Abstract

There have been many recent developments in 3D display technology to provide correct accommodation cues over an extended focus range. To this end, those displays rely on scene decomposition algorithms to reproduce accurate occlusion boundaries as well asretinal defocus blur. Recently, tomographic displays have been proposed with improved trade-offs of focus range, spatial resolution, and exit-pupil. The advantage of the system partly stems from a high-speed backlight modulation system based on a digital micromirror device, which only supports 1-bit images. However, its inherent binary constraint hinders achieving the optimal scene decomposition, thus leaving boundary artifacts. In this work, we present a technique for synthesizing optimal imagery of general 3D scenes with occlusion on tomographic displays. Requiring no prior knowledge of the scene geometry, our technique addresses the blending issue via non-convex optimization, inspired by recent studies in discrete tomography. Also, we present a general framework for this rendering algorithm and demonstrate the utility of the technique for volumetric display systems with binary representation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2019 (2)

H. Yu, M. Bemana, M. Wernikowski, M. Chwesiuk, O. T. Tursun, G. Singh, K. Myszkowski, R. Mantiuk, H.-P. Seidel, and P. Didyk, “A perception-driven hybrid decomposition for multi-layer accommodative displays,” IEEE Trans. Vis. Comput. Graph. 25(5), 1940–1950 (2019).
[Crossref] [PubMed]

S. Lee, Y. Jo, D. Yoo, J. Cho, D. Lee, and B. Lee, “Tomographic near-eye displays,” Nat. Commun. 10, 2497 (2019).
[Crossref] [PubMed]

2018 (4)

K. Rathinavel, H. Wang, A. Blate, and H. Fuchs, “An extended depth-at-field volumetric near-eye augmented reality display,” IEEE Trans. Vis. Comput. Graph. 24(11), 2857–2866 (2018).
[Crossref] [PubMed]

J.-H. R. Chang, B. V. K. V. Kumar, and A. C. Sankaranarayanan, “Towards multifocal displays with dense focal stacks,” ACM Trans. Graph. 37(6), 198 (2018).
[Crossref]

M. Yan, “A new primal–dual algorithm for minimizing the sum of three functions with a linear operator,” J. Sci. Comput. 76(3), 1698–1717 (2018).
[Crossref]

D. Kim, S. Lee, S. Moon, J. Cho, Y. Jo, and B. Lee, “Hybrid multi-layer displays providing accommodation cues,” Opt. Express 26(13), 17170–17184 (2018).
[Crossref] [PubMed]

2017 (2)

D. Davis and W. Yin, “A three-operator splitting scheme and its optimization applications,” Set-valued and variational analysis 25(4), 829–858 (2017).
[Crossref]

O. Mercier, Y. Sulai, K. Mackenzie, M. Zannoli, J. Hillis, D. Nowrouzezahrai, and D. Lanman, “Fast gaze-contingent optimal decompositions for multifocal displays,” ACM Trans. Graph. 36(6), 237 (2017).
[Crossref]

2016 (8)

S. Lee, C. Jang, S. Moon, J. Cho, and B. Lee, “Additive light field displays: realization of augmented reality with holographic optical elements,” ACM Trans. Graph. 35(4), 60 (2016).
[Crossref]

T. Pock and S. Sabach, “Inertial proximal alternating linearized minimization (ipalm) for nonconvex and nonsmooth problems,” SIAM J. Imaging Sci. 9(4), 1756–1787 (2016).
[Crossref]

P. Wang, C. Shen, A. van den Hengel, and P. H. Torr, “Large-scale binary quadratic optimization using semidefinite relaxation and applications,” IEEE Trans. Pattern Anal. Mach. Intell. 39(3), 470–485 (2016).
[Crossref] [PubMed]

M. Hong, Z.-Q. Luo, and M. Razaviyayn, “Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems,” SIAM J. Optim. 26(1), 337–364 (2016).
[Crossref]

A. Chambolle and T. Pock, “An introduction to continuous optimization for imaging,” Acta Numer. 25, 161–319 (2016).
[Crossref]

M. Zisler, J. H. Kappes, C. Schnörr, S. Petra, and C. Schnörr, “Non-binary discrete tomography by continuous non-convex optimization,” IEEE Trans. Comput. Imaging 2(3), 335–347 (2016).
[Crossref]

T. Lukić and P. Balázs, “Binary tomography reconstruction based on shape orientation,” Pattern Recognit. Lett. 79, 18–24 (2016).
[Crossref]

C.-K. Lee, S. Moon, S. Lee, D. Yoo, J.-Y. Hong, and B. Lee, “Compact three-dimensional head-mounted display system with savart plate,” Opt. Express 24(17), 19531–19544 (2016).
[Crossref] [PubMed]

2015 (3)

Y. Lou, T. Zeng, S. Osher, and J. Xin, “A weighted difference of anisotropic and isotropic total variation model for image processing,” SIAM J. Imaging Sci. 8(3), 1798–1823 (2015).
[Crossref]

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34(4), 59 (2015).
[Crossref]

F.-C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: Immersive computer graphics via factored near-eye light field displays with focus cues,” ACM Trans. Graph. 34(4), 60 (2015).
[Crossref]

2014 (3)

L. Wang, B. Sixou, and F. Peyrin, “Binary tomography reconstructions with stochastic level-set methods,” IEEE Signal Process. Lett. 22(7), 920–924 (2014).
[Crossref]

S. Roux, H. Leclerc, and F. Hild, “Efficient binary tomographic reconstruction,” J. Math. Imaging Vis. 49(2), 335–351 (2014).
[Crossref]

J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Math. Program. 146(1–2), 459–494 (2014).
[Crossref]

2013 (2)

L. Condat, “A primal–dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms,” J. Optim. Theory Appl. 158(2), 460–479 (2013).
[Crossref]

D. Lanman and D. Luebke, “Near-eye light field displays,” ACM Trans. Graph. 32(6), 220 (2013).
[Crossref]

2012 (1)

G. Wetzstein, D. Lanman, M. Hirsch, and R. Raskar, “Tensor displays: Compressive light field synthesis using multilayer displays with directional backlighting,” ACM Trans. Graph. 31(4), 80 (2012).
[Crossref]

2011 (5)

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vis. 40(1), 120–145 (2011).
[Crossref]

K. J. Batenburg and J. Sijbers, “Dart: a practical reconstruction algorithm for discrete tomography,” IEEE Trans. Image Process. 20(9), 2542–2553 (2011).
[Crossref] [PubMed]

G. Wetzstein, D. Lanman, W. Heidrich, and R. Raskar, “Layered 3d: Tomographic image synthesis for attenuation-based light field and high dynamic range displays,” ACM Trans. Graph. 30(4), 95 (2011).
[Crossref]

D. Lanman, G. Wetzstein, M. Hirsch, W. Heidrich, and R. Raskar, “Polarization fields: Dynamic light field display using multi-layer lcds,” ACM Trans. Graph. 30(6), 186 (2011).
[Crossref]

R. Mantiuk, K. J. Kim, A. G. Rempel, and W. Heidrich, “Hdr-vdp-2: A calibrated visual metric for visibility and quality predictions in all luminance conditions,” ACM Trans. Graph. 30(4), 40 (2011).
[Crossref]

2010 (1)

K. J. MacKenzie, D. M. Hoffman, and S. J. Watt, “Accommodation to multiple-focal-plane displays: Implications for improving stereoscopic displays and for accommodation control,” J. Vis. 10(8), 22 (2010).
[Crossref] [PubMed]

2008 (2)

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence–accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 8(3), 33 (2008).
[Crossref] [PubMed]

K. J. Batenburg, “A network flow algorithm for reconstructing binary images from continuous x-rays,” J. Math. Imaging Vis. 30(3), 231–248 (2008).
[Crossref]

2006 (1)

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Trans. Graph. 25(3), 924–934 (2006).
[Crossref]

2005 (1)

T. Schüle, C. Schnörr, S. Weber, and J. Hornegger, “Discrete tomography by convex–concave regularization and dc programming,” Discret. Appl. Math. 151(1–3), 229–243 (2005).
[Crossref]

2004 (2)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

K. Akeley, S. J. Watt, A. R. Girshick, and M. S. Banks, “A stereo display prototype with multiple focal distances,” ACM Trans. Graph. 23(3), 804–813 (2004).
[Crossref]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenom. 60(1–4), 259–268 (1992).
[Crossref]

1984 (1)

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (sart): a superior implementation of the art algorithm,” Ultrason. Imaging 6(1), 81–94 (1984).
[Crossref] [PubMed]

Adams, A.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” ACM Trans. Graph. 25(3), 924–934 (2006).
[Crossref]

Akeley, K.

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence–accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 8(3), 33 (2008).
[Crossref] [PubMed]

K. Akeley, S. J. Watt, A. R. Girshick, and M. S. Banks, “A stereo display prototype with multiple focal distances,” ACM Trans. Graph. 23(3), 804–813 (2004).
[Crossref]

Albert, R. A.

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34(4), 59 (2015).
[Crossref]

Andersen, A. H.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (sart): a superior implementation of the art algorithm,” Ultrason. Imaging 6(1), 81–94 (1984).
[Crossref] [PubMed]

Balázs, P.

T. Lukić and P. Balázs, “Binary tomography reconstruction based on shape orientation,” Pattern Recognit. Lett. 79, 18–24 (2016).
[Crossref]

Banks, M. S.

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34(4), 59 (2015).
[Crossref]

D. M. Hoffman, A. R. Girshick, K. Akeley, and M. S. Banks, “Vergence–accommodation conflicts hinder visual performance and cause visual fatigue,” J. Vis. 8(3), 33 (2008).
[Crossref] [PubMed]

K. Akeley, S. J. Watt, A. R. Girshick, and M. S. Banks, “A stereo display prototype with multiple focal distances,” ACM Trans. Graph. 23(3), 804–813 (2004).
[Crossref]

Baraniuk, R.

T. Goldstein, M. Li, X. Yuan, E. Esser, and R. Baraniuk, “Adaptive primal-dual hybrid gradient methods for saddle-point problems,” arXiv preprint arXiv:1305.0546 (2013).

Batenburg, K. J.

K. J. Batenburg and J. Sijbers, “Dart: a practical reconstruction algorithm for discrete tomography,” IEEE Trans. Image Process. 20(9), 2542–2553 (2011).
[Crossref] [PubMed]

K. J. Batenburg, “A network flow algorithm for reconstructing binary images from continuous x-rays,” J. Math. Imaging Vis. 30(3), 231–248 (2008).
[Crossref]

Bemana, M.

H. Yu, M. Bemana, M. Wernikowski, M. Chwesiuk, O. T. Tursun, G. Singh, K. Myszkowski, R. Mantiuk, H.-P. Seidel, and P. Didyk, “A perception-driven hybrid decomposition for multi-layer accommodative displays,” IEEE Trans. Vis. Comput. Graph. 25(5), 1940–1950 (2019).
[Crossref] [PubMed]

Blate, A.

K. Rathinavel, H. Wang, A. Blate, and H. Fuchs, “An extended depth-at-field volumetric near-eye augmented reality display,” IEEE Trans. Vis. Comput. Graph. 24(11), 2857–2866 (2018).
[Crossref] [PubMed]

Bolte, J.

J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Math. Program. 146(1–2), 459–494 (2014).
[Crossref]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Bulbul, A.

R. Narain, R. A. Albert, A. Bulbul, G. J. Ward, M. S. Banks, and J. F. O’Brien, “Optimal presentation of imagery with focus cues on multi-plane displays,” ACM Trans. Graph. 34(4), 59 (2015).
[Crossref]

Chambolle, A.

A. Chambolle and T. Pock, “An introduction to continuous optimization for imaging,” Acta Numer. 25, 161–319 (2016).
[Crossref]

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vis. 40(1), 120–145 (2011).
[Crossref]

T. Pock and A. Chambolle, “Diagonal preconditioning for first order primal-dual algorithms in convex optimization,” in in Proceedings of IEEE Conference on Computer Vision (IEEE, 2011), pp. 1762–1769.

Chang, J.-H. R.

J.-H. R. Chang, B. V. K. V. Kumar, and A. C. Sankaranarayanan, “Towards multifocal displays with dense focal stacks,” ACM Trans. Graph. 37(6), 198 (2018).
[Crossref]

Chapman, M.

L. Xiao, A. Kaplanyan, A. Fix, M. Chapman, and D. Lanman, “Deepfocus: learned image synthesis for computational displays,” in SIGGRAPH Asia 2018 Technical Papers, (ACM, 2018), p. 200.

Chen, K.

F.-C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: Immersive computer graphics via factored near-eye light field displays with focus cues,” ACM Trans. Graph. 34(4), 60 (2015).
[Crossref]

Chen, Y.

Y. Chen and X. Ye, “Projection onto a simplex,” arXiv preprint arXiv:1101.6081 (2011).

Cho, J.

S. Lee, Y. Jo, D. Yoo, J. Cho, D. Lee, and B. Lee, “Tomographic near-eye displays,” Nat. Commun. 10, 2497 (2019).
[Crossref] [PubMed]

D. Kim, S. Lee, S. Moon, J. Cho, Y. Jo, and B. Lee, “Hybrid multi-layer displays providing accommodation cues,” Opt. Express 26(13), 17170–17184 (2018).
[Crossref] [PubMed]

S. Lee, C. Jang, S. Moon, J. Cho, and B. Lee, “Additive light field displays: realization of augmented reality with holographic optical elements,” ACM Trans. Graph. 35(4), 60 (2016).
[Crossref]

Chwesiuk, M.

H. Yu, M. Bemana, M. Wernikowski, M. Chwesiuk, O. T. Tursun, G. Singh, K. Myszkowski, R. Mantiuk, H.-P. Seidel, and P. Didyk, “A perception-driven hybrid decomposition for multi-layer accommodative displays,” IEEE Trans. Vis. Comput. Graph. 25(5), 1940–1950 (2019).
[Crossref] [PubMed]

Condat, L.

L. Condat, “A primal–dual splitting method for convex optimization involving lipschitzian, proximable and linear composite terms,” J. Optim. Theory Appl. 158(2), 460–479 (2013).
[Crossref]

Davis, D.

D. Davis and W. Yin, “A three-operator splitting scheme and its optimization applications,” Set-valued and variational analysis 25(4), 829–858 (2017).
[Crossref]

Didyk, P.

H. Yu, M. Bemana, M. Wernikowski, M. Chwesiuk, O. T. Tursun, G. Singh, K. Myszkowski, R. Mantiuk, H.-P. Seidel, and P. Didyk, “A perception-driven hybrid decomposition for multi-layer accommodative displays,” IEEE Trans. Vis. Comput. Graph. 25(5), 1940–1950 (2019).
[Crossref] [PubMed]

Esser, E.

T. Goldstein, M. Li, X. Yuan, E. Esser, and R. Baraniuk, “Adaptive primal-dual hybrid gradient methods for saddle-point problems,” arXiv preprint arXiv:1305.0546 (2013).

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenom. 60(1–4), 259–268 (1992).
[Crossref]

Fix, A.

L. Xiao, A. Kaplanyan, A. Fix, M. Chapman, and D. Lanman, “Deepfocus: learned image synthesis for computational displays,” in SIGGRAPH Asia 2018 Technical Papers, (ACM, 2018), p. 200.

Footer, M.

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Figures (13)

Fig. 1
Fig. 1 Principle of tomographic displays. A tomographic display depicts a 3D scene by synchronizing fast spatially adjustable backlight and a focus-tunable lens. (a) When it renders the scene based on its depth information (binary blending), users may notice boundary artifacts since it merges the planes in an additive manner. It can be moderated with a scene decomposition algorithm (b) using SART; but there still remain some defects (red arrow), (c) using non-convex optimization algorithm. (d) The ground truth.
Fig. 2
Fig. 2 Illustration of artifacts which occur in tomographic displays. (a) As the viewpoint shifts, the user may notice black artifacts through the gap between voxels. (b) In real environment, the light from a rear plane is blocked by front objects. (c) In additive 3D displays, each plane cannot conceal the light from real planes.
Fig. 3
Fig. 3 Comparison on convergence rate with various blending techniques. Averaged PSNRs over 80 focal planes between 0.0D to 5.5D on scene A are evaluated after each iteration. We plot two graphs according to the initial condition based on (a) the RGB-D image and (b) uniform gray-level of 0.5. The continuous backlight values, which are represented as solid lines, are rounded using EPR after the 90th iteration. The dashed lines indicate PSNRs rendered with backlights after EPR at each iteration whereas the dotted lines represent roughly rounded ones.
Fig. 4
Fig. 4 Comparison of the HDR-VDP-2 metric for different blending methods applied to various scenes. We average the metric over the depth range on the right side. The color maps correspond to the probability of detection of differences between reconstructed and original ground truth focal stacks. The boundary artifacts are significantly decreased in PALM-based method. The numbers on the right side denote the number and range of focal slices used for each scene, according to Table 1. (Source image courtesy: "Interior Scene" and "SimplePoly Urban", www.cgtrader.com)
Fig. 5
Fig. 5 A reconstructed focal plane of scene A at 1.81D (top) and insets (bottom). The average PSNR is calculated from 80 synthesized retinal images within 5.5D and 0.0D. Insets highlight that our PALM-based algorithm recovers the 3D scene with the highest accuracy and correct occlusion boundaries (red arrows).
Fig. 6
Fig. 6 Comparison of experimental and simulation results from our prototype display with various blending methods on (a) scene B and (b) scene C. We captured the scenes with a CCD camera, of which focal distances are denoted at the topleft.
Fig. 7
Fig. 7 Optimizing the coefficient of H(x, Y). (a) Averaged PSNR over 80 focal planes between 0.0D and 5.5D is evaluated, as a function of initial and final values of α (αi and αf, respectively), for scene A. (b) For the points that arrows are pointing, we plotted a convergence graph. The dotted lines denote the metrics of focal planes rendered with rounded binary backlight each iteration using EPR, thus can be interpreted as the effective PSNRduring optimization. Here, we used resized images of 128×128 resolution with a simple GPU-based implementation and without intermediate rounding operations.
Fig. 8
Fig. 8 (a) Insets of scene D at 0.35 diopters, using binary blending method and PALM-based blending with β = 0.1,0.4 and M = 80. In binary blending, by turning on pixels on adjacent planes, the boundary artifacts are mitigated (orange box) whereas details are blurred (purple box). (b,c) Trade-off relationship in designing tomographic displays with the two methods. averaged PSNR over 80 focal planes is evaluated as a function of the number of layers M and brightness β for scene D and the parameters in Section 5.
Fig. 9
Fig. 9 Retinal contrast as a function of accommodation distance. Normalized contrast ratio is obtained for spatial frequency of 4, 6, and 8 cycles per degree (cpd) with samples collected within 1 cpd interval. The vertical dotted line indicates average depth of the patch based on depth-map.
Fig. 10
Fig. 10 Application of our non-convex method on the display proposed by Rathinavel et al. [3] for scene D. (a) Insets demonstrate that the PALM based method can correct boundary artifacts as well as color distortions. (b) Voxel-oriented decomposed images. We show images for their system without and with optimization. Note that higher brightness (t = 5) can be achieved with our method.
Fig. 11
Fig. 11 Rendered focal slices at 2.85D using the PALM-based method with λ = 0 and 3, on scene D. It demonstrates the effect of the TV regularization. By penalizing 2D gradient of backlight image, sparse backlights are obtained (bottom row) and the noises are suppressed (arrows).
Fig. 12
Fig. 12 (a) Binary, (b) occlusion, (c) DART-based, and (d) PALM-based blended RGB image and binary backlight sequences for tomographic displays on scene B.
Fig. 13
Fig. 13 (a) Binary, (b) occlusion, (c) DART-based, and (d) PALM-based blended RGB image and binary backlight sequences for tomographic displays on scene C.

Tables (8)

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Algorithm 1 Overall optimization scheme for tomographic displays

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Algorithm 2 Display update using the primal-dual algorithm

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Algorithm 3 Backlight update using PALM

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Algorithm 4 Proximal operator using primal-dual three-operator splitting [35]

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Table 1 Specification of rendered scenes

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Table 2 Average PSNR (dB) and SSIM of different optimization methods for tomographic displays on four scenes. We initialized values based on the RGB-D image from the center viewpoint.

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Algorithm 5 Backlight update using SART [12, 27]

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Algorithm 6 Backlight update using DART [20]

Equations (22)

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min x max y g ( x ) + Kx , y h * ( x ) ,
min x f ( x ) + g ( x ) + h ( Kx ) ,
min x , y Ψ ( x , y ) : = F ( x ) + G ( y ) + H ( x , y ) ,
S B : = { Y B M N 2 × 2 : Y i 1 + Y i 2 = 1 ,   f o r   i = 1 , M N 2 } ,   δ S ( x ) = { 0 i f   x S o t h e r w i s e .
minimize u , x Az t b 2 , s . t .   z = u M x ,   0 u M 1 ,   x B M N 2 .
Az = A ( u M x ) = AUx = K U x ,
  = A ( x u M ) = AX u r = K X u r .
minimize z Az b 2 = minimize u r AX u r b 2 , s . t .   0 u r 1.
f = 0 , g ( u ) = δ [ 0 , 1 ] ( u ) , h ( v ) = 1 2 v b 2 , K = AX .
prox γ g ( u ) = P [ 0 , 1 ] ( u ) = min  ( max  ( 0 , u ) , 1 ) ,
prox δ h ( u ) = ( u + δ b ) / ( 1 + δ ) .
minimize z 1 2 Az b 2 = minimize x 1 2 Kx b 2 , s . t .   x B M N 2 .
d ( x , B M N 2 ) : = min w B M N 2 x w 2 = i = 1 M N 2 min w i B ( x i w i ) 2 .
E b i n a r y ( x , Y ) : = d ( x , B M N 2 ) = min Y S B k = 1 2 i = 1 M N 2 Y i k 2 ( x i B k ) 2 .
E ( x , Y ) : = 1 2 Kx b 2 + δ [ 0 , 1 ] ( x ) F ( x )   +   δ S [ 0 , 1 ] ( Y ) G ( Y )   +   α 2 k = 1 2 i = 1 M N 2 Y i k 2 ( x i B k ) 2 H ( x , Y ) .
prox τ F ( p ) = argmin x 1 2 τ x p 2 + 1 2 AUx b 2 + δ [ 0 , 1 ] ( x ) ,
prox σ G ( Q ) = P S [ 0 , 1 ] ( Q ) .
f ( x ) = 1 2 τ x p 2 , g ( x ) = δ [ 0 , 1 ] ( x ) , h ( y ) = 1 2 y b 2 , K = AU .
prox τ F ( p ) = argmin x 1 2 τ x p 2 + 1 2 AUx b 2 + δ [ 0 , 1 ] ( x ) + λ | x | 1 .
e p r ( X ) = P S ( X ) , S = { W | s i j = m M W i j m and W i j m B , for 1 i , j N , 1 m M } .
γ = [ γ 1 , , γ M N 2 ] T , δ = [ δ 1 , , δ k 2 N 2 ] T ,   for   γ j = ( i = 1 k 2 N 2 K i j ) 1 , δ i = ( j = 1 M N 2 K i j ) 1 .
( x H ( x n , Y n ) ) i = α k = 1 2 Y i k 2 ( x i B k ) , ( Y H ( x n , Y n ) ) i k = α Y i k ( x i B k ) 2 .