Abstract

We demonstrate a compact, easy-to-build computational camera for single-shot three-dimensional (3D) imaging. Our lensless system consists solely of a diffuser placed in front of an image sensor. Every point within the volumetric field-of-view projects a unique pseudorandom pattern of caustics on the sensor. By using a physical approximation and simple calibration scheme, we solve the large-scale inverse problem in a computationally efficient way. The caustic patterns enable compressed sensing, which exploits sparsity in the sample to solve for more 3D voxels than pixels on the 2D sensor. Our 3D reconstruction grid is chosen to match the experimentally measured two-point optical resolution, resulting in 100 million voxels being reconstructed from a single 1.3 megapixel image. However, the effective resolution varies significantly with scene content. Because this effect is common to a wide range of computational cameras, we provide a new theory for analyzing resolution in such systems.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (4)

A. Singh, G. Pedrini, M. Takeda, and W. Osten, “Scatter-plate microscope for lensless microscopy with diffraction limited resolution,” Sci. Rep. 7, 10687 (2017).
[Crossref]

A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).
[Crossref]

Y. Kashter, A. Vijayakumar, and J. Rosen, “Resolving images by blurring: superresolution method with a scattering mask between the observed objects and the hologram recorder,” Optica 4, 932–939 (2017).
[Crossref]

A. Sinha, J. Lee, S. Li, and G. Barbastathis, “Lensless computational imaging through deep learning,” Optica 4, 1117–1125 (2017).
[Crossref]

2016 (3)

N. C. Pégard, H.-Y. Liu, N. Antipa, M. Gerlock, H. Adesnik, and L. Waller, “Compressive light-field microscopy for 3D neural activity recording,” Optica 3, 517–524 (2016).
[Crossref]

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6, 33558 (2016).
[Crossref]

K. Lee and Y. Park, “Exploiting the speckle-correlation scattering matrix for a compact reference-free holographic image sensor,” Nat. Commun. 7, 13359 (2016).
[Crossref]

2015 (1)

2014 (6)

A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Looking through a diffuser and around an opaque surface: a holographic approach,” Opt. Express 22, 7694–7701 (2014).
[Crossref]

W. Harm, C. Roider, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Lensless imaging through thin diffusive media,” Opt. Express 22, 22146–22156 (2014).
[Crossref]

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[Crossref]

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref]

D. G. Stork and P. R. Gill, “Optical, mathematical, and computational foundations of lensless ultra-miniature diffractive imagers and sensors,” Int. J. Adv. Syst. Meas. 7, 201–208 (2014).

F. Krahmer, S. Mendelson, and H. Rauhut, “Suprema of chaos processes and the restricted isometry property,” Commun. Pur. Appl. Math. 67, 1877–1904 (2014).
[Crossref]

2013 (3)

M. S. C. Almeida and M. Figueiredo, “Deconvolving images with unknown boundaries using the alternating direction method of multipliers,” IEEE Trans. Image Process. 22, 3074–3086 (2013).
[Crossref]

A. Matakos, S. Ramani, and J. A. Fessler, “Accelerated edge-preserving image restoration without boundary artifacts,” IEEE Trans. Image Process. 22, 2019–2029 (2013).
[Crossref]

M. Broxton, L. Grosenick, S. Yang, N. Cohen, A. Andalman, K. Deisseroth, and M. Levoy, “Wave optics theory and 3-D deconvolution for the light field microscope,” Opt. Express 21, 25418–25439 (2013).
[Crossref]

2011 (1)

2010 (3)

W. Bishara, T.-W. Su, A. F. Coskun, and A. Ozcan, “Lensfree on-chip microscopy over a wide field-of-view using pixel super-resolution,” Opt. Express 18, 11181–11191 (2010).
[Crossref]

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[Crossref]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

2009 (2)

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
[Crossref]

D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[Crossref]

2008 (4)

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imag. Sci. 1, 248–272 (2008).
[Crossref]

T. F. Holekamp, D. Turaga, and T. E. Holy, “Fast three-dimensional fluorescence imaging of activity in neural populations by objective-coupled planar illumination microscopy,” Neuron 57, 661–672 (2008).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

2007 (1)

R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, “Three-dimensional information acquisition using a compound imaging system,” Opt. Rev. 14, 347–350 (2007).
[Crossref]

2004 (1)

H. Faulkner and J. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

2001 (1)

1992 (1)

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

1990 (1)

W. Denk, J. Strickler, and W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref]

1988 (1)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[Crossref]

Adams, A.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in ACM Trans. Graph. (Proc. SIGGRAPH) (2006), Vol. 25.

J. Ragan-Kelley, C. Barnes, A. Adams, S. Paris, F. Durand, and S. Amarasinghe, “Halide: a language and compiler for optimizing parallelism, locality, and recomputation in image processing pipelines,” in ACM SIGPLAN Notices (2013), Vol. 48, pp. 519–530.

Adesnik, H.

Afonso, M. V.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[Crossref]

Almeida, M. S. C.

M. S. C. Almeida and M. Figueiredo, “Deconvolving images with unknown boundaries using the alternating direction method of multipliers,” IEEE Trans. Image Process. 22, 3074–3086 (2013).
[Crossref]

Amarasinghe, S.

J. Ragan-Kelley, C. Barnes, A. Adams, S. Paris, F. Durand, and S. Amarasinghe, “Halide: a language and compiler for optimizing parallelism, locality, and recomputation in image processing pipelines,” in ACM SIGPLAN Notices (2013), Vol. 48, pp. 519–530.

Andalman, A.

Antipa, N.

N. C. Pégard, H.-Y. Liu, N. Antipa, M. Gerlock, H. Adesnik, and L. Waller, “Compressive light-field microscopy for 3D neural activity recording,” Optica 3, 517–524 (2016).
[Crossref]

N. Antipa, S. Necula, R. Ng, and L. Waller, “Single-shot diffuser-encoded light field imaging,” in IEEE International Conference on Computational Photography (ICCP) (2016), pp. 1–11.

G. Kuo, N. Antipa, R. Ng, and L. Waller, “DiffuserCam: diffuser-based lensless cameras,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–2.

Asif, M. S.

M. S. Asif, A. Ayremlou, A. Veeraraghavan, R. Baraniuk, and A. Sankaranarayanan, “Flatcam: replacing lenses with masks and computation,” in IEEE International Conference on Computer Vision Workshop (ICCVW) (IEEE, 2015), pp. 663–666.

Ayremlou, A.

M. S. Asif, A. Ayremlou, A. Veeraraghavan, R. Baraniuk, and A. Sankaranarayanan, “Flatcam: replacing lenses with masks and computation,” in IEEE International Conference on Computer Vision Workshop (ICCVW) (IEEE, 2015), pp. 663–666.

Baraniuk, R.

M. S. Asif, A. Ayremlou, A. Veeraraghavan, R. Baraniuk, and A. Sankaranarayanan, “Flatcam: replacing lenses with masks and computation,” in IEEE International Conference on Computer Vision Workshop (ICCVW) (IEEE, 2015), pp. 663–666.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Barbastathis, G.

Barnes, C.

J. Ragan-Kelley, C. Barnes, A. Adams, S. Paris, F. Durand, and S. Amarasinghe, “Halide: a language and compiler for optimizing parallelism, locality, and recomputation in image processing pipelines,” in ACM SIGPLAN Notices (2013), Vol. 48, pp. 519–530.

Beck, A.

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
[Crossref]

Bernet, S.

Bioucas-Dias, J. M.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[Crossref]

Bishara, W.

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

Brady, D.

Bredif, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” , Stanford University, 2005), pp. 3418–3421.

Broxton, M.

Candès, E. J.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

Carron, I.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref]

Chardon, G.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref]

Chi, W.

Choi, K.

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

Cohen, N.

Coskun, A. F.

Daudet, L.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
[Crossref]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Deisseroth, K.

Denk, W.

W. Denk, J. Strickler, and W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Durand, F.

J. Ragan-Kelley, C. Barnes, A. Adams, S. Paris, F. Durand, and S. Amarasinghe, “Halide: a language and compiler for optimizing parallelism, locality, and recomputation in image processing pipelines,” in ACM SIGPLAN Notices (2013), Vol. 48, pp. 519–530.

Duval, G.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” , Stanford University, 2005), pp. 3418–3421.

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

Edrei, E.

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6, 33558 (2016).
[Crossref]

Erickson, E.

P. R. Gill, J. Tringali, A. Schneider, S. Kabir, D. G. Stork, E. Erickson, and M. Kellam, “Thermal escher sensors: pixel-efficient lensless imagers based on tiled optics,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–3.

Fatemi, E.

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

Faulkner, H.

H. Faulkner and J. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref]

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[Crossref]

Fergus, R.

R. Fergus, A. Torralba, and W. T. Freeman, “Random lens imaging,” (Massachusetts Institute of Technology, 2006).

Fessler, J. A.

A. Matakos, S. Ramani, and J. A. Fessler, “Accelerated edge-preserving image restoration without boundary artifacts,” IEEE Trans. Image Process. 22, 2019–2029 (2013).
[Crossref]

Figueiredo, M.

M. S. C. Almeida and M. Figueiredo, “Deconvolving images with unknown boundaries using the alternating direction method of multipliers,” IEEE Trans. Image Process. 22, 3074–3086 (2013).
[Crossref]

Figueiredo, M. A. T.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[Crossref]

Fink, M.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[Crossref]

Footer, M.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in ACM Trans. Graph. (Proc. SIGGRAPH) (2006), Vol. 25.

Freeman, W. T.

R. Fergus, A. Torralba, and W. T. Freeman, “Random lens imaging,” (Massachusetts Institute of Technology, 2006).

George, N.

Gerlock, M.

Gigan, S.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
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A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
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D. G. Stork and P. R. Gill, “Optical, mathematical, and computational foundations of lensless ultra-miniature diffractive imagers and sensors,” Int. J. Adv. Syst. Meas. 7, 201–208 (2014).

P. R. Gill, J. Tringali, A. Schneider, S. Kabir, D. G. Stork, E. Erickson, and M. Kellam, “Thermal escher sensors: pixel-efficient lensless imagers based on tiled optics,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–3.

Grosenick, L.

Hanrahan, P.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” , Stanford University, 2005), pp. 3418–3421.

Harm, W.

Heidmann, P.

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
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T. F. Holekamp, D. Turaga, and T. E. Holy, “Fast three-dimensional fluorescence imaging of activity in neural populations by objective-coupled planar illumination microscopy,” Neuron 57, 661–672 (2008).
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T. F. Holekamp, D. Turaga, and T. E. Holy, “Fast three-dimensional fluorescence imaging of activity in neural populations by objective-coupled planar illumination microscopy,” Neuron 57, 661–672 (2008).
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R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, “Three-dimensional information acquisition using a compound imaging system,” Opt. Rev. 14, 347–350 (2007).
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R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” , Stanford University, 2005), pp. 3418–3421.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in ACM Trans. Graph. (Proc. SIGGRAPH) (2006), Vol. 25.

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K. Tajima, T. Shimano, Y. Nakamura, M. Sao, and T. Hoshizawa, “Lensless light-field imaging with multi-phased Fresnel zone aperture,” in IEEE International Conference on Computational Photography (ICCP) (2017), pp. 76–82.

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R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, “Three-dimensional information acquisition using a compound imaging system,” Opt. Rev. 14, 347–350 (2007).
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Jesacher, A.

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P. R. Gill, J. Tringali, A. Schneider, S. Kabir, D. G. Stork, E. Erickson, and M. Kellam, “Thermal escher sensors: pixel-efficient lensless imagers based on tiled optics,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–3.

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A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
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O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
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P. R. Gill, J. Tringali, A. Schneider, S. Kabir, D. G. Stork, E. Erickson, and M. Kellam, “Thermal escher sensors: pixel-efficient lensless imagers based on tiled optics,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–3.

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
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G. Kuo, N. Antipa, R. Ng, and L. Waller, “DiffuserCam: diffuser-based lensless cameras,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–2.

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M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
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K. Lee and Y. Park, “Exploiting the speckle-correlation scattering matrix for a compact reference-free holographic image sensor,” Nat. Commun. 7, 13359 (2016).
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S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
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Li, S.

Lim, S.

Liu, H.-Y.

Liutkus, A.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
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Martina, D.

A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
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N. Antipa, S. Necula, R. Ng, and L. Waller, “Single-shot diffuser-encoded light field imaging,” in IEEE International Conference on Computational Photography (ICCP) (2016), pp. 1–11.

M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in ACM Trans. Graph. (Proc. SIGGRAPH) (2006), Vol. 25.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” , Stanford University, 2005), pp. 3418–3421.

G. Kuo, N. Antipa, R. Ng, and L. Waller, “DiffuserCam: diffuser-based lensless cameras,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–2.

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R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, “Three-dimensional information acquisition using a compound imaging system,” Opt. Rev. 14, 347–350 (2007).
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L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
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A. Singh, G. Pedrini, M. Takeda, and W. Osten, “Scatter-plate microscope for lensless microscopy with diffraction limited resolution,” Sci. Rep. 7, 10687 (2017).
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A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).
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A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Looking through a diffuser and around an opaque surface: a holographic approach,” Opt. Express 22, 7694–7701 (2014).
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K. Lee and Y. Park, “Exploiting the speckle-correlation scattering matrix for a compact reference-free holographic image sensor,” Nat. Commun. 7, 13359 (2016).
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A. Singh, G. Pedrini, M. Takeda, and W. Osten, “Scatter-plate microscope for lensless microscopy with diffraction limited resolution,” Sci. Rep. 7, 10687 (2017).
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A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).
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A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Looking through a diffuser and around an opaque surface: a holographic approach,” Opt. Express 22, 7694–7701 (2014).
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Pégard, N. C.

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
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A. Stylianou and R. Pless, “Sparklegeometry: glitter imaging for 3D point tracking,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops (2016), pp. 10–17.

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A. Liutkus, D. Martina, S. Popoff, G. Chardon, O. Katz, G. Lerosey, S. Gigan, L. Daudet, and I. Carron, “Imaging with nature: compressive imaging using a multiply scattering medium,” Sci. Rep. 4, 5552 (2014).
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J. Ragan-Kelley, C. Barnes, A. Adams, S. Paris, F. Durand, and S. Amarasinghe, “Halide: a language and compiler for optimizing parallelism, locality, and recomputation in image processing pipelines,” in ACM SIGPLAN Notices (2013), Vol. 48, pp. 519–530.

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A. Matakos, S. Ramani, and J. A. Fessler, “Accelerated edge-preserving image restoration without boundary artifacts,” IEEE Trans. Image Process. 22, 2019–2029 (2013).
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F. Krahmer, S. Mendelson, and H. Rauhut, “Suprema of chaos processes and the restricted isometry property,” Commun. Pur. Appl. Math. 67, 1877–1904 (2014).
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L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
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M. S. Asif, A. Ayremlou, A. Veeraraghavan, R. Baraniuk, and A. Sankaranarayanan, “Flatcam: replacing lenses with masks and computation,” in IEEE International Conference on Computer Vision Workshop (ICCVW) (IEEE, 2015), pp. 663–666.

Sao, M.

K. Tajima, T. Shimano, Y. Nakamura, M. Sao, and T. Hoshizawa, “Lensless light-field imaging with multi-phased Fresnel zone aperture,” in IEEE International Conference on Computational Photography (ICCP) (2017), pp. 76–82.

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E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6, 33558 (2016).
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P. R. Gill, J. Tringali, A. Schneider, S. Kabir, D. G. Stork, E. Erickson, and M. Kellam, “Thermal escher sensors: pixel-efficient lensless imagers based on tiled optics,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–3.

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K. Tajima, T. Shimano, Y. Nakamura, M. Sao, and T. Hoshizawa, “Lensless light-field imaging with multi-phased Fresnel zone aperture,” in IEEE International Conference on Computational Photography (ICCP) (2017), pp. 76–82.

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A. Singh, G. Pedrini, M. Takeda, and W. Osten, “Scatter-plate microscope for lensless microscopy with diffraction limited resolution,” Sci. Rep. 7, 10687 (2017).
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A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).
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A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Looking through a diffuser and around an opaque surface: a holographic approach,” Opt. Express 22, 7694–7701 (2014).
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S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
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D. G. Stork and P. R. Gill, “Optical, mathematical, and computational foundations of lensless ultra-miniature diffractive imagers and sensors,” Int. J. Adv. Syst. Meas. 7, 201–208 (2014).

P. R. Gill, J. Tringali, A. Schneider, S. Kabir, D. G. Stork, E. Erickson, and M. Kellam, “Thermal escher sensors: pixel-efficient lensless imagers based on tiled optics,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–3.

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A. Stylianou and R. Pless, “Sparklegeometry: glitter imaging for 3D point tracking,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops (2016), pp. 10–17.

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Sun, T.

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
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K. Tajima, T. Shimano, Y. Nakamura, M. Sao, and T. Hoshizawa, “Lensless light-field imaging with multi-phased Fresnel zone aperture,” in IEEE International Conference on Computational Photography (ICCP) (2017), pp. 76–82.

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M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
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A. Singh, G. Pedrini, M. Takeda, and W. Osten, “Scatter-plate microscope for lensless microscopy with diffraction limited resolution,” Sci. Rep. 7, 10687 (2017).
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A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).
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A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Looking through a diffuser and around an opaque surface: a holographic approach,” Opt. Express 22, 7694–7701 (2014).
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P. R. Gill, J. Tringali, A. Schneider, S. Kabir, D. G. Stork, E. Erickson, and M. Kellam, “Thermal escher sensors: pixel-efficient lensless imagers based on tiled optics,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–3.

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T. F. Holekamp, D. Turaga, and T. E. Holy, “Fast three-dimensional fluorescence imaging of activity in neural populations by objective-coupled planar illumination microscopy,” Neuron 57, 661–672 (2008).
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M. S. Asif, A. Ayremlou, A. Veeraraghavan, R. Baraniuk, and A. Sankaranarayanan, “Flatcam: replacing lenses with masks and computation,” in IEEE International Conference on Computer Vision Workshop (ICCVW) (IEEE, 2015), pp. 663–666.

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G. Kuo, N. Antipa, R. Ng, and L. Waller, “DiffuserCam: diffuser-based lensless cameras,” in Computational Optical Sensing and Imaging (Optical Society of America, 2017), paper CTu3B–2.

N. Antipa, S. Necula, R. Ng, and L. Waller, “Single-shot diffuser-encoded light field imaging,” in IEEE International Conference on Computational Photography (ICCP) (2016), pp. 1–11.

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Appl. Opt. (1)

Commun. Pur. Appl. Math. (1)

F. Krahmer, S. Mendelson, and H. Rauhut, “Suprema of chaos processes and the restricted isometry property,” Commun. Pur. Appl. Math. 67, 1877–1904 (2014).
[Crossref]

Found. Trends Mach. Learn. (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[Crossref]

IEEE Signal Process. Mag. (2)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
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IEEE Trans. Image Process. (4)

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A. Matakos, S. Ramani, and J. A. Fessler, “Accelerated edge-preserving image restoration without boundary artifacts,” IEEE Trans. Image Process. 22, 2019–2029 (2013).
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A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
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Int. J. Adv. Syst. Meas. (1)

D. G. Stork and P. R. Gill, “Optical, mathematical, and computational foundations of lensless ultra-miniature diffractive imagers and sensors,” Int. J. Adv. Syst. Meas. 7, 201–208 (2014).

Light Sci. Appl. (1)

A. Singh, D. Naik, G. Pedrini, M. Takeda, and W. Osten, “Exploiting scattering media for exploring 3D objects,” Light Sci. Appl. 6, e16219 (2017).
[Crossref]

Nat. Commun. (1)

K. Lee and Y. Park, “Exploiting the speckle-correlation scattering matrix for a compact reference-free holographic image sensor,” Nat. Commun. 7, 13359 (2016).
[Crossref]

Nat. Photonics (1)

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
[Crossref]

Neuron (1)

T. F. Holekamp, D. Turaga, and T. E. Holy, “Fast three-dimensional fluorescence imaging of activity in neural populations by objective-coupled planar illumination microscopy,” Neuron 57, 661–672 (2008).
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Opt. Express (7)

Opt. Rev. (1)

R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, “Three-dimensional information acquisition using a compound imaging system,” Opt. Rev. 14, 347–350 (2007).
[Crossref]

Optica (3)

Phys. Rev. Lett. (2)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61, 834–837 (1988).
[Crossref]

H. Faulkner and J. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
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Physica D (1)

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

Sci. Rep. (3)

E. Edrei and G. Scarcelli, “Memory-effect based deconvolution microscopy for super-resolution imaging through scattering media,” Sci. Rep. 6, 33558 (2016).
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Supplementary Material (1)

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Figures (7)

Fig. 1.
Fig. 1. DiffuserCam setup and reconstruction pipeline. Our lensless system consists of a diffuser placed in front of a sensor (bumps on the diffuser are exaggerated for illustration). The system encodes a 3D scene into a 2D image on the sensor. A one-time calibration consists of scanning a point source axially while capturing images. Images are reconstructed computationally by solving a nonlinear inverse problem with a sparsity prior. The result is a 3D image reconstructed from a single 2D measurement.

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Fig. 2.
Fig. 2. Caustic pattern shifts with lateral shifts of a point source in the scene and scales with axial shifts. (a) Ray-traced renderings of caustics as a point source moves laterally. For large shifts, part of the pattern is clipped by the sensor. (b) The caustics magnify as the source is brought closer.

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Fig. 3.
Fig. 3. Experimentally determined field-of-view (FoV) and resolution. (a) System architecture with design parameters. (b) Angular pixel response of our sensor. We define the angular cutoff (αc) as the angle at which the response falls to 20%. (c) Reconstructed images of two points (captured separately) at varying separations laterally and axially, near the z=20  mm depth plane. Points are considered resolved if they are separated by a dip of at least 20%. (d) To-scale nonuniform voxel grid for 3D reconstruction. The chosen voxel grid is based on the system geometry and Nyquist-sampled two-point resolution over the entire FoV. For visualization purposes, each box represents 20×20 voxels, as shown in red.

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Fig. 4.
Fig. 4. Our computational camera has object-dependent performance, such that the resolution depends on the number of points. (a) To illustrate, we show here a situation with two points successfully resolved at the two-point resolution limit (Δx,Δz)=(45  μm,336  μm) at a depth of approximately 20 mm. (c) When the object consists of more points (16 points in a 4×4 grid in the x–z plane) at the same spacing, however, the reconstruction fails. (b) and (d) Increasing the separation to (Δx,Δz)=(75  μm,448  μm) gives successful reconstructions. (e) and (f) A close-up of the raw data shows noticeable splitting of the caustic lines for the 16-point case, making the points distinguishable. Heuristically, the 16-point resolution cutoff is a good indicator of resolution for real-world objects.

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Fig. 5.
Fig. 5. Our local condition number theory shows how the resolution varies with the object complexity. (a) Virtual point sources are simulated on a fixed grid and moved by integer numbers of voxels to change the separation distance. (b) Local condition numbers are plotted for sub-matrices corresponding to grids of neighboring point sources with varying separation (at a depth 20 mm from the sensor). As the number of sources increases, the condition number approaches a limit, indicating that resolution for complex objects can be approximated by a limited number (but more than two) sources.

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Fig. 6.
Fig. 6. Experimental validation of the convolution model. (a)–(c) Close-ups of registered experimental PSFs for sources at 0°, 15°, and 30°. The PSF at 15° is visually similar to that on-axis, while the PSF at 30° has subtle differences. (d) Inner product between the on-axis PSF and registered off-axis PSFs as a function of source position. (e) Resulting spot size (normalized by on-axis spot). The convolution model holds well up to ±15°, beyond which resolution degrades (solid). Exhaustive calibration would improve the resolution (dashed), at the expense of complexity in computation and calibration.

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Fig. 7.
Fig. 7. Experimental 3D reconstructions. (a) Tilted resolution target, which was reconstructed on a 4.2 MP lateral grid with 128 z-planes and cropped to 640×640×50 voxels. The large panel shows the max projection over z. Note that the spatial scale is not isotropic. Inset is a magnification of group 2 with an intensity cutline, showing that we resolve element 5 at a distance of 24 mm, which corresponds to a feature size of 79 μm (approximately twice the lateral voxel size of 35 μm at this depth). The degraded resolution matches our 16-point distinguishability (75 μm at 20 mm depth). Lower panels show depth slices from the recovered volume. (b) Reconstruction of a small plant, cropped to 480×320×128 voxels, rendered from multiple angles.

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Equations (9)

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b=Hv,
v^=argminv≥0 12‖b−Hv‖22+τ‖Ψv‖1.
b(x′,y′)=∑(x,y,z)v(x,y,z)h(x′,y′;x,y,z).
b(x′,y′)=∑z∑(x,y)v(x,y,z)h(x′+mx,y′+my;z)=C∑z[v(−x′m,−y′m,z)*h(x′,y′;z)].
v^=argminw≥0,u,v 12‖b−Dv‖22+τ‖u‖1s.t.  v=Mv,u=Ψv,w=v,
uk+1←Tτμ2(Ψvk+ηk/μ2)vk+1←(D⊺D+μ1I)−1(ξk+μ1Mvk+D⊺b)wk+1←max(ρk/μ3+vk,0)vk+1←(μ1M⊺M+μ2Ψ⊺Ψ+μ3I)−1rkξk+1←ξk+μ1(Mvk+1−vk+1)ηk+1←ηk+μ2(Ψvk+1−uk+1)ρk+1←ρk+μ3(vk+1−wk+1),
rk=(μ3wk+1−ρk)+Ψ⊺(μ2uk+1−ηk)+M⊺(μ1vk+1−ξk).
FoV=β+min[αc,tan−1(l+w2d)].
c=1z1−1z2.

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