Experimental comparison of single-pixel imaging algorithms

Liheng Bian; Jinli Suo; Qionghai Dai; Feng Chen

doi:10.1364/JOSAA.35.000078

Journal of the Optical Society of America A
Vol. 35,
Issue 1,
pp. 78-87
(2018)
•https://doi.org/10.1364/JOSAA.35.000078

Experimental comparison of single-pixel imaging algorithms

Liheng Bian, Jinli Suo, Qionghai Dai, and Feng Chen

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Liheng Bian, Jinli Suo, Qionghai Dai, and Feng Chen, "Experimental comparison of single-pixel imaging algorithms," J. Opt. Soc. Am. A 35, 78-87 (2018)

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Abstract

Single-pixel imaging (SPI) is a novel technique that captures 2D images using a photodiode, instead of conventional 2D array sensors. SPI has high signal-to-noise ratio, wide spectral range, low cost, and robustness to light scattering. Various algorithms have been proposed for SPI reconstruction, including linear correlation methods, the alternating projection (AP) method, and compressive sensing (CS) based methods. However, there has been no comprehensive review discussing respective advantages, which is important for SPI’s further applications and development. In this paper, we review and compare these algorithms in a unified reconstruction framework. We also propose two other SPI algorithms, including a conjugate gradient descent (CGD) based method and a Poisson maximum-likelihood-based method. Both simulations and experiments validate the following conclusions: to obtain comparable reconstruction accuracy, the CS-based total variation (TV) regularization method requires the fewest measurements and consumes the least running time for small-scale reconstruction, the CGD and AP methods run fastest in large-scale cases, and the TV and AP methods are the most robust to measurement noise. In a word, there are trade-offs in capture efficiency, computational complexity, and robustness to noise among different SPI algorithms. We have released our source code for non-commercial use.

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Methods		Perspective	Principles
No iteration	Matrix inversion	Formation fitting	$x = {(A^{T} A)}^{- 1} A^{T} b$
No iteration	Conventional correlation [3]	Measurement: scene–pattern correlation	Eq. (2): ${b_{i} a_{i}} - {b_{i}} {a_{i}}$
	DGI [5]		Eq. (3): ${b_{i} a_{i}} - \frac{{b_{i}}}{{s_{i}}} {s_{i} a_{i}}$
Linear iteration			Problem description	Gradient $p$	Step $Δ_{x}$
	Gradient descent	Formation fitting	Eq. (4): $\min {‖ Ax - b ‖}_{l_{2}}^{2}$	Eq. (5): $A^{T} (Ax - b)$	Eq. (7): $- \frac{p^{T} A^{T} r}{p^{T} A^{T} Ap}$
	Conjugate gradient descent	Formation fitting	Eq. (8): $A^{T} Ax = A^{T} b$	Eq. (9): $- r^{(k - 1)} - \frac{r^{(k - 1) T} r^{(k - 1)}}{r^{(k - 2) T} r^{(k - 2)}} p$	Eq. (10): $\frac{r^{T} r}{p^{T} A^{T} Ap}$
	Poisson maximum likelihood	Signal statistic	Eq. (12): min $\sum_{i = 1}^{m} (a_{i} x - b_{i} \log (a_{i} x))$	Eq. (13): $A^{T} (\frac{Ax - b}{Ax})$	Alg. (1)
	Alternating projection [2]	Measurement: zero-spatial-frequency coefficient	Eq. (1): $Ax = b$	Eq. (14): $\frac{a_{i}^{T} ⊙ (a_{i}^{T} ⊙ x)}{\max {(a_{i})}^{2}} \frac{a_{i} x - b_{i}}{a_{i} x}$	1
Non-linear iteration	Sparse representation [1]	Image prior	Eq. (15): $\min {‖ c ‖}_{l_{0}}$ $s.t. Dx = c, Ax = b$	See Eqs. (16)–(22) in Section 2.C.1
Non-linear iteration	Total variation	Image prior	Eq. (15): $\min {‖ c ‖}_{l_{0}}$ $s.t. Dx = c, Ax = b$	See Eqs. (16)–(22) in Section 2.C.1

Methods		Perspective	Principles
No iteration	Matrix inversion	Formation fitting	$x = {(A^{T} A)}^{- 1} A^{T} b$
No iteration	Conventional correlation [3]	Measurement: scene–pattern correlation	Eq. (2): ${b_{i} a_{i}} - {b_{i}} {a_{i}}$
	DGI [5]		Eq. (3): ${b_{i} a_{i}} - \frac{{b_{i}}}{{s_{i}}} {s_{i} a_{i}}$
Linear iteration			Problem description	Gradient $p$	Step $Δ_{x}$
	Gradient descent	Formation fitting	Eq. (4): $\min {‖ Ax - b ‖}_{l_{2}}^{2}$	Eq. (5): $A^{T} (Ax - b)$	Eq. (7): $- \frac{p^{T} A^{T} r}{p^{T} A^{T} Ap}$
	Conjugate gradient descent	Formation fitting	Eq. (8): $A^{T} Ax = A^{T} b$	Eq. (9): $- r^{(k - 1)} - \frac{r^{(k - 1) T} r^{(k - 1)}}{r^{(k - 2) T} r^{(k - 2)}} p$	Eq. (10): $\frac{r^{T} r}{p^{T} A^{T} Ap}$
	Poisson maximum likelihood	Signal statistic	Eq. (12): min $\sum_{i = 1}^{m} (a_{i} x - b_{i} \log (a_{i} x))$	Eq. (13): $A^{T} (\frac{Ax - b}{Ax})$	Alg. (1)
	Alternating projection [2]	Measurement: zero-spatial-frequency coefficient	Eq. (1): $Ax = b$	Eq. (14): $\frac{a_{i}^{T} ⊙ (a_{i}^{T} ⊙ x)}{\max {(a_{i})}^{2}} \frac{a_{i} x - b_{i}}{a_{i} x}$	1
Non-linear iteration	Sparse representation [1]	Image prior	Eq. (15): $\min {‖ c ‖}_{l_{0}}$ $s.t. Dx = c, Ax = b$	See Eqs. (16)–(22) in Section 2.C.1
Non-linear iteration	Total variation	Image prior	Eq. (15): $\min {‖ c ‖}_{l_{0}}$ $s.t. Dx = c, Ax = b$	See Eqs. (16)–(22) in Section 2.C.1

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Journal of the Optical Society of America A