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Experimental comparison of single-pixel imaging algorithms

Liheng Bian, Jinli Suo, Qionghai Dai, Feng Chen

arXiv:1707.03164v2cs.CVphysics.optics

TL;DR

SPI lacks a comprehensive comparison of reconstruction algorithms despite their importance for choosing methods and advancing applications. This paper reviews and evaluates existing algorithms in a unified framework, adds two reconstruction methods, and finds trade-offs among measurement efficiency, computation, and noise robustness.

  • Problem

    Existing SPI reconstruction algorithms had not been comprehensively compared under a unified framework, limiting evidence for choosing appropriate methods.

  • Method

    The paper reviews SPI and GI algorithms, proposes conjugate-gradient-descent and Poisson-maximum-likelihood methods, and evaluates them on simulated and real data.

  • Results

    The study compares capture efficiency, computational complexity, and robustness to measurement noise across algorithms at comparable reconstruction accuracy.

  • Takeaways & Limitations

    Algorithm choice involves trade-offs among capture efficiency, computational complexity, and noise robustness, so methods should be selected for specific SPI configurations and applications.

Abstract

from arXiv · show

Single-pixel imaging (SPI) is a novel technique capturing 2D images using a photodiode, instead of conventional 2D array sensors. SPI owns high signal-to-noise ratio, wide spectrum range, low cost, and robustness to light scattering. Various algorithms have been proposed for SPI reconstruction, including the linear correlation methods, the alternating projection method (AP), and the compressive sensing 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 reviewed and compared these algorithms in a unified reconstruction framework. Besides, we proposed two other SPI algorithms including a conjugate gradient descent based method (CGD) and a Poisson maximum likelihood based method. Both simulations and experiments validate the following conclusions: to obtain comparable reconstruction accuracy, the compressive sensing based total variation regularization method (TV) requires the least measurements and consumes the least running time for small-scale reconstruction; the CGD and AP methods run fastest in large-scale cases; the TV and AP methods are the most robust to measurement noise. In a word, there are trade-offs between capture efficiency, computational complexity and robustness to noise among different SPI algorithms. We have released our source code for non-commercial use.

1. INTRODUCTION

SPI reconstructs 2D scenes from one-dimensional photodiode measurements and offers advantages including high SNR, broad spectral range, low cost, and robustness to scattering. This paper addresses the need for unified algorithm comparison by reviewing and testing methods across experimental conditions.

  • SPI and its applications: SPI produces 2D images with a photodiode instead of a conventional 2D sensor array.Light modulation patterns and corresponding one-dimensional measurements support computational reconstruction.
  • SPI and its applications: SPI offers high signal-to-noise ratio, wide spectrum range, low cost, flexible configuration, and robustness to light scattering.The technique has been applied to multispectral imaging, 3D modeling, encryption, remote sensing, tracking, and imaging through turbulence.
  • Related imaging approaches: SPI and ghost imaging multiplex scene information into one-dimensional measurements and computationally demultiplex the 2D target image.Their historical developments span computer science and optics, including classical thermal-light ghost imaging and SLM-based modulation.
  • Motivation and contribution: A unified comparison of reconstruction algorithms is needed to help researchers choose methods for experiments and support SPI development and applications.The paper compares capture efficiency, computational complexity, and noise robustness under varied settings.
  • Motivation and contribution: The paper reviews SPI and GI algorithms, adds conjugate-gradient-descent and Poisson-maximum-likelihood methods, and tests them on simulated and real data.Experiments vary sampling ratio, image size, and noise level while comparing efficiency, complexity, and robustness at comparable reconstruction accuracy.

2. SPI ALGORITHMS

SPI reconstruction is formulated as a linear inverse problem: recover the vectorized target scene from modulation patterns and their corresponding measurements. The paper derives algorithms within this common formulation.

  • Measurement model: The single-pixel imaging scheme is modeled as a linear system.The measurement equation is written as Ax = b.
  • Measurement model: A contains m modulation patterns with n pixels, x is the vectorized target scene, and b contains m measurements.The dimensions are A ∈ R^(m×n), x ∈ R^(n×1), and b ∈ R^(m×1).
  • Reconstruction framework: SPI reconstruction calculates x from the modulation patterns A and corresponding measurements b.The paper groups the resulting algorithms into non-iterative, linear iterative, and non-linear iterative methods.

A. Non-iterative methods

Non-iterative reconstruction directly estimates the scene without iteration, while correlation-based methods use measurements as weights on modulation patterns. DGI additionally normalizes illumination using total intensity.

  • Direct reconstruction: Non-iterative methods directly reconstruct the target scene without iteration.Inverse-matrix reconstruction becomes unavailable when m < n because A^T A is non-full rank.
  • Direct reconstruction: When m ≥ n, inverse-matrix reconstruction is equivalent to minimizing ||Ax − b||_l2; when m < n, the method does not work.The normal-equation estimate is x = (A^T A)^−1 A^T b.
  • Correlation methods: Correlation reconstruction forms a weighted summation of modulation patterns, using corresponding measurements as weights.Larger measurements indicate greater similarity between a pattern and the target scene, producing larger reconstruction weights.
  • Differential ghost imaging: DGI accounts for illumination fluctuations by using an additional detector to measure each illumination pattern’s total intensity.The method normalizes illumination patterns and can improve the final reconstruction’s SNR.

B. Linear iterative methods

Linear iterative methods reconstruct SPI images by reducing measurement-estimation error or optimizing photon-count likelihood, while alternating projection switches between Fourier and spatial domains to impose constraints.

  • Gradient descent: Gradient descent iteratively updates the target image using the gradient of a quadratic measurement-error objective.The residual is r = b − Ax, and iterations stop when the objective falls below a threshold.
  • Conjugate gradient descent: The conjugate gradient descent method solves the quadratic minimization problem after transforming the system with A′ = A^T A and b′ = A^T b.CGD requires a symmetric positive-definite system and uses conjugate gradients across iterations.
  • Conjugate gradient descent: CGD converges faster than conventional gradient descent and theoretically requires no more than n iterations for an n-pixel image.Its gradients are designed to be mutually conjugate.
  • Poisson maximum likelihood: The Poisson maximum likelihood method estimates the image by maximizing the likelihood of measurements modeled with Poisson photon statistics.Because its gradient is nonlinear, backtracking line search determines each step size.
  • Alternating projection: Alternating projection treats each photodiode measurement as a zero-spatial-frequency constraint and alternates Fourier- and spatial-domain updates.All measurements are sequentially used to update the target image in each iteration.

C. Non-linear iterative methods

Non-linear iterative SPI methods use signal priors, including sparsity and total variation, to reconstruct images from underdetermined measurements through augmented-Lagrangian optimization.

  • Prior-based reconstruction: Compressive sensing reduces SPI measurements by introducing priors such as sparse representations and total variation regularization.Natural images can be sparse in overcomplete or orthogonal bases, while TV uses image-gradient structure.
  • Sparse representation prior: The sparse-representation model minimizes the number of nonzero coefficients, using l1 norm to approximate the l0 sparsity objective.The optimization is solved with an augmented Lagrange multiplier framework.
  • Optimization procedure: The augmented-Lagrangian scheme alternately updates coefficients, the image, Lagrange multipliers, and balancing parameters while holding other variables fixed.Thresholding is used in the coefficient update, and multiplier parameters control the optimization balance.
  • Total variation regularization prior: Total variation represents image gradients with a matrix and applies an l1 norm to their integral, yielding an optimization form solvable by the sparse-representation procedure.The paper identifies the TV and sparse-representation problems as sharing the same optimization form.
  • Algorithm comparison: The paper summarizes the optimization models and reconstruction principles of the SPI algorithms in Table 1.The comparison covers the methods introduced in the reconstruction framework.

B. Simulations

The simulations use standardized target images and repeated Matlab evaluations to compare SPI algorithms under controlled sampling and image-size settings.

  • Simulation data: 10 widely used Standard test images, normalized to [0, 1], serve as simulation target scenes.Average reconstruction RMSE across these images quantitatively characterizes performance.
  • Simulation data: Random modulation synthesizes measurements, and reconstructed cameraman images are shown in the figures to conserve space.The target-scene set is broader than the single exemplar image displayed in later figures.
  • Experiment settings: Sampling ratio is the number of measurements divided by the number of reconstructed pixels and determines capture efficiency.The simulations vary sampling ratio while generally keeping image size at 64 × 64 pixels.
  • Evaluation procedure: Each simulation is repeated 20 times and averaged, with algorithms implemented in Matlab on an Intel Core i7-4790 3.6 GHz CPU and 16G RAM.The stated platform is a 64-bit Windows 7 system.

B.1. Sampling ratio

As sampling ratio increases, reconstruction error improves most up to a ratio of 1, while algorithm choice determines measurement efficiency and computational behavior.

  • Reconstruction accuracy: Reconstruction error largely decreases as sampling ratio rises to 1, then decreases more slowly at higher ratios.The result indicates diminishing error reduction beyond a sampling ratio of 1.
  • Computational and capture cost: Iteration number and running time change little with sampling ratio, but capture time increases as more measurements are acquired.This separates computational cost from the measurement-acquisition cost affected by sampling ratio.
  • Algorithm comparison: The linear correlation, Poisson maximum likelihood, and gradient descent methods retain large reconstruction errors even at a sampling ratio of 5.The paper therefore characterizes these methods as having low capture efficiency.
  • Algorithm comparison: The TV method needs only 50% measurements of the signals to achieve normalized RMSE smaller than 0.03.Nonlinear iterative methods, especially TV, use fewer measurements for comparable quality and converge faster than most linear iterative methods except CGD.

B.2. Image size

Increasing image size reduces reconstruction error but raises computational demands. CGD and AP offer the best combination of reconstruction quality and running time for large-scale SPI.

  • B.2. Image size: Larger images produce less reconstruction error but require more iterations and running time.The comparison spans simulated images from 32×32 to 160×160 pixels at a constant sampling ratio of 1.
  • B.2. Image size: Non-linear iterative methods become increasingly slower as image size grows because they perform non-linear calculations and update multiple variables.
  • B.2. Image size: Non-linear iterative methods do not require more iterations as image size increases when the sampling ratio is high enough.The paper attributes this convergence behavior to introduced image priors.
  • B.2. Image size: CGD and AP provide the best combined reconstruction quality and computational complexity for large-scale SPI.DGI, CGD, and AP are fastest beyond 160×160 pixels, but DGI has much larger final reconstruction error.

B.3. Noise

The study evaluates Gaussian white measurement noise by varying its standard-deviation-to-pixel-number ratio. Increasing noise degrades every reconstruction, but TV and AP are most robust.

  • B.3. Noise: Measurement noise is modeled as Gaussian white noise, with noise level defined as standard deviation divided by pixel number.The ratio is varied from 0 to 3e-3 using a sampling ratio of 1 and 64×64-pixel images.
  • B.3. Noise: All reconstruction methods degrade as measurement noise increases.
  • B.3. Noise: CGD and compressive-sensing sparse-representation methods degrade faster than the other algorithms under increasing noise.
  • B.3. Noise: TV and AP are the two most robust methods for attenuating measurement noise.

C. Experiment

Real-data experiments compare SPI algorithms across sampling ratios using a projector, DMD, and photodetector setup. The experimental results reproduce the conclusions observed in simulation.

  • C. Experiment: The experiment uses a commercial projector illumination module and a DMD to project 64×64-pixel patterns onto a printed transmissive film.The setup uses a 10 kHz sampling rate and synchronizes the DMD with the detector.
  • C. Experiment: The real-data experiments arrive at the same conclusions as the simulations.

4. CONCLUSIONS

The paper unifies and compares SPI reconstruction algorithms, adding CGD and Poisson maximum likelihood methods. Results reveal trade-offs among measurement efficiency, computational complexity, and noise robustness.

  • 4. CONCLUSIONS: The study reviews DGI, GD, AP, sparse representation, and TV, and proposes CGD and Poisson maximum likelihood methods.
  • 4. CONCLUSIONS: The algorithms use different reconstruction perspectives, including correlation, formation fitting, photon-count statistics, spatial spectrum, and natural-image priors.
  • 4. CONCLUSIONS: Non-iterative methods generally require the most measurements but the least running time, whereas non-linear iterative methods require the fewest measurements.The algorithms are grouped into non-iterative, linear iterative, and non-linear iterative categories.
  • 4. CONCLUSIONS: DGI and AP have high storage efficiency because they do not need all patterns and measurements during each update.
  • 4. CONCLUSIONS: TV requires the fewest measurements for comparable quality; TV and CGD are fastest at small scale, while CGD and AP are fastest at large scale.TV and AP are the most robust to measurement noise, showing trade-offs among capture efficiency, computational complexity, and noise robustness.
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