) 𝒞 ( c ) .
𝒟 ( c ) = k ψ ( g [ k ] γ [ k ] ) ,
ψ ( t ) = π 2 M { t , t < 0 M 1 arctan ( Mt ) , otherwise .
𝓡 ( c ) = k θ c [ k ] ,
θ c [ k ] = ( ( c φ x 1 ) [ k ] 2 + ( c φ x 2 ) [ k ] 2 + υ 2 ) 1 2 ,
φ x 1 [ k ] = φ ( k 1 + 1 2 ) φ ( k 2 ) ,
φ x 2 [ k ] = φ ( k 2 + 1 2 ) φ ( k 1 ) ,
P 𝒞 ( · ) = P ( A T Γ ψ ( Γ A · ) + λ ( R 1 T θ R 1 + R 2 T θ R 2 ) . ) ,
ψ i ( t ) = { 1 , t i < 0 ( 1 + M 2 t i 2 ) 1 , otherwise .
P= 𝓚 P ( D ( 𝓜 ) A χ T A χ U ( 𝓜 ) + σR ) 1 ,

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