Abstract

Single-photon counting imaging is a novel computational imaging technique that counts every photon collected by reflective light; it has target detection capability under extremely low-light conditions and thus has elicited increasing research interest. However, a low single-photon counting number and considerable noise will significantly affect image quality under low-light conditions. To improve the quality of single-photon counting image efficiently, we propose a robust single-photon counting imaging method with spatially correlated and total variation (TV) constraints. A robust Poisson negative log-likelihood function is introduced as a data fidelity term, which is robust to some spatial points that have extremely large background count in real situations. The TV regularization constraint is adopted to reduce noise. Considering that the reflectivity of several spatially correlated points may be similar, we suggest adding another constraint based on the counting information from these points rather than a single point for estimating reflectivity in each pixel. This approach will be helpful in reducing truncation errors. The proposed imaging model is formulated on the basis of the aforementioned factors. The alternative direction multiplier method is used to solve the optimization problem. The superiority of the proposed method over state-of-the-art techniques is verified on simulated and real captured experimental datasets under different conditions.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (3)

S. Li, Z. Zhang, Y. Ma, H. Zeng, P. Zhao, and W. Zhang, “Ranging performance models based on negative-binomial (NB) distribution for photon-counting lidars,” Opt. Express 27(12), A861–A877 (2019).
[Crossref]

P. W. Connolly, X. Ren, R. K. Henderson, and G. S. Buller, “Hot pixel classification of single-photon avalanche diode detector arrays using a log-normal statistical distribution,” Electron. Lett. 55(18), 1004–1006 (2019).
[Crossref]

J. Ma, J. Zhao, J. Jiang, H. Zhou, and X. Guo, “Locality preserving matching,” Int. J. Comput. Vis. 127(5), 512–531 (2019).
[Crossref]

2018 (6)

D. B. Lindell, M. O’Toole, and G. Wetzstein, “Single-photon 3D imaging with deep sensor fusion,” ACM Trans. Graph. 37(4), 1–12 (2018).
[Crossref]

H. Fan, C. Li, Y. Guo, G. Kuang, and J. Ma, “Spatial–spectral total variation regularized low-rank tensor decomposition for hyperspectral image denoising,” IEEE Trans. Geosci. Remote Sensing 56(10), 6196–6213 (2018).
[Crossref]

S. Nie, C. Wang, X. Xi, S. Luo, G. Li, J. Tian, and H. Wang, “Estimating the vegetation canopy height using micro-pulse photon-counting LiDAR data,” Opt. Express 26(10), A520–A540 (2018).
[Crossref]

H. Ikoma, M. Broxton, T. Kudo, and G. Wetzstein, “A convex 3D deconvolution algorithm for low photon count fluorescence imaging,” Sci. Rep. 8(1), 11489 (2018).
[Crossref]

S. Liu, Z. Zhang, J. Zheng, L. Xu, C. Kuang, and X. Liu, “Parallelized fluorescence lifetime imaging microscopy (FLIM) based on photon reassignment,” Opt. Commun. 421, 83–89 (2018).
[Crossref]

F. Heide, S. Diamond, D. B. Lindell, and G. Wetzstein, “Sub-picosecond photon-efficient 3D imaging using single-photon sensors,” Sci. Rep. 8(1), 17726 (2018).
[Crossref]

2017 (3)

M. Itzler, G. Salzano, M. Entwistle, X. Jiang, M. Owens, B. Piccione, S. Wilton, K. Slomkowski, S. C. Roszko, and E. Wei, “Asynchronous Geiger-mode APD cameras with free-running InGaAsP pixels,” Proc. SPIE 10212, 102120K (2017).
[Crossref]

D. Shin, J. H. Shapiro, and V. K. Goyal, “Photon-efficient super-resolution laser radar,” Proc. SPIE 10394, 1039409 (2017).
[Crossref]

J. Jiang, J. Ma, C. Chen, X. Jiang, and Z. Wang, “Noise robust face image super-resolution through smooth sparse representation,” IEEE Trans. Cybern. 47(11), 3991–4002 (2017).
[Crossref]

2016 (8)

L. Azzari and A. Foi, “Variance stabilization for noisy+estimate combination in iterative Poisson denoising,” IEEE Signal Processing Letters 23(8), 1086–1090 (2016).
[Crossref]

D. Shin, F. Xu, D. Venkatraman, R. Lussana, F. Villa, F. Zappa, V. K. Goyal, F. N. Wong, and J. H. Shapiro, “Photon-efficient imaging with a single-photon camera,” Nat. Commun. 7(1), 12046 (2016).
[Crossref]

I. Gyongy, A. Davies, N. A. Dutton, R. R. Duncan, C. Rickman, R. K. Henderson, and P. A. Dalgarno, “Smart-aggregation imaging for single molecule localisation with SPAD cameras,” Sci. Rep. 6(1), 37349 (2016).
[Crossref]

J. Ma, C. Chen, C. Li, and J. Huang, “Infrared and visible image fusion via gradient transfer and total variation minimization,” Inf. Fusion 31, 100–109 (2016).
[Crossref]

D. Shin, F. Xu, F. N. C. Wong, J. H. Shapiro, and V. K. Goyal, “Computational multi-depth single-photon imaging,” Opt. Express 24(3), 1873–1888 (2016).
[Crossref]

Y. Altmann, X. Ren, A. McCarthy, G. S. Buller, and S. McLaughlin, “Lidar waveform-based analysis of depth images constructed using sparse single-photon data,” IEEE Trans. on Image Process. 25(5), 1935–1946 (2016).
[Crossref]

L. Ye, G. Gu, W. He, H. Dai, J. Lin, and Q. Chen, “Adaptive target profile acquiring method for photon counting 3-D imaging lidar,” IEEE Photonics J. 8(6), 1–10 (2016).
[Crossref]

J. Ma, J. Zhao, and A. L. Yuille, “Non-rigid point set registration by preserving global and local structures,” IEEE Trans. on Image Process. 25(1), 53–64 (2016).
[Crossref]

2015 (2)

G. Gariepy, N. Krstajic, R. Henderson, C. Li, R. R. Thomson, G. S. Buller, B. Heshmat, R. Raskar, J. Leach, and D. Faccio, “Single-photon sensitive light-in-fight imaging,” Nat. Commun. 6(1), 6021 (2015).
[Crossref]

D. Shin, A. Kirmani, V. K. Goyal, and J. H. Shapiro, “Photon-efficient computational 3-D and reflectivity imaging with single-photon detectors,” IEEE Trans. Comput. Imaging 1(2), 112–125 (2015).
[Crossref]

2014 (3)

J. Salmon, Z. Harmany, C.-A. Deledalle, and R. Willett, “Poisson noise reduction with non-local PCA,” J. Math. Imaging Vis. 48(2), 279–294 (2014).
[Crossref]

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. C. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343(6166), 58–61 (2014).
[Crossref]

N. Parikh and S. Boyd, “Proximal algorithms,” Foundations Trends Optim. 1(3), 127–239 (2014).
[Crossref]

2013 (1)

2012 (1)

Z. T. Harmany, R. F. Marcia, and R. M. Willett, “This is SPIRAL-TAP: Sparse Poisson intensity reconstruction algorithms-theory and practice,” IEEE Trans. on Image Process. 21(3), 1084–1096 (2012).
[Crossref]

2011 (1)

M. Makitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. on Image Process. 20(1), 99–109 (2011).
[Crossref]

2010 (1)

2009 (1)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2(1), 183–202 (2009).
[Crossref]

2008 (1)

Y. Wen, M. K. Ng, and Y. Huang, “Efficient total variation minimization methods for color image restoration,” IEEE Trans. on Image Process. 17(11), 2081–2088 (2008).
[Crossref]

2007 (1)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. on Image Process. 16(8), 2080–2095 (2007).
[Crossref]

2006 (1)

X. Michalet, O. Siegmund, J. Vallerga, P. Jelinsky, J. Millaud, and S. Weiss, “Photon-counting H33D detector for biological fluorescence imaging,” Nucl. Instruments Methods Phys. Res. Sect. A: Accel. Spectrometers, Detect. Assoc. Equip. 567(1), 133–136 (2006).
[Crossref]

2003 (1)

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19(6), S165–S187 (2003).
[Crossref]

1999 (1)

E. D. Kolaczyk, “Wavelet shrinkage estimation of certain Poisson intensity signals using corrected thresholds,” Stat. Sinica 9, 119–135 (1999).

1997 (2)

J. Lu and M. L. Liou, “A simple and efficient search algorithm for block-matching motion estimation,” IEEE Trans. Circuits Syst. Video Technol. 7(2), 429–433 (1997).
[Crossref]

A. G. Weber, “Usc-sipi image database version 5,” USC-SIPI Rep. 315, 1–24 (1997).

1995 (1)

F. Dufaux and F. Moscheni, “Motion estimation techniques for digital TV: A review and a new contribution,” Proc. IEEE 83(6), 858–876 (1995).
[Crossref]

Altmann, Y.

Y. Altmann, X. Ren, A. McCarthy, G. S. Buller, and S. McLaughlin, “Lidar waveform-based analysis of depth images constructed using sparse single-photon data,” IEEE Trans. on Image Process. 25(5), 1935–1946 (2016).
[Crossref]

Arlt, J.

Azzari, L.

L. Azzari and A. Foi, “Variance stabilization for noisy+estimate combination in iterative Poisson denoising,” IEEE Signal Processing Letters 23(8), 1086–1090 (2016).
[Crossref]

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2(1), 183–202 (2009).
[Crossref]

Boyd, S.

N. Parikh and S. Boyd, “Proximal algorithms,” Foundations Trends Optim. 1(3), 127–239 (2014).
[Crossref]

Bronstein, A. M.

T. Remez, O. Litany, R. Giryes, and A. M. Bronstein, “Deep convolutional denoising of low-light images,” arXiv preprint arXiv:1701.01687 (2017).

Broxton, M.

H. Ikoma, M. Broxton, T. Kudo, and G. Wetzstein, “A convex 3D deconvolution algorithm for low photon count fluorescence imaging,” Sci. Rep. 8(1), 11489 (2018).
[Crossref]

Buller, G. S.

P. W. Connolly, X. Ren, R. K. Henderson, and G. S. Buller, “Hot pixel classification of single-photon avalanche diode detector arrays using a log-normal statistical distribution,” Electron. Lett. 55(18), 1004–1006 (2019).
[Crossref]

Y. Altmann, X. Ren, A. McCarthy, G. S. Buller, and S. McLaughlin, “Lidar waveform-based analysis of depth images constructed using sparse single-photon data,” IEEE Trans. on Image Process. 25(5), 1935–1946 (2016).
[Crossref]

G. Gariepy, N. Krstajic, R. Henderson, C. Li, R. R. Thomson, G. S. Buller, B. Heshmat, R. Raskar, J. Leach, and D. Faccio, “Single-photon sensitive light-in-fight imaging,” Nat. Commun. 6(1), 6021 (2015).
[Crossref]

Buts, A.

Chan, T.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19(6), S165–S187 (2003).
[Crossref]

Charbon, E.

Chen, C.

J. Jiang, J. Ma, C. Chen, X. Jiang, and Z. Wang, “Noise robust face image super-resolution through smooth sparse representation,” IEEE Trans. Cybern. 47(11), 3991–4002 (2017).
[Crossref]

J. Ma, C. Chen, C. Li, and J. Huang, “Infrared and visible image fusion via gradient transfer and total variation minimization,” Inf. Fusion 31, 100–109 (2016).
[Crossref]

Chen, Q.

L. Ye, G. Gu, W. He, H. Dai, J. Lin, and Q. Chen, “Adaptive target profile acquiring method for photon counting 3-D imaging lidar,” IEEE Photonics J. 8(6), 1–10 (2016).
[Crossref]

Colaço, A.

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. C. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343(6166), 58–61 (2014).
[Crossref]

A. Kirmani, D. Venkatraman, A. Colaço, F. N. Wong, and V. K. Goyal, “High photon efficiency computational range imaging using spatio-temporal statistical regularization,” Proceedings of IEEE Conference on Lasers and Electro-Optics, (IEEE, 2013), pp. 1–2.

Connolly, P. W.

P. W. Connolly, X. Ren, R. K. Henderson, and G. S. Buller, “Hot pixel classification of single-photon avalanche diode detector arrays using a log-normal statistical distribution,” Electron. Lett. 55(18), 1004–1006 (2019).
[Crossref]

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. on Image Process. 16(8), 2080–2095 (2007).
[Crossref]

Dai, H.

L. Ye, G. Gu, W. He, H. Dai, J. Lin, and Q. Chen, “Adaptive target profile acquiring method for photon counting 3-D imaging lidar,” IEEE Photonics J. 8(6), 1–10 (2016).
[Crossref]

Dalgarno, P. A.

I. Gyongy, A. Davies, N. A. Dutton, R. R. Duncan, C. Rickman, R. K. Henderson, and P. A. Dalgarno, “Smart-aggregation imaging for single molecule localisation with SPAD cameras,” Sci. Rep. 6(1), 37349 (2016).
[Crossref]

Davies, A.

I. Gyongy, A. Davies, N. A. Dutton, R. R. Duncan, C. Rickman, R. K. Henderson, and P. A. Dalgarno, “Smart-aggregation imaging for single molecule localisation with SPAD cameras,” Sci. Rep. 6(1), 37349 (2016).
[Crossref]

Deledalle, C.-A.

J. Salmon, Z. Harmany, C.-A. Deledalle, and R. Willett, “Poisson noise reduction with non-local PCA,” J. Math. Imaging Vis. 48(2), 279–294 (2014).
[Crossref]

Diamond, S.

F. Heide, S. Diamond, D. B. Lindell, and G. Wetzstein, “Sub-picosecond photon-efficient 3D imaging using single-photon sensors,” Sci. Rep. 8(1), 17726 (2018).
[Crossref]

M. O’Toole, F. Heide, D. B. Lindell, K. Zang, S. Diamond, and G. Wetzstein, “Reconstructing transient images from single-photon sensors,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, (IEEE, 2017), pp. 1539–1547.

Dufaux, F.

F. Dufaux and F. Moscheni, “Motion estimation techniques for digital TV: A review and a new contribution,” Proc. IEEE 83(6), 858–876 (1995).
[Crossref]

Duncan, R. R.

I. Gyongy, A. Davies, N. A. Dutton, R. R. Duncan, C. Rickman, R. K. Henderson, and P. A. Dalgarno, “Smart-aggregation imaging for single molecule localisation with SPAD cameras,” Sci. Rep. 6(1), 37349 (2016).
[Crossref]

Dutton, N. A.

I. Gyongy, A. Davies, N. A. Dutton, R. R. Duncan, C. Rickman, R. K. Henderson, and P. A. Dalgarno, “Smart-aggregation imaging for single molecule localisation with SPAD cameras,” Sci. Rep. 6(1), 37349 (2016).
[Crossref]

Egiazarian, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. on Image Process. 16(8), 2080–2095 (2007).
[Crossref]

Entwistle, M.

M. Itzler, G. Salzano, M. Entwistle, X. Jiang, M. Owens, B. Piccione, S. Wilton, K. Slomkowski, S. C. Roszko, and E. Wei, “Asynchronous Geiger-mode APD cameras with free-running InGaAsP pixels,” Proc. SPIE 10212, 102120K (2017).
[Crossref]

Faccio, D.

G. Gariepy, N. Krstajic, R. Henderson, C. Li, R. R. Thomson, G. S. Buller, B. Heshmat, R. Raskar, J. Leach, and D. Faccio, “Single-photon sensitive light-in-fight imaging,” Nat. Commun. 6(1), 6021 (2015).
[Crossref]

Fan, H.

H. Fan, C. Li, Y. Guo, G. Kuang, and J. Ma, “Spatial–spectral total variation regularized low-rank tensor decomposition for hyperspectral image denoising,” IEEE Trans. Geosci. Remote Sensing 56(10), 6196–6213 (2018).
[Crossref]

Foi, A.

L. Azzari and A. Foi, “Variance stabilization for noisy+estimate combination in iterative Poisson denoising,” IEEE Signal Processing Letters 23(8), 1086–1090 (2016).
[Crossref]

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D. Shin, J. H. Shapiro, and V. K. Goyal, “Computational single-photon depth imaging without transverse regularization,” in Proceedings of IEEE International Conference on Image Processing, (IEEE, 2016), pp. 973–977.

D. Shin, A. Kirmani, V. K. Goyal, and J. H. Shapiro, “Computational 3D and reflectivity imaging with high photon efficiency,” in Proceedings of IEEE International Conference on Image Processing, (IEEE, 2014), pp. 46–50.

M. O’Toole, F. Heide, D. B. Lindell, K. Zang, S. Diamond, and G. Wetzstein, “Reconstructing transient images from single-photon sensors,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, (IEEE, 2017), pp. 1539–1547.

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Figures (7)

Fig. 1.
Fig. 1. Schematics of SPCIS and APCIS. (a) SPCIS. (b) APCIS.
Fig. 2.
Fig. 2. Motivation of the spatially correlated constraint. (a) single-photon counting image. (b) grayscale image.
Fig. 3.
Fig. 3. Visual comparison results of different algorithms ($p=100$). (a) Ground truth. (b) Simulated photon counting image without noise. (c) Simulated photon counting image with noise. (d) SPIRAL-ONB. (e) Binomial SPIRAL-TV. (f) NLSPCA. (g) VST+BM3D. (h) Proposed method.
Fig. 4.
Fig. 4. Reconstruction results of different methods at a repetition rate of 5 MHz. (a) Photograph of the target (a Chinese character). (b) Photon counting raw data of (a). (c) SPIRAL-ONB of (b). (d) Binomial SPIRAL-TV of (b). (e) NLSPCA of (b). (f) VST+BM3D of (b). (g) Proposed method of (b). (h) Photograph of the target (a ceramic cup). (i) Photon counting raw data of (h). (j) SPIRAL-ONB of (i). (k) Binomial SPIRAL-TV of (i). (l) NLSPCA of (i). (m) VST+BM3D of (i). (n) Proposed method of (i).
Fig. 5.
Fig. 5. Reconstruction results of different methods at a repetition rate of 1 MHz. (a) Photograph of the target (a Chinese character). (b) Photon counting raw data of (a). (c) SPIRAL-ONB of (b). (d) Binomial SPIRAL-TV of (b). (e) NLSPCA of (b). (f) VST+BM3D of (b). (g) Proposed method of (b). (h) Photograph of the target (a ceramic cup). (i) Photon counting raw data of (h). (j) SPIRAL-ONB of (i). (k) Binomial SPIRAL-TV of (i). (l) NLSPCA of (i). (m) VST+BM3D of (i). (n) Proposed method of (i).
Fig. 6.
Fig. 6. Targets for quantitative analysis. (a) Single-photon counting image. (b) Corresponding photograph.
Fig. 7.
Fig. 7. Objective comparison results under different SNR conditions. (a) PSNR. (b) SSIM.

Tables (1)

Tables Icon

Table 1. Objective comparison results with different numbers of emitted pulses for each pixel: p = 200 and p = 100 .

Equations (15)

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arg min F J ( F ) + K ( F ) ,
L ( F ( i , j ) ; G ( i , j ) ) = G ( i , j ) log { 1 e x p ( ( η S ( i , j ) F ( i , j ) + B ( i , j ) ) ) } + η S ( i , j ) ( p G ( i , j ) ) F ( i , j ) + B ( i , j ) ( p G ( i , j ) ) ,
L ( D ( i , j ) , F ( i , j ) ; G ( i , j ) ) = G ( i , j ) log { 1 e x p ( ( η S ( i , j ) F ( i , j ) + B ( i , j ) ) ) } + η S ( i , j ) ( p G ( i , j ) ) F ( i , j ) + B ( i , j ) ( p G ( i , j ) ) , s . t .     D 0 l ,
K 1 ( F ) := F T V = D h F 1 + D v F 1 ,
{ D h F = F ( i + 1 , j ) F ( i , j ) , D v F = F ( i , j + 1 ) F ( i , j ) ,
M A D = 1 M × N i = 1 M j = 1 N | C ( i , j ) C ¯ ( i , j ) | .
K 2 ( F ) := G p ( 1 e x p ( ( η S F + B ) ) ) F 2 G p ( η S F + B ) F 2 ,
arg min F , D L ( D , F ; G ) + λ F T V + γ D 1 + α 2 G p ( η S F + b + D ) F 2 ,
L ( F , D , Z , Y ) = G log Z + ( p G ) Z + α 2 G p Z F 2 + ρ 2 Z ( η S F + b + D ) F 2 Y , Z ( η S F + b + D ) + γ D 1 + λ F T V ,
{ F t + 1 = arg min F ρ 2 Z t ( η S F + b + D t ) F 2 Y t , Z t ( η S F + b + D t ) + λ F T V , ( 10 ) D t + 1 = arg min D ρ 2 Z t ( η S F t + 1 + b + D ) Y t ρ F 2 + γ D 1 , ( 11 ) Z t + 1 = arg min Z G log Z + ( p G ) Z + ρ 2 Z ( η S F t + 1 + b + D t + 1 ) Y t ρ F 2 + α 2 G p Z F 2 , ( 12 ) Y t + 1 = Y t ρ ( Z t + 1 ( η S F t + 1 + b + D t + 1 ) ) . ( 13 )
F t + 1 = arg min F ρ 2 Z t ( η S F + b + D t ) Y t ρ F 2 + λ F T V ,
D t + 1 = max { | V | γ ρ , 0 } s i g n ( V ) ,
( ρ + α p 2 ) Z 2 + ( p G α p G ρ η S F t + 1 ρ b ρ D t + 1 Y t ) Z G = 0 ,
Z t + 1 = W + W 2 + 4 ( ρ + α p 2 ) G 2 ( ρ + α p 2 ) ,
{ P S N R = 10 lg M × N × I M a x 2 i = 1 M j = 1 N ( Y ( i , j ) X ( i , j ) ) 2 , S S I M = ( 2 μ X μ Y + c 1 ) ( 2 δ X Y + c 2 ) ( μ X 2 + μ Y 2 + c 1 ) ( δ X 2 + δ Y 2 + c 2 ) .

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