Abstract

We apply adaptive sensing techniques to the problem of locating sparse metallic scatterers using high-resolution, frequency modulated continuous wave W-band RADAR. Using a single detector, a frequency stepped source, and a lateral translation stage, inverse synthetic aperture RADAR reconstruction techniques are used to search for one or two wire scatterers within a specified range, while an adaptive algorithm determined successive sampling locations. The two-dimensional location of each scatterer is thereby identified with sub-wavelength accuracy in as few as 1/4 the number of lateral steps required for a simple raster scan. The implications of applying this approach to more complex scattering geometries are explored in light of the various assumptions made.

© 2014 Optical Society of America

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  1. M. S. Heimbeck, D. L. Marks, D. Brady, H. O. Everitt, “Terahertz interferometric synthetic aperture tomography for confocal imaging systems,” Opt. Lett. 37, 1316–1318 (2012).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in “Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on,” (2012), pp. 778–785.
  21. D. J. Brady, Optical Imaging and Spectroscopy (Wiley-interscience, New Jersey, USA, 2008).
  22. M. E. Tipping, “Sparse bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).
  23. S. Ji, Y. Xue, L. Carin, “Bayesian compressive sensing,” IEEE Trans. Sig. Process. 56, 2346–2356 (2008).
    [CrossRef]

2013

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013).
[CrossRef]

2012

2011

M. Duarte, Y. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans. Sig. Proc. 59, 4053–4085 (2011).
[CrossRef]

2010

W. U. Bajwa, R. Calderbank, S. Jafarpour, “Why gabor frames? two fundamental measures of coherence and their role in model selection,” J. Commun. Netw. 12, 289–307 (2010).
[CrossRef]

C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt. 49, E67–E82 (2010).
[CrossRef] [PubMed]

E. Lebed, P. J. Mackenzie, M. V. Sarunic, F. M. Beg, “Rapid volumetric oct image acquisition using compressive sampling,” Opt. Express 18, 21003–21012 (2010).
[CrossRef] [PubMed]

L. Li, W. Zhang, F. Li, “Derivation and discussion of the sar migration algorithm within inverse scattering problem: Theoretical analysis,” Geoscience and Remote Sensing, IEEE Transactions on 48, 415–422 (2010).
[CrossRef]

2009

2008

S. Ji, Y. Xue, L. Carin, “Bayesian compressive sensing,” IEEE Trans. Sig. Process. 56, 2346–2356 (2008).
[CrossRef]

E. Candes, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique 346, 589–592 (2008).
[CrossRef]

E. Candes, M. Wakin, “An introduction to compressive sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008).
[CrossRef]

W. L. Chan, M. L. Moravec, R. G. Baraniuk, D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008).
[CrossRef] [PubMed]

2006

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

T. S. Ralston, D. L. Marks, P. S. Carney, S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23, 1027–1037 (2006).
[CrossRef]

E. Candes, T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

2004

P. Potuluri, M. Gehm, M. Sullivan, D. Brady, “Measurement-efficient optical wavemeters,” Opt. Express 12, 6219–6229 (2004).
[CrossRef] [PubMed]

Y. Zhang, X. Liao, L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface uxo,” IEEE Trans. Geosci. Remote Sensing 42, 2535–2543 (2004).
[CrossRef]

2001

M. E. Tipping, “Sparse bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).

1992

D. J. MacKay, “Information-based objective functions for active data selection,” Neural Computation 4, 590–604 (1992).
[CrossRef]

Ahn, C.-B.

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013).
[CrossRef]

Bajwa, W. U.

W. U. Bajwa, R. Calderbank, S. Jafarpour, “Why gabor frames? two fundamental measures of coherence and their role in model selection,” J. Commun. Netw. 12, 289–307 (2010).
[CrossRef]

Baraniuk, R. G.

Beg, F. M.

Bhaskar, B.

G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in “Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on,” (2012), pp. 778–785.

Boppart, S. A.

Brady, D.

Brady, D. J.

Buma, T.

Calderbank, R.

W. U. Bajwa, R. Calderbank, S. Jafarpour, “Why gabor frames? two fundamental measures of coherence and their role in model selection,” J. Commun. Netw. 12, 289–307 (2010).
[CrossRef]

Candes, E.

E. Candes, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique 346, 589–592 (2008).
[CrossRef]

E. Candes, M. Wakin, “An introduction to compressive sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008).
[CrossRef]

E. Candes, T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

Carin, L.

S. Ji, Y. Xue, L. Carin, “Bayesian compressive sensing,” IEEE Trans. Sig. Process. 56, 2346–2356 (2008).
[CrossRef]

Y. Zhang, X. Liao, L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface uxo,” IEEE Trans. Geosci. Remote Sensing 42, 2535–2543 (2004).
[CrossRef]

Carney, P. S.

Chan, W. L.

Choi, K.

Cull, C. F.

Donoho, D.

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

Duarte, M.

M. Duarte, Y. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans. Sig. Proc. 59, 4053–4085 (2011).
[CrossRef]

Duarte-Carvajalino, J.

J. Duarte-Carvajalino, G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18, 1395–1408 (2009).
[CrossRef] [PubMed]

Eldar, Y.

M. Duarte, Y. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans. Sig. Proc. 59, 4053–4085 (2011).
[CrossRef]

Everitt, H. O.

Gehm, M.

Ham, W.-G.

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013).
[CrossRef]

Heimbeck, M. S.

Horisaki, R.

Jafarpour, S.

W. U. Bajwa, R. Calderbank, S. Jafarpour, “Why gabor frames? two fundamental measures of coherence and their role in model selection,” J. Commun. Netw. 12, 289–307 (2010).
[CrossRef]

Ji, S.

S. Ji, Y. Xue, L. Carin, “Bayesian compressive sensing,” IEEE Trans. Sig. Process. 56, 2346–2356 (2008).
[CrossRef]

Kim, K.

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013).
[CrossRef]

Ku, J.

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013).
[CrossRef]

Lebed, E.

Lee, D.-G.

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013).
[CrossRef]

Lee, S.-H.

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013).
[CrossRef]

Li, F.

L. Li, W. Zhang, F. Li, “Derivation and discussion of the sar migration algorithm within inverse scattering problem: Theoretical analysis,” Geoscience and Remote Sensing, IEEE Transactions on 48, 415–422 (2010).
[CrossRef]

Li, L.

L. Li, W. Zhang, F. Li, “Derivation and discussion of the sar migration algorithm within inverse scattering problem: Theoretical analysis,” Geoscience and Remote Sensing, IEEE Transactions on 48, 415–422 (2010).
[CrossRef]

Liao, X.

Y. Zhang, X. Liao, L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface uxo,” IEEE Trans. Geosci. Remote Sensing 42, 2535–2543 (2004).
[CrossRef]

Lim, S.

MacKay, D. J.

D. J. MacKay, “Information-based objective functions for active data selection,” Neural Computation 4, 590–604 (1992).
[CrossRef]

Mackenzie, P. J.

Mait, J. N.

Marks, D. L.

Mattheiss, M.

Mittleman, D. M.

Moravec, M. L.

Park, H.

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013).
[CrossRef]

Potuluri, P.

Ralston, T. S.

Recht, B.

G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in “Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on,” (2012), pp. 778–785.

Sapiro, G.

J. Duarte-Carvajalino, G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18, 1395–1408 (2009).
[CrossRef] [PubMed]

Sarunic, M. V.

Shah, P.

G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in “Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on,” (2012), pp. 778–785.

Son, J.-H.

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013).
[CrossRef]

Sullivan, M.

Tang, G.

G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in “Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on,” (2012), pp. 778–785.

Tao, T.

E. Candes, T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

Tipping, M. E.

M. E. Tipping, “Sparse bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).

Wakin, M.

E. Candes, M. Wakin, “An introduction to compressive sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008).
[CrossRef]

Wikner, D. A.

Xue, Y.

S. Ji, Y. Xue, L. Carin, “Bayesian compressive sensing,” IEEE Trans. Sig. Process. 56, 2346–2356 (2008).
[CrossRef]

Zhang, W.

L. Li, W. Zhang, F. Li, “Derivation and discussion of the sar migration algorithm within inverse scattering problem: Theoretical analysis,” Geoscience and Remote Sensing, IEEE Transactions on 48, 415–422 (2010).
[CrossRef]

Zhang, Y.

Y. Zhang, X. Liao, L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface uxo,” IEEE Trans. Geosci. Remote Sensing 42, 2535–2543 (2004).
[CrossRef]

Zhang, Z.

Appl. Opt.

Comptes Rendus Mathematique

E. Candes, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique 346, 589–592 (2008).
[CrossRef]

Geoscience and Remote Sensing, IEEE Transactions on

L. Li, W. Zhang, F. Li, “Derivation and discussion of the sar migration algorithm within inverse scattering problem: Theoretical analysis,” Geoscience and Remote Sensing, IEEE Transactions on 48, 415–422 (2010).
[CrossRef]

IEEE Sig. Proc. Mag.

E. Candes, M. Wakin, “An introduction to compressive sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing

Y. Zhang, X. Liao, L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface uxo,” IEEE Trans. Geosci. Remote Sensing 42, 2535–2543 (2004).
[CrossRef]

IEEE Trans. Image Process.

J. Duarte-Carvajalino, G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18, 1395–1408 (2009).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

E. Candes, T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[CrossRef]

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

IEEE Trans. Sig. Proc.

M. Duarte, Y. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans. Sig. Proc. 59, 4053–4085 (2011).
[CrossRef]

IEEE Trans. Sig. Process.

S. Ji, Y. Xue, L. Carin, “Bayesian compressive sensing,” IEEE Trans. Sig. Process. 56, 2346–2356 (2008).
[CrossRef]

IEEE trans. Terahertz Sci. Technol.

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013).
[CrossRef]

J. Commun. Netw.

W. U. Bajwa, R. Calderbank, S. Jafarpour, “Why gabor frames? two fundamental measures of coherence and their role in model selection,” J. Commun. Netw. 12, 289–307 (2010).
[CrossRef]

J. Mach. Learn. Res.

M. E. Tipping, “Sparse bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).

J. Opt. Soc. Am. A

Neural Computation

D. J. MacKay, “Information-based objective functions for active data selection,” Neural Computation 4, 590–604 (1992).
[CrossRef]

Opt. Express

Opt. Lett.

Other

G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in “Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on,” (2012), pp. 778–785.

D. J. Brady, Optical Imaging and Spectroscopy (Wiley-interscience, New Jersey, USA, 2008).

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Figures (7)

Fig. 1
Fig. 1

The system diagram. A transceiver generates a Gaussian beam transmit and receive gain pattern focused by two lenses in a 4f configuration. A translation stage then moves the target through the beam to generate multiple lateral measurements.

Fig. 2
Fig. 2

Cross sections of the three objective functions in Eq. (5) with the model replaced with the model in Eq. (2) (plots (a) & (b)), the depth demodulated model in Eq. (15) (plots (c) & (d)), and the fully-demodulated model in Eq. (16) (plots (e) & (f)). The true location of the scatterer is indicated by the star. The left column is for a single point scatterer in which the scattering density is held constant and the two position variables of the objective function are varied. The right column searches for the first of two point scatterers holding constant the scattering density of the first scatterer and all parameters of the second scatterer. The true location of the sought scatterer is x1 = 3 mm and zi = 4 cm for all scenarios. Note the reduced oscillations in (b) and (c) relative to (a), and in (e) and (f) relative to (d), as well as the inaccurate lateral location of the minimum in (f).

Fig. 3
Fig. 3

The optimal hopping geometry showing the previously sampled region (red), the oversampled region (blue), the newly interrogated region of interest (green), and the region of possible interference for P = 2(orange). Operating farther from the focus yields more efficient sampling, but lower SNR.

Fig. 4
Fig. 4

A typical measurement path for a simulated (a) and fully adaptive acquisition (c), and the magnitude of the first and second derivative of the precision matrix versus measurement for that path (b) and (d), respectively.

Fig. 5
Fig. 5

The lateral estimates of a single point located at 0 mm from an optimal hopping window(a) and wide window(b) after eight lateral measurements, and of the weaker (greater distance from focus) of two point scatterers from an optimal hopping window (c) after six lateral measurements and a wide window (d) after seven lateral measurements.

Fig. 6
Fig. 6

A weighted histogram showing an energy distribution of the estimated points when P = 5 is assumed and P = 1 is the true model representing the quantity ξ defined in Eq. (19). The bin width is 2mm (±1mm) An overwhelming amount of the energy is placed in the true object location.

Fig. 7
Fig. 7

The regression line specifying the true location of the target (a) and the standard deviation from that line vs. iteration (b) for the fully adaptive experiment.

Equations (19)

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

g ( u n ) = M ( u n ; w ) + η n
M 0 ( x n , k n , NA ; { x i , z i , q i } i = 1 P ) = i = 1 P [ W 0 W exp [ ( x i x n ) 2 W 2 j k z i j k ( x i x n ) 2 2 R + j tan 1 z i z R ] ] 2 q i
p ( D N | w ) exp [ β 2 n = 1 N | M ( u n ; w ) g ( u n ) | 2 ] .
p ( w | D N ) p ( D N | w ) p 0 ( w ) ,
w ^ = argmin w ˜ β 2 n = 1 N | g ( u n ) M ( u n ; w ˜ ) | 2 log p 0 ( w ˜ ) ,
p ( w | D N ) exp [ 1 2 ( w w ^ ) H A N ( w w ^ ) ] .
log p ( w | D N ) = 1 2 ( w w ^ ) H A N ( w w ^ ) + C
A N = w w log p ( w | D N ) ,
A N = w w ( log ( p ( D N | w ) ) + log ( p 0 ( w ) ) ) .
M ( u n ; w ) M ( u n ; w ^ ) + ( w w ^ ) H w M ( u n ; w ) | w ^
v ( u n ; w ^ ) = w M ( u n ; w ) | w ^ ,
A N β n = 1 N v ( u n ; w ^ ) v ( u n ; w ^ ) H w w log p 0 ( w )
A N + 1 = A N + β v ( u N + 1 ; w ^ ) v ( u N + 1 ; w ^ ) H
u N + 1 = argmax u ^ N + 1 v ( u N + 1 ; w ^ ) H A N 1 v ( u N + 1 ; w ^ ) .
M ^ ( x n , k n , NA ; { x i , z i , q i } i = 1 P ) = i = 1 P [ W 0 W exp [ ( x i x n ) 2 W 2 j ( k k min ) z i j k ( x i x n ) 2 2 R + j tan 1 z i z R ] ] 2 q i
M ˜ ( x n , k n , NA ; x i , z i , q i ) = i = 1 P [ W 0 W exp [ ( x i x n ) 2 W 2 j ( k k min ) z i j ( k k min ) ( x i x n ) 2 2 R + j tan 1 z i z R ] ] 2 q i .
g ( x n , k n , NA ; x 1 , z 1 , q 1 ) = g ( x n , k n , NA ; x 1 , z 1 , q 1 ) G ( k n )
G ( k n ) = exp [ k max 2 k n 2 ] .
ξ t = j = 1 J i = 1 P ˜ rect ( ( x i , j b t ) / bw ) | q i , j | 2 ,

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