Abstract

Computational ghost imaging relies on the decomposition of an image into patterns that are summed together with weights that measure the overlap of each pattern with the scene being imaged. These tasks rely on a computer. Here we demonstrate that the computational integration can be performed directly with the human eye. This builds upon the known persistence time of the human eye and we use our ghost imaging approach as an alternative to evaluate the temporal response of the eye. We verify that the image persistence time is of order 20 ms, followed by a further 20 ms exponential decay. These persistence times are consistent with previous studies but can now potentially be extended to include a more precise characterisation of visual stimuli and provide a new experimental tool for the study of visual perception.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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  23. P. Lincoln, A. Blate, M. Singh, T. Whitted, A. State, A. Lastra, and H. Fuchs, “From motion to photons in 80 microseconds: Towards minimal latency for virtual and augmented reality,” IEEE Trans. Vis. Comput. Graphics 22, 1367–1376 (2016).
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2017 (2)

M. J. Padgett and R. W. Boyd, “An introduction to ghost imaging: quantum and classical,” Philos. Trans. R. Soc. London A 375, 2099 (2017).
[Crossref]

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref] [PubMed]

2016 (3)

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Ghost imaging in the time domain,” Nat. Photonics 10, 167 (2016).
[Crossref]

F. Devaux, P.-A. Moreau, S. Denis, and E. Lantz, “Computational temporal ghost imaging,” Optica 3, 698–701 (2016).
[Crossref]

P. Lincoln, A. Blate, M. Singh, T. Whitted, A. State, A. Lastra, and H. Fuchs, “From motion to photons in 80 microseconds: Towards minimal latency for virtual and augmented reality,” IEEE Trans. Vis. Comput. Graphics 22, 1367–1376 (2016).
[Crossref] [PubMed]

2015 (2)

2011 (2)

2010 (1)

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

2009 (1)

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[Crossref]

2008 (1)

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[Crossref]

2007 (1)

C. Koch and N. Tsuchiya, “Attention and consciousness: two distinct brain processes,” Trends Cognit. Sci. 11, 16–22 (2007).
[Crossref]

2004 (1)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

2003 (1)

F. Crick and C. Koch, “A framework for consciusness,” Nat. Neuroscience 6, 119 (2003).
[Crossref]

2002 (1)

J. R. Brockmole, R. F. Wang, and D. E. Irwin, “Temporal integration between visual images and visual percepts,” J. Exp. Psychol. Hum. Percept. Perform. 28, 315–334 (2002).

1998 (1)

G. Loftus and D. Irwin, “On the relations among different measures of visible and informational persistence,” Cogn. Psychol. 35, 135–199 (1998).
[Crossref] [PubMed]

1995 (1)

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref]

1980 (2)

V. D. Lollo, “Temporal integration in visual memory,” J. Exp. Psychol. 109, 75–97 (1980).
[Crossref]

M. Coltheart, “Iconic memory and visible persistence,” Percep. Psychophys. 27, 183–228 (1980).
[Crossref]

1970 (1)

R. Efron, “The relationship between the duration of a stimulus and the duration of a perception,” Neuropsychologia 8, 37–55 (1970).
[Crossref] [PubMed]

1967 (1)

C. Eriksen and J. Collins, “Some temporal characteristics of visual pattern perception,” J. Exp. Psychology 74, 476–484 (1967).

Aspden, R. S.

Bache, M.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

Barbier, M.

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Ghost imaging in the time domain,” Nat. Photonics 10, 167 (2016).
[Crossref]

Barnett, S. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref] [PubMed]

Blate, A.

P. Lincoln, A. Blate, M. Singh, T. Whitted, A. State, A. Lastra, and H. Fuchs, “From motion to photons in 80 microseconds: Towards minimal latency for virtual and augmented reality,” IEEE Trans. Vis. Comput. Graphics 22, 1367–1376 (2016).
[Crossref] [PubMed]

P. Lincoln, A. Blate, M. Singh, A. State, M. Whitton, T. Whitted, and H. Fuchs, “Scene-adaptive high dynamic range display for low latency augmented reality,” in “Proceedings of the 21st ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games,” (ACM, 2017), I3D 17.

Block, N.

N. Block, “The puzzle of perceptual precision,” in “Open MIND,”, T. K. Metzinger and J. M. Windt, eds., (MIND Group, 2015), chap. 5.

Boyd, R. W.

Brambilla, E.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

Brockmole, J. R.

J. R. Brockmole, R. F. Wang, and D. E. Irwin, “Temporal integration between visual images and visual percepts,” J. Exp. Psychol. Hum. Percept. Perform. 28, 315–334 (2002).

Bromberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[Crossref]

Buller, G. S.

Chen, P.-X.

Collins, J.

C. Eriksen and J. Collins, “Some temporal characteristics of visual pattern perception,” J. Exp. Psychology 74, 476–484 (1967).

Coltheart, M.

M. Coltheart, “Iconic memory and visible persistence,” Percep. Psychophys. 27, 183–228 (1980).
[Crossref]

Crick, F.

F. Crick and C. Koch, “A framework for consciusness,” Nat. Neuroscience 6, 119 (2003).
[Crossref]

Dai, H.-Y.

Deacon, K. S.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98, 111115 (2011).
[Crossref]

Denis, S.

Devaux, F.

Dixon, P. B.

Dudley, J. M.

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Ghost imaging in the time domain,” Nat. Photonics 10, 167 (2016).
[Crossref]

Edgar, M. P.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref] [PubMed]

Efron, R.

R. Efron, “The relationship between the duration of a stimulus and the duration of a perception,” Neuropsychologia 8, 37–55 (1970).
[Crossref] [PubMed]

Eriksen, C.

C. Eriksen and J. Collins, “Some temporal characteristics of visual pattern perception,” J. Exp. Psychology 74, 476–484 (1967).

Friberg, A. T.

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Ghost imaging in the time domain,” Nat. Photonics 10, 167 (2016).
[Crossref]

Fuchs, H.

P. Lincoln, A. Blate, M. Singh, T. Whitted, A. State, A. Lastra, and H. Fuchs, “From motion to photons in 80 microseconds: Towards minimal latency for virtual and augmented reality,” IEEE Trans. Vis. Comput. Graphics 22, 1367–1376 (2016).
[Crossref] [PubMed]

P. Lincoln, A. Blate, M. Singh, A. State, M. Whitton, T. Whitted, and H. Fuchs, “Scene-adaptive high dynamic range display for low latency augmented reality,” in “Proceedings of the 21st ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games,” (ACM, 2017), I3D 17.

Gatti, A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

Gemmell, N. R.

Genty, G.

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Ghost imaging in the time domain,” Nat. Photonics 10, 167 (2016).
[Crossref]

Gibson, G. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref] [PubMed]

Gong, W.

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Hadfield, R. H.

Han, S.

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Howell, J. C.

Howland, G. A.

Irwin, D.

G. Loftus and D. Irwin, “On the relations among different measures of visible and informational persistence,” Cogn. Psychol. 35, 135–199 (1998).
[Crossref] [PubMed]

Irwin, D. E.

J. R. Brockmole, R. F. Wang, and D. E. Irwin, “Temporal integration between visual images and visual percepts,” J. Exp. Psychol. Hum. Percept. Perform. 28, 315–334 (2002).

Katz, O.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[Crossref]

Kirkwood, R. A.

Koch, C.

C. Koch and N. Tsuchiya, “Attention and consciousness: two distinct brain processes,” Trends Cognit. Sci. 11, 16–22 (2007).
[Crossref]

F. Crick and C. Koch, “A framework for consciusness,” Nat. Neuroscience 6, 119 (2003).
[Crossref]

Lantz, E.

Lastra, A.

P. Lincoln, A. Blate, M. Singh, T. Whitted, A. State, A. Lastra, and H. Fuchs, “From motion to photons in 80 microseconds: Towards minimal latency for virtual and augmented reality,” IEEE Trans. Vis. Comput. Graphics 22, 1367–1376 (2016).
[Crossref] [PubMed]

Li, Q.

Lincoln, P.

P. Lincoln, A. Blate, M. Singh, T. Whitted, A. State, A. Lastra, and H. Fuchs, “From motion to photons in 80 microseconds: Towards minimal latency for virtual and augmented reality,” IEEE Trans. Vis. Comput. Graphics 22, 1367–1376 (2016).
[Crossref] [PubMed]

P. Lincoln, A. Blate, M. Singh, A. State, M. Whitton, T. Whitted, and H. Fuchs, “Scene-adaptive high dynamic range display for low latency augmented reality,” in “Proceedings of the 21st ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games,” (ACM, 2017), I3D 17.

Liu, W.-T.

Loftus, G.

G. Loftus and D. Irwin, “On the relations among different measures of visible and informational persistence,” Cogn. Psychol. 35, 135–199 (1998).
[Crossref] [PubMed]

Lollo, V. D.

V. D. Lollo, “Temporal integration in visual memory,” J. Exp. Psychol. 109, 75–97 (1980).
[Crossref]

Lugiato, L. A.

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classicalcorrelation,” Phys. Rev. Lett. 93, 093602 (2004).
[Crossref]

Mertens, L.

Meyers, R. E.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98, 111115 (2011).
[Crossref]

Moreau, P.-A.

Morris, P. A.

Padgett, M. J.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref] [PubMed]

M. J. Padgett and R. W. Boyd, “An introduction to ghost imaging: quantum and classical,” Philos. Trans. R. Soc. London A 375, 2099 (2017).
[Crossref]

R. S. Aspden, N. R. Gemmell, P. A. Morris, D. S. Tasca, L. Mertens, M. G. Tanner, R. A. Kirkwood, A. Ruggeri, A. Tosi, R. W. Boyd, G. S. Buller, R. H. Hadfield, and M. J. Padgett, “Photon-sparse microscopy: visible light imaging using infrared illumination,” Optica 2, 1049–1052 (2015).
[Crossref]

Phillips, D. B.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref] [PubMed]

Pittman, T.

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref]

Ruggeri, A.

Ryczkowski, P.

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Ghost imaging in the time domain,” Nat. Photonics 10, 167 (2016).
[Crossref]

Sergienko, A.

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref]

Shapiro, J. H.

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[Crossref]

Shen, X.

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Shih, Y.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98, 111115 (2011).
[Crossref]

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref]

Silberberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[Crossref]

Singh, M.

P. Lincoln, A. Blate, M. Singh, T. Whitted, A. State, A. Lastra, and H. Fuchs, “From motion to photons in 80 microseconds: Towards minimal latency for virtual and augmented reality,” IEEE Trans. Vis. Comput. Graphics 22, 1367–1376 (2016).
[Crossref] [PubMed]

P. Lincoln, A. Blate, M. Singh, A. State, M. Whitton, T. Whitted, and H. Fuchs, “Scene-adaptive high dynamic range display for low latency augmented reality,” in “Proceedings of the 21st ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games,” (ACM, 2017), I3D 17.

State, A.

P. Lincoln, A. Blate, M. Singh, T. Whitted, A. State, A. Lastra, and H. Fuchs, “From motion to photons in 80 microseconds: Towards minimal latency for virtual and augmented reality,” IEEE Trans. Vis. Comput. Graphics 22, 1367–1376 (2016).
[Crossref] [PubMed]

P. Lincoln, A. Blate, M. Singh, A. State, M. Whitton, T. Whitted, and H. Fuchs, “Scene-adaptive high dynamic range display for low latency augmented reality,” in “Proceedings of the 21st ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games,” (ACM, 2017), I3D 17.

Strekalov, D.

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429 (1995).
[Crossref]

Sun, M.-J.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref] [PubMed]

Tanner, M. G.

Tasca, D. S.

Taylor, J. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3, e1601782 (2017).
[Crossref] [PubMed]

Tosi, A.

Tsuchiya, N.

C. Koch and N. Tsuchiya, “Attention and consciousness: two distinct brain processes,” Trends Cognit. Sci. 11, 16–22 (2007).
[Crossref]

Wang, R. F.

J. R. Brockmole, R. F. Wang, and D. E. Irwin, “Temporal integration between visual images and visual percepts,” J. Exp. Psychol. Hum. Percept. Perform. 28, 315–334 (2002).

Whitted, T.

P. Lincoln, A. Blate, M. Singh, T. Whitted, A. State, A. Lastra, and H. Fuchs, “From motion to photons in 80 microseconds: Towards minimal latency for virtual and augmented reality,” IEEE Trans. Vis. Comput. Graphics 22, 1367–1376 (2016).
[Crossref] [PubMed]

P. Lincoln, A. Blate, M. Singh, A. State, M. Whitton, T. Whitted, and H. Fuchs, “Scene-adaptive high dynamic range display for low latency augmented reality,” in “Proceedings of the 21st ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games,” (ACM, 2017), I3D 17.

Whitton, M.

P. Lincoln, A. Blate, M. Singh, A. State, M. Whitton, T. Whitted, and H. Fuchs, “Scene-adaptive high dynamic range display for low latency augmented reality,” in “Proceedings of the 21st ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games,” (ACM, 2017), I3D 17.

Xu, Y.-K.

Zhang, E.-F.

Zhang, P.

P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[Crossref]

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98, 111115 (2011).
[Crossref]

Cogn. Psychol. (1)

G. Loftus and D. Irwin, “On the relations among different measures of visible and informational persistence,” Cogn. Psychol. 35, 135–199 (1998).
[Crossref] [PubMed]

IEEE Trans. Vis. Comput. Graphics (1)

P. Lincoln, A. Blate, M. Singh, T. Whitted, A. State, A. Lastra, and H. Fuchs, “From motion to photons in 80 microseconds: Towards minimal latency for virtual and augmented reality,” IEEE Trans. Vis. Comput. Graphics 22, 1367–1376 (2016).
[Crossref] [PubMed]

J. Exp. Psychol. (1)

V. D. Lollo, “Temporal integration in visual memory,” J. Exp. Psychol. 109, 75–97 (1980).
[Crossref]

J. Exp. Psychol. Hum. Percept. Perform. (1)

J. R. Brockmole, R. F. Wang, and D. E. Irwin, “Temporal integration between visual images and visual percepts,” J. Exp. Psychol. Hum. Percept. Perform. 28, 315–334 (2002).

J. Exp. Psychology (1)

C. Eriksen and J. Collins, “Some temporal characteristics of visual pattern perception,” J. Exp. Psychology 74, 476–484 (1967).

Nat. Neuroscience (1)

F. Crick and C. Koch, “A framework for consciusness,” Nat. Neuroscience 6, 119 (2003).
[Crossref]

Nat. Photonics (1)

P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, and G. Genty, “Ghost imaging in the time domain,” Nat. Photonics 10, 167 (2016).
[Crossref]

Neuropsychologia (1)

R. Efron, “The relationship between the duration of a stimulus and the duration of a perception,” Neuropsychologia 8, 37–55 (1970).
[Crossref] [PubMed]

Opt. Express (1)

Optica (2)

Percep. Psychophys. (1)

M. Coltheart, “Iconic memory and visible persistence,” Percep. Psychophys. 27, 183–228 (1980).
[Crossref]

Philos. Trans. R. Soc. London A (1)

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Figures (5)

Fig. 1
Fig. 1 Experimental ghost imaging with the human eye. LED1 illuminates the DMD which projects Hadamard patterns at 20 kHz onto an object. The reflected light is collected by a single-pixel detector. The output modulates the intensity of LED2 which also illuminates the DMD and is subject to the same patterns as LED1. The intensity-weighted Hadamard patterns are viewed on the DMD by eye or projected onto a screen. Human vision integrates over the patterns when these are projected for much shorter durations than the eye’s persistence time. As a result, although only black and white patterns are projected, the eye effectively perceives a “ghost” image of the object.
Fig. 2
Fig. 2 Macro-pixel method for increased spatial resolution. The image on the left shows a computer simulation of the method with the original images divided in 64 macro-pixels, each of which is sampled with the same 256 Hadamard patterns. On the right we show the same image projected onto a screen and photographed with 50 ms exposure time.
Fig. 3
Fig. 3 Flow chart for estimating the response time of the human visual system.
Fig. 4
Fig. 4 (a) Decomposition of a single white pixel into a reduced set of Hadamard patterns (boxed in red). (b) Example of reduced set of Hadamard patterns for a 3x3 pixel “zero”. The red arrows indicate the sequence with which each subset of Hadamard patterns (and hence the relative white pixels from the image) are projected by the DMD.
Fig. 5
Fig. 5 Examples of simulated images used to evaluate visible persistence times. The top row shows the chose images (numbers and letters) together with the number of white pixels and the total number of Hadamard patterns required to represent the images following the recipe explained in Fig. 4. The following rows show the reconstructed images with varying exponential decay times (σ) of the function f ( t ) used to weigh the projection of each individual Hadamard pattern.

Equations (4)

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O ( x , y ) = i = 1 N A i H i ( x , y )
O ( x , y , t ) = R ( t t ) I ( x , y , t ) d t
O ( x , y ) = A ( t ) H ( x , y , t ) d t .
O ( x , y , t ) = δ ( t t ) A ( t ) H ( x , y , t ) d t ,

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