Bayesian Sequential Optimal Experimental Design for Single-Pixel Imaging using Reinforcement Learning

Marcos Antonio Obando; Tianyuan Wang; Tanja Tarvainen; Tristan van Leeuwen; Felix Lucka

cs.CVNov 2, 2026

Bayesian Sequential Optimal Experimental Design for Single-Pixel Imaging using Reinforcement Learning

Authors: Marcos Antonio Obando, Tianyuan Wang, Tanja Tarvainen, Tristan van Leeuwen, Felix Lucka

Abstract

Adaptive compressed sensing (CS) aims to select the most informative measurements sequentially, which is crucial for many modern imaging systems. This task can be formulated as a Bayesian sequential optimal experimental design (sOED) problem, where each new acquisition is chosen to maximally reduce uncertainty about the unknown image given prior information and subsequent data provided by acquisitions. However, in single-pixel imaging (SPI), the large design space consists of all possible binary illumination patterns. Directly optimizing the Bayesian sOED objective over this space using conventional methods is generally challenging. In this work, we reformulate sOED for SPI as a reinforcement learning (RL) problem. In RL, an agent learns a policy through repeated experimental trials to maximize a reward, here given as the expected information gain, the most complete measure of uncertainty reduction in Bayesian OED. This avoids the need to directly optimize complex information-theoretic objectives during a measurement to obtain the next design. Instead, the result is a learned policy function that directly maps the current posterior belief about the image to the next optimal binary measurement pattern, in a computationally efficient way. Using a simple yet representative SPI imaging scenario, this work evaluates the scalability of the learned policy under two different policy parameterizations. Furthermore, we investigate how performance deteriorates when only approximate posterior information is available or the measurements are impaired by noise, and propose several strategies to introduce additional design constraints motivated by biomedical applications. The numerical experiments demonstrate that the proposed Bayesian sOED-RL framework remains robust and can be extended to more complex, large-scale SPI scenarios.

Figures & tables

Figure 1: The posterior is being updated using measurements of the ground truth

\mathbf{x}

via the proposed patterns

\mathbf{h}_{n}

(in green). Bisection and raster scan are considered as two possible strategies to identify the ground truth from measurements via one-active-pixel prior knowledge.

Figure 2: For a uniform one-active-pixel, strategies that illuminate either one or all but one site are suboptimal in an EIG sense. This can be seen from the unbalanced posterior at the different possible outcomes

\mathbf{y}

. For the bisection strategy, measurements 0 or 1 have equal probability and the posterior support halves, which produces the optimal one-step EIG.

Figure 3: Total normalized EIG for categorical (lighter tone, square marker) and Bernoulli (darker tone, circle marker) policies trained over different image sizes $M=\{4,8,12,16\}$ and same experimental step size $N=2$ . Optimal and random baselines are drawn as red and gray lines, respectively.

Figure 4: Total expected information gain for policy training under sample-based state approximations, with $N_{s}=\{1,2,4,8,16\}$ . First row figure displays mean total EIG, second row shows $95\%$ confidence interval width. Curve for learning with no approximation, using exact state representation is illustrated in yellow for comparison of the sampling impact. Optimal policy reward is computed exactly and displayed in red for both plots.

Figure 5: Total expected information gain using a sample-based reward approximation, with $N_{r}=\{2,4,8,16,32\}$ samples and a exact state representation. Optimal rewards for each sample quantity are illustrated in red, with mean values in the first row plot and $95\%$ confidence value widths over the second row. For comparison, EIG reward calculation using the posterior distribution representation without approximation (’No Approximation’) is displayed in yellow.

Figure 6: Total normalized EIG for three noisy measurement levels

\sigma\in\{0.2,0.4,0.8\}

. First column plot displays normalized EIG and second column plot

95\%

confidence interval widths, averaged across 100 categorical policies realizations. Curves are normalized using random and optimal baselines for each noise level, and represented in red and green, respectively.

Figure 7: Total normalized EIG for three noisy measurement levels under constant and a linear curriculum learning for noise scheduling. Columns represent noise measurement levels

\sigma\in\{0.2,0.4,0.8\}

and rows display the normalized means and

95\%

confidence interval widths. Curves are normalized using random and optimal baselines for each noise level, and are drawn in gray and green lines, respectively.

Figure 8: Total normalized EIG under dose budget constraints $B\in\{1,\dots,8\}$ for conditional Bernoulli policies. First and second row display mean and $95\%$ confidence interval width, respectively. Random baseline, in red, is computed using a unconstrained Bernoulli policy. Optimal baseline is drawn in green.

Figure 9: From top to bottom, total normalized EIG, dose budgets $B$ and normalized $95\%$ confidence interval widths. Dose constraints $\lambda\in\{0.01,0.02,0.04,0.08,0.1,0.2,0.4\}$ are used, with random and optimal baseline strategies in gray and red, respectively. For the dose budget plot, baselines represent a fixed (gray) or an adaptive (red) policy design choice.

Figure 10: Impact of reward approximation on policy learning. Training curves showing normalized cumulative $EIG$ for categorical policy learning under a Monte Carlo reward approximation with $N_{s},N_{r}\in\{2,4,8,16,32\}$ posterior samples. The same samples are used to compute both state and reward approximations. Curves are averaged across 100 independent policy training realizations.

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The Lift Spectrum: How Measurement-to-Space Adaptivity Shapes Robustness in Image-Free Single-Pixel Sensing

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Image Segmentation Representation Learning Computational Imaging Noise Robustness

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Single-pixel sensing encodes a scene as a short sequence of coded measurements, and image-free methods infer the task directly from that sequence. We show that removing image reconstruction relocates the central design problem to the lift: how 1D measurements become a 2D task representation. We organize this choice as a lift spectrum from a fixed-physics inverse, through a learned static projection, to content-adaptive retrieval. These are not interchangeable forms of reconstruction: the fixed-physics route reconstructs an image consumed at inference, whereas our spatiotemporal soft-fusion (STSF) network lifts measurements directly into task features, and task-prioritized loss scheduling (TPLS) uses a separate learned reconstruction branch only as scheduled training supervision. A probe-selected recurrent encoder and a parameter-matched lift ablation identify the STSF design. In simulation, STSF+TPLS exceeds the prior image-free baseline on three datasets at 3.13% sampling (+3.2 to +9.9 pp foreground mIoU) and remains competitive down to 0.39%. The strongest clean-trained reconstruct-then-segment baseline wins without measurement noise, but measurement noise reverses the ranking: the reconstructed task input carries a 20-70x larger normalized relative perturbation than the measurements themselves. Stressed to failure, the three lift regions exhibit distinct dominant signatures--collapse, imprinting, and coarsening. STSF+TPLS transfers without fine-tuning to a real single-pixel bench, where the reversal reappears as a proof of concept; inference takes about 14 ms per mask on an RTX 4090. Within the tested fixed-acquisition regime, measurement-to-space adaptivity therefore organizes both the clean-to-noisy operating envelope and the failure a system encounters. Code and pretrained weights: https://github.com/Hanyuyuan6/STSF-TPLS.

Abstract

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