Abstract

A hardware implementation of a real-time compressed-domain image acquisition system is demonstrated. The system performs front-end computational imaging, whereby the inner product between an image and an arbitrarily-specified mask is implemented in silicon. The acquisition system is based on an intelligent readout integrated circuit (iROIC) that is capable of providing independent bias voltages to individual detectors, which enables implementation of spatial multiplication with any prescribed mask through a bias-controlled response-modulation mechanism. The modulated pixels are summed up in the image grabber to generate the compressed samples, namely aperture-coded coefficients, of an image. A rigorous bias-selection algorithm is presented to the readout circuit, which exploits the bias-dependent nature of the imager’s responsivity. Proven functionality of the hardware in transform coding compressed image acquisition, silicon-level compressive sampling, in pixel nonuniformity correction and hardware-level implementation of region-based enhancement is demonstrated.

© 2017 Optical Society of America

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References

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2016 (2)

F. Shao, W. Lin, G. Jiang, and Q. Dai, “Models of monocular and binocular visual perception in quality assessment of stereoscopic images,” IEEE T. Comput. Imag. 2, 123–135 (2016).
[Crossref]

A. Mehrish, A. Subramanyam, and S. Emmanuel, “Sensor pattern noise estimation using probabilistically estimated RAW values,” IEEE Signal Process. Lett. 23, 693–697 (2016).
[Crossref]

2015 (4)

S. V. Venkatakrishnan, L. F. Drummy, M. Jackson, M. De Graef, J. Simmons, and C. A. Bouman, “Model-based iterative reconstruction for bright-field electron tomography,” IEEE T. Comput. Imag. 1, 1–15 (2015).
[Crossref]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref]

A. Gandomi and M. Haider, “Beyond the hype: Big data concepts, methods, and analytics,” Int. J. Inform. Manage. 35, 137–144 (2015).
[Crossref]

I. Cevik, X. Huang, H. Yu, M. Yan, and S. U. Ay, “An ultra-low power CMOS image sensor with on-chip energy harvesting and power management capability,” Sensors 15, 5531–5554 (2015).
[Crossref] [PubMed]

2013 (4)

M. Dadkhah, M. J. Deen, and S. Shirani, “Compressive sensing image sensors-hardware implementation,” Sensors 13, 4961–4978 (2013).
[Crossref] [PubMed]

A. J. Cooper, “Improved photo response non-uniformity (PRNU) based source camera identification,” Forensic Sci. Int. 226, 132–141 (2013).
[Crossref] [PubMed]

A. Piva, “An overview on image forensics,” ISRN Sig. Proc. 2013, 496701 (2013).

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, 10526–10545 (2013).
[Crossref] [PubMed]

2011 (2)

C. Zuo, Q. Chen, G. Gu, and X. Sui, “Scene-based nonuniformity correction algorithm based on interframe registration,” JOSA A 28, 1164–1176 (2011).
[Crossref] [PubMed]

M. Sheng, J. Xie, and Z. Fu, “Calibration-based NUC method in real-time based on IRFPA,” Physics Procedia 22, 372–380 (2011).
[Crossref]

2009 (1)

Y.-M. Zhou, C. Zhang, and Z.-K. Zhang, “An efficient fractal image coding algorithm using unified feature and DCT,” Chaos, Solitons & Fractals 39, 1823–1830 (2009).
[Crossref]

2008 (3)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag. 25, 83 (2008).
[Crossref]

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Sig. Proc. Mag. 25, 72–82 (2008).
[Crossref]

2007 (2)

M. Lustig, D. Donoho, and J. M. Pauly, “MRI Sparse: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[Crossref] [PubMed]

R. G. Baraniuk, “Compressive sensing,” IEEE Sig. Proc. Mag. 24, 118 (2007).
[Crossref]

2006 (1)

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
[Crossref]

2004 (2)

M. Prastawa, E. Bullitt, S. Ho, and G. Gerig, “A brain tumor segmentation framework based on outlier detection,” Med. Image Anal. 8, 275–283 (2004).
[Crossref] [PubMed]

Y. Oike, M. Ikeda, and K. Asada, “Design and implementation of real-time 3-D image sensor with 640× 480 pixel resolution,” IEEE J. Solid-St. Circ. 39, 622–628 (2004).
[Crossref]

2001 (1)

D. Litwiller, “CCD vs. CMOS,” Photon. Spectra 35, 154–158 (2001).

2000 (2)

K. Yonemoto and H. Sumi, “A CMOS image sensor with a simple fixed-pattern-noise-reduction technology and a hole accumulation diode,” IEEE J. Solid-St. Circ. 35, 2038–2043 (2000).
[Crossref]

S. Saha, “Image compression-from DCT to wavelets: a review,” Crossroads 6, 12–21 (2000).
[Crossref]

1999 (1)

J. Ribas-Corbera and S. Lei, “Rate control in DCT video coding for low-delay communications,” IEEE T. Circuits Syst. Video Technol. 9, 172–185 (1999).
[Crossref]

1998 (1)

N. Nakano, R. Nishimura, H. Sai, A. Nishizawa, and H. Komatsu, “Digital still camera system for megapixel CCD,” IEEE T. Consum. Electron. 44, 581–586 (1998).
[Crossref]

1997 (1)

B. E. Stine, D. S. Boning, and J. E. Chung, “Analysis and decomposition of spatial variation in integrated circuit processes and devices,” IEEE T. Semicond. Manuf. 10, 24–41 (1997).
[Crossref]

1994 (2)

S. Mendis, S. E. Kemeny, and E. R. Fossum, “CMOS active pixel image sensor,” IEEE T. Electron. Dev. 41, 452–453 (1994).
[Crossref]

J. B. Sampsell, “Digital micromirror device and its application to projection displays,” J. Vac. Sci. Technol. B 12, 3242–3246 (1994).
[Crossref]

1993 (2)

J. G. Daugman, “High confidence visual recognition of persons by a test of statistical independence,” IEEE T. Pattern Anal. 15, 1148–1161 (1993).
[Crossref]

D. L. Perry and E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1854–1859 (1993).
[Crossref]

Asada, K.

Y. Oike, M. Ikeda, and K. Asada, “Design and implementation of real-time 3-D image sensor with 640× 480 pixel resolution,” IEEE J. Solid-St. Circ. 39, 622–628 (2004).
[Crossref]

Ay, S. U.

I. Cevik, X. Huang, H. Yu, M. Yan, and S. U. Ay, “An ultra-low power CMOS image sensor with on-chip energy harvesting and power management capability,” Sensors 15, 5531–5554 (2015).
[Crossref] [PubMed]

Bahl, P.

R. LiKamWa, B. Priyantha, M. Philipose, L. Zhong, and P. Bahl, “Energy characterization and optimization of image sensing toward continuous mobile vision,” in “Proceeding of the 11th annual international conference on Mobile systems, applications, and services,” (ACM, 2013), pp. 69–82.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag. 25, 83 (2008).
[Crossref]

R. G. Baraniuk, “Compressive sensing,” IEEE Sig. Proc. Mag. 24, 118 (2007).
[Crossref]

Barrett, P. T.

P. T. Barrett, “Method for image compression on a personal computer,” (1994). US Patent5,287,420.

Bhattarai, M.

M. Bhattarai, J. Ghasemi, G. R. Fiorante, P. Zarkesh-Ha, S. Krishna, and M. M. Hayat, “Intelligent bias-selection method for computational imaging on a CMOS imager,” in “2016 IEEE Photonics Conference,” (2016).

Bigas, M.

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
[Crossref]

Boning, D. S.

B. E. Stine, D. S. Boning, and J. E. Chung, “Analysis and decomposition of spatial variation in integrated circuit processes and devices,” IEEE T. Semicond. Manuf. 10, 24–41 (1997).
[Crossref]

Bouman, C. A.

S. V. Venkatakrishnan, L. F. Drummy, M. Jackson, M. De Graef, J. Simmons, and C. A. Bouman, “Model-based iterative reconstruction for bright-field electron tomography,” IEEE T. Comput. Imag. 1, 1–15 (2015).
[Crossref]

Bowman, R. W.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref]

Brady, D. J.

Bullitt, E.

M. Prastawa, E. Bullitt, S. Ho, and G. Gerig, “A brain tumor segmentation framework based on outlier detection,” Med. Image Anal. 8, 275–283 (2004).
[Crossref] [PubMed]

Cabruja, E.

M. Bigas, E. Cabruja, J. Forest, and J. Salvi, “Review of CMOS image sensors,” Microelectron. J. 37, 433–451 (2006).
[Crossref]

Caldwell, L. V.

M. J. Schulz and L. V. Caldwell, “Nonuniformity correction and correctability of infrared focal plane arrays,” in “SPIE’s 1995 Symposium on OE/Aerospace Sensing and Dual Use Photonics,” (International Society for Optics and Photonics, 1995), pp. 200–211.

Candes, E.

E. Candes and J. Romberg, “l1-magic: Recovery of sparse signals via convex programming,” URL: www.acm.caltech.edu/l1magic/downloads/l1magic.pdf 4, 14 (2005).

Candès, E. J.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008).
[Crossref]

Carin, L.

Cevik, I.

I. Cevik, X. Huang, H. Yu, M. Yan, and S. U. Ay, “An ultra-low power CMOS image sensor with on-chip energy harvesting and power management capability,” Sensors 15, 5531–5554 (2015).
[Crossref] [PubMed]

Chen, Q.

C. Zuo, Q. Chen, G. Gu, and X. Sui, “Scene-based nonuniformity correction algorithm based on interframe registration,” JOSA A 28, 1164–1176 (2011).
[Crossref] [PubMed]

Chowdhury, A.

A. Chowdhury, R. Darveaux, J. Tome, R. Schoonejongen, M. Reifel, A. De Guzman, S. S. Park, Y. W. Kim, and H. W. Kim, “Challenges of megapixel camera module assembly and test,” in “Proceedings Electronic Components and Technology, 2005. ECTC’05.”, (IEEE, 2005), pp. 1390–1401.

Chung, J. E.

B. E. Stine, D. S. Boning, and J. E. Chung, “Analysis and decomposition of spatial variation in integrated circuit processes and devices,” IEEE T. Semicond. Manuf. 10, 24–41 (1997).
[Crossref]

Codreanu, M.

M. Leinonen, M. Codreanu, and M. Juntti, “Compressed acquisition and progressive reconstruction of multidimensional correlated data in wireless sensor networks,” in “2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP),” (IEEE, 2014), pp. 6449–6453.

Cooper, A. J.

A. J. Cooper, “Improved photo response non-uniformity (PRNU) based source camera identification,” Forensic Sci. Int. 226, 132–141 (2013).
[Crossref] [PubMed]

Dadkhah, M.

M. Dadkhah, M. J. Deen, and S. Shirani, “Compressive sensing image sensors-hardware implementation,” Sensors 13, 4961–4978 (2013).
[Crossref] [PubMed]

Dai, Q.

F. Shao, W. Lin, G. Jiang, and Q. Dai, “Models of monocular and binocular visual perception in quality assessment of stereoscopic images,” IEEE T. Comput. Imag. 2, 123–135 (2016).
[Crossref]

Darveaux, R.

A. Chowdhury, R. Darveaux, J. Tome, R. Schoonejongen, M. Reifel, A. De Guzman, S. S. Park, Y. W. Kim, and H. W. Kim, “Challenges of megapixel camera module assembly and test,” in “Proceedings Electronic Components and Technology, 2005. ECTC’05.”, (IEEE, 2005), pp. 1390–1401.

Daugman, J. G.

J. G. Daugman, “High confidence visual recognition of persons by a test of statistical independence,” IEEE T. Pattern Anal. 15, 1148–1161 (1993).
[Crossref]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag. 25, 83 (2008).
[Crossref]

De Graef, M.

S. V. Venkatakrishnan, L. F. Drummy, M. Jackson, M. De Graef, J. Simmons, and C. A. Bouman, “Model-based iterative reconstruction for bright-field electron tomography,” IEEE T. Comput. Imag. 1, 1–15 (2015).
[Crossref]

De Guzman, A.

A. Chowdhury, R. Darveaux, J. Tome, R. Schoonejongen, M. Reifel, A. De Guzman, S. S. Park, Y. W. Kim, and H. W. Kim, “Challenges of megapixel camera module assembly and test,” in “Proceedings Electronic Components and Technology, 2005. ECTC’05.”, (IEEE, 2005), pp. 1390–1401.

Deen, M. J.

M. Dadkhah, M. J. Deen, and S. Shirani, “Compressive sensing image sensors-hardware implementation,” Sensors 13, 4961–4978 (2013).
[Crossref] [PubMed]

Dereniak, E. L.

D. L. Perry and E. L. Dereniak, “Linear theory of nonuniformity correction in infrared staring sensors,” Opt. Eng. 32, 1854–1859 (1993).
[Crossref]

Dierickx, B.

N. Ricquier and B. Dierickx, “Active pixel CMOS image sensor with on-chip non-uniformity correction,” in “Proc. IEEE Workshop Charge-Coupled Devices and Advanced Image Sensors,” (1995), pp. 20–22.

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, “MRI Sparse: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[Crossref] [PubMed]

Donoho, D. L.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Sig. Proc. Mag. 25, 72–82 (2008).
[Crossref]

Drummy, L. F.

S. V. Venkatakrishnan, L. F. Drummy, M. Jackson, M. De Graef, J. Simmons, and C. A. Bouman, “Model-based iterative reconstruction for bright-field electron tomography,” IEEE T. Comput. Imag. 1, 1–15 (2015).
[Crossref]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. E. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag. 25, 83 (2008).
[Crossref]

Edgar, M. P.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5, 10669 (2015).
[Crossref]

El Gamal, A.

Y. Oike and A. El Gamal, “A 256× 256 CMOS image sensor with ∆Σ-based single-shot compressed sensing,” in “2012 IEEE International Solid-State Circuits Conference,” (IEEE, 2012), pp. 386–388.

Emmanuel, S.

A. Mehrish, A. Subramanyam, and S. Emmanuel, “Sensor pattern noise estimation using probabilistically estimated RAW values,” IEEE Signal Process. Lett. 23, 693–697 (2016).
[Crossref]

Evans, J. M.

C. F. Weiman and J. M. Evans, “Digital image compression employing a resolution gradient,” (1992). US Patent5,103,306,.

Fiorante, G. R.

M. Bhattarai, J. Ghasemi, G. R. Fiorante, P. Zarkesh-Ha, S. Krishna, and M. M. Hayat, “Intelligent bias-selection method for computational imaging on a CMOS imager,” in “2016 IEEE Photonics Conference,” (2016).

Fiorante, G. R. C.

G. R. C. Fiorante, P. Zarkesh-Ha, J. Ghasemi, and S. Krishna, “Spatio-temporal tunable pixels for multi-spectral infrared imagers,” in “2013 IEEE 56th International Midwest Symposium on Circuits and Systems (MWSCAS),” (IEEE, 2013), pp. 317–320.

Forest, J.

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Figures (12)

Fig. 1
Fig. 1

a) A system-level block diagram of a conventional imaging system, which includes image acquisition, storage, and post-processing stages. b) Block diagram of the intelligent readout integrated circuit we propose for on-chip image acquisition and compression.

Fig. 2
Fig. 2

Block diagram of the individual pixel bias tunable readout integrated circuit and the CTIA-based unitcell at the extended view. The extra circuitry added to the CTIA-based unitcell enables setting independent bias voltages for each individual pixel while the previously integrated voltage is being read out.

Fig. 3
Fig. 3

a) A cross section of the n+/nwell/psub photodetector used in this chip, b) the measured photoresponse of n+/nwell/psub photodetector as a function of the applied bias voltages at different illumination levels, and c) the same measured results that are scaled to one. In this experiment, a green LED is used as the illumination source and the dimension of the photodetector is 100 µm×100 µm.

Fig. 4
Fig. 4

a) Switch level implementation of iROIC unitcell. The unitcell includes 15 transistors and three capacitors, b) the video switches, c) the row/column select peripherals, and d) a sample timing diagram of a single unitcell.

Fig. 5
Fig. 5

Demonstration of the normalized modulation function of the system to a uniform illumination level. The graph reflects the system’s response to the modulation of the detector’s bias.

Fig. 6
Fig. 6

a) A microphotograph of the fabricated ROIC, the row and column select, and the test devices. The unitcell is shown in the extended view. b) A block diagram of the experimental setup, which includes a Raspberry Pi board as the main controller of the system, an ADC and a DAC to set the bias voltage of the detectors and grabs the readout of the imager. All communication between the controller and a remote machine is over SSH.

Fig. 7
Fig. 7

a) The result of imaging a white paper with uniform biasing, while the illumination is not uniform. Defects and other sources of nonuniformity also contribute to the variation across the image. The stack of three graphs demonstrates (I) camera output image, (II) illumination contour, and (III) 3D view of the intensities. b) Another white paper is imaged with the same illumination condition using the implemented nonuniformity correction. The graph has the same scale as part (a), and the legend in the middle is for part (II). c) and d) show the histogram for the measured results of part (a) and (b), respectively.

Fig. 8
Fig. 8

Acquisition and compression processes, which include mapping k mask matrices to their corresponding bias voltages. The mapping is based on the system’s response-modulation function shown in Fig. 5. Then the bias matrices that are stored in the Raspberry Pi memory are loaded to the imager and projected to the object’s reflectance function. The resultant dot product is optionally summed up in the hardware, and the k resulting coefficients are sent to the remote computer for reconstruction.

Fig. 9
Fig. 9

a) Distribution of 8 × 8 block-based DCT mask coefficients for naïve method, b) distribution of bias for naïve method, c) distribution of 8 × 8 block-based DCT mask coefficients for MMSE method, and d) distribution of bias for MMSE method.

Fig. 10
Fig. 10

The resulting images reconstructed using a) naïve DCT, b) minimum-mean-square error based DCT, c) compressive sensing, and d) ideal DCT. e) The performance of different method is compared in terms of the mean square error between the reconstructed image and the original image.

Fig. 11
Fig. 11

Four images that are taken using iROIC camera in normal mode. a) phantom, b) a cell, c) some rice grains, and d) UNM logo.

Fig. 12
Fig. 12

a) Original white matter image used for imaging. b) Image is taken using iROIC with a uniform biasing for all of the pixels where some of the pixels are saturated due to the high intensity. In c), d), e), and, f) the same scene is imaged using proper biasing for the different areas that normally are at the noise floor of the imager.

Tables (1)

Tables Icon

Table 1 Comparison between different configuration for preamplifier used in an imager. Due to the need for good bias control, high injection efficiency, and sufficient charge storage, we have selected CTIA configuration for iROIC.

Equations (34)

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Y i j k = g i j k I i j k + o i j k ,
I i j k = w i j k Y i j k + b i j k ,
R = g ( I , V ) ,
y u v = 2 M N i = 0 M 1 j = 0 N 1 [ C ( u ) C ( v ) I i j cos π ( 2 i + 1 ) u 2 N cos π ( 2 j + 1 ) v 2 N ] ,
C ( u ) , C ( v ) = { 1 2 if u , v = 0 1 otherwise .
I i j = 2 M N u = 0 M 1 v = 0 N 1 [ y u v cos π ( 2 i + 1 ) u 2 N cos π ( 2 j + 1 ) v 2 N ]
y u v = 2 M N C ( u ) C ( v ) i = 0 M 1 j = 0 N 1 [ I i j M a s k u v ( i , j ) ] ,
u , v = 0 , 1 , , N 1 .
M a s k u v ( i , j ) = cos π ( 2 m + 1 ) u 2 N cos π ( 2 n + 1 ) v 2 N .
I = [ I 1 , 1 I 1 , 96 I 96 , 1 I 96 , 96 ] , Y = [ y ( 1 ) y 96 y ( 96 2 96 ) y ( 96 2 ) ]
B ( k ) = [ b ˜ 1 , 1 ( k ) b ˜ 1 , 96 ( k ) y 96 , 1 ( k ) b ˜ 96 , 96 ( k ) ] .
R ˜ ( k ) = [ r ˜ 1 , 1 ( k ) r ˜ 1 , 96 ( k ) r ˜ 96 , 1 ( k ) r ˜ 96 , 96 ( k ) ] ,
r ˜ ( v ) = r ( v ) + η ( μ , σ v 2 ) ,
y R ˜ ( k ) = i = 1 96 j = 1 96 I i , j r ˜ i , j ( k ) ( v ) ,
y e r r ( k ) = y i d l ( k ) y R ˜ ( k ) = i = 1 96 j = 1 96 I i , j ( b i , j ( k ) r ˜ i , j ( k ) ( v ) ) ,
y i d l ( k ) = i j I i , j b i , j ( k ) .
f ( v ) = ( b r ˜ ( v ) ) 2 ,
minimize v f ( v ) subject to E ( f ( v ) ) = 0 ,
v o p t = argmin v E ( f ( v ) ) ,
E ( f ( v ) ) = ( b r ( v ) ) 2 2 μ ( b r ( v ) ) + μ 2 + σ v 2 ,
r ( v o p t ) = b μ σ d d v σ v o p t d d v r ( v o p t ) ,
v o p t = σ v o p t 2 [ ( d d v σ v o p t d d v r ( v o p t ) ) 2 + 1 ] ,
y i j k = i j I i j b i j k .
y i j k = i j I i j r i j k = i j I i j ( m b i j + c ) = m i j I i j b i j + c i j I i j = m y i j k + c i j I i j .
y i j k m c i j I i j = y i j k .
y o p t k = i = 1 96 i = 1 96 I i , j r ˜ i , j k ( v o p t ) .
I k R o p t k y o p t k .
x = i = 1 P s i ϕ i ,
x = ϕ s .
M c K log ( P K ) ,
( 1 σ k ) | x | 2 2 | ψ x | 2 2 ( 1 + σ k ) | x | 2 2 ,
x ^ = min â Ą ą x 1 ,
ψ x = y ,
x 1 = i x i .

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