Single Pixel Imaging and Compressive Sensing: A Practical Tutorial

TL;DR

This paper addresses image reconstruction in Single Pixel Imaging (SPI) using a bucket detector by encoding the illumination with measurement bases and applying compressive sensing (CS) and deep learning (DL) to reduce data and enable faster imaging. It presents both deterministic CS (Basis Pursuit via SPGL1) and DL (a linear network) reconstruction pipelines, detailing the experimental setup with DMD/SLM devices and the measurement models $y = \boldsymbol{\uPhi} x$ and $\Theta = \boldsymbol{\uPhi} \Psi$. The authors compare the influence of basis choice (canonical, Hadamard, Walsh, cake-cutting, random Gaussian) and reconstruction algorithms on reconstruction quality (RMSE, PSNR, SSIM), showing Hadamard–Walsh often optimizes performance for deterministic CS, while random Gaussian measurements paired with a linear network can outperform others at low compression. They emphasize reproducibility through Python notebooks and demonstrate results on CIFAR-10 data and a 32×32 image, highlighting practical guidance for choosing bases and algorithms in SPI applications. The work advances practical SPI by offering concrete guidance on measurement bases, reconstruction methods, and training strategies for real-time or in vivo imaging across wavelengths where conventional sensors are limited.

Abstract

Single Pixel Imaging is an emerging imaging technique that employs a bucket detector (photodiode) to sample a spatially modulated light field, rather than measuring the spatial distribution with an array of detectors. This approach provides a low-cost alternative for imaging at unconventional wavelengths and enables improved signal collection in noisy measurement environments. Furthermore, it allows the application of compressive sensing to reduce the amount of acquired data and measurement time, facilitating live or in vivo imaging applications. This tutorial presents the experimental implementation of measurement bases and compressive sensing reconstruction methods, including both deterministic algorithms and deep learning approaches. Accompanying Python notebooks guide readers through the reproduction of the presented results and support the application of the methods to their own work.

Figures (5)

• Figure 1: A) Basic setup for Single Pixel Imaging. The input field $x$ is represented by the dog picture. On the SLM a row vector of the Hadamard basis is projected, selecting only designated parts of the input field. The selected field is then projected by a lens onto a detector where the intensity $y_i$ is measured by integrating the intensity over the whole detector surface. B) Different measurement bases of size $N = 16$ and the projection of the second measurement vector onto a $(4 \times 4)$ 2d grid for sampling with a SLM or DMD.
• Figure 2: A) Original image. B) Measurement bases: Canonical, Hadamard, and random Gaussian. C) One-dimensional vector representations of the original image (black) and sampled signals using the canonical (blue), Hadamard (green), and random Gaussian (orange) bases. D) Reconstructed image using Eq. \ref{['eq:SPI']}.
• Figure 3: A) Reconstructed images using the SPGL1 basis pursuit algorithm, with increasing compression ratio along the columns. Rows show, from top to bottom, the natural Hadamard, Hadamard-Walsh, Cake-Cutting, and random Gaussian orderings. B) PSNR, C) RMSE, and D) SSIM of the reconstructed images for the natural Hadamard (blue), Hadamard-Walsh (orange), Cake-Cutting (green), and random Gaussian (red) bases as a function of the compression ratio.
• Figure 4: A) Reconstructed images sampled with the Hadamard-Walsh basis with increasing compression ratio along the columns. Rows show, from top to bottom, the basic basis pursuit, SPGL1 basis pursuit, and SPGL1 LASSO algorithms. B) PSNR, C) RMSE, and D) SSIM of the reconstructed images for the basic basis pursuit (blue), SPGL1 basis pursuit (orange), and SPGL1 LASSO (green) algorithms as a function of the compression ratio.
• Figure 5: A) Reconstructed images using trained linear neural networks, with increasing compression ratio along the columns. Rows show, from top to bottom, the natural Hadamard, Hadamard-Walsh, Cake-Cutting, and random Gaussian orderings. B) PSNR, C) RMSE, and D) SSIM of the reconstructed images for the natural Hadamard (blue), Hadamard-Walsh (orange), Cake-Cutting (green), and random Gaussian (red) bases as a function of the compression ratio.