Noise-resilient single-pixel compressive sensing with single photon counting

The study investigates noise limitations in single-photon, single-pixel compressive sensing systems used for imaging and classification. Conventional single-pixel approaches, while cost-effective and flexible, are highly sensitive to noise at low photon levels, including ambient background light, detector dark counts, and inherent Poisson fluctuations. This sensitivity hampers applications such as LiDAR under daylight conditions. The authors aim to quantify how photon-counting and ambient noises affect classification accuracy and to demonstrate a noise-resilient approach based on quantum parametric mode sorting (QPMS) with frequency upconversion. By selectively converting photons in a target spatiotemporal mode and shifting the wavelength to allow low-dark-count silicon detection, QPMS is expected to significantly increase signal-to-noise ratio and robustness to in-band noise. The work’s purpose is to provide a practical pathway to high-accuracy, low-light single-photon sensing and classification under strong ambient noise conditions, using MNIST digit recognition as a testbed.

The paper situates its contribution within single-pixel compressive sensing and imaging literature, including applications in microscopy, terahertz/short-wave IR imaging, imaging through scattering media, and 3D sensing. Foundational compressive sensing theory (Candes, Tao; Donoho) and single-pixel architectures are referenced, along with structured pattern encoding (Fourier, Hadamard, Walsh) and ordering strategies for efficient sampling. Recent advances using AI and deep learning for enhanced reconstruction and classification with compressed data are noted (e.g., deep learning early stopping, multiscale reconstruction, recurrent networks). For mitigating background noise, mid- and near-IR upconversion detection is reviewed as a superior alternative to direct IR thermal detection due to lower noise and higher sensitivity in the visible with Si-SPDs. Prior work in photon-starved imaging, coincidence-pumping upconversion, mid-IR photon counting, and Hadamard-coded upconversion imaging is cited. QPMS (mode-selective upconversion) is highlighted as enabling spatiotemporal mode selectivity and improved SNR, with relevance to LiDAR and noise-tolerant single-photon imaging.

Experimental setup: A femtosecond mode-locked laser (CALMAR FPL-03CFF) at ~1550 nm (FWHM ~60 nm), 50 MHz repetition rate, is WDM-filtered into probe and pump arms. After further filtering and amplification, both pulses are ~6 ps. The QPMS waveguide’s phase-matching bandwidth is ~1 nm, enabling high mode selectivity with broadband pump pulses. Probe and pump are amplified via EDFAs and filtered. Probe power is attenuated to nanowatt levels via a variable attenuator and 99:1 splitter, then collimated to a DMD showing Walsh 2D patterns and relayed to an SLM that encodes MNIST digit images as binary phase patterns. MNIST 28×28 images are resized to 240×240 to match the probe diameter. The modulated probe is coupled to single-mode fiber and split 50:50 into two detection paths. Direct detection (DD) path: Photons are directly detected by an InGaAs single-photon detector (ID210) with 1 ns effective gate width and 20% quantum efficiency; including fiber/filter losses, total detection efficiency is 4.2%. QPMS upconversion path: The other probe arm is combined with the pump in fiber (WDM) and coupled into a fiber-coupled MgO:PPLN waveguide module for sum-frequency generation (mode-selective upconversion). Pump delay (ODL) and polarization controllers are used to maximize conversion. Upconverted output at ~779.59 nm is detected by a silicon SPD (Excelitas) with 66% QE. The PPLN module has maximum normalized internal conversion efficiency 207.7% W⁻¹ cm⁻²; accounting for losses, the total QPMS detection efficiency is 12.0%. Both SPD outputs are recorded by a Swabian Time Tagger Ultra. Noise injection: In-band background noise is simulated by ASE generated from an EDFA with no input, WDM-filtered around the probe band, re-amplified, and power-adjusted with a mechanical attenuator. The noise is introduced via a 50:50 fiber coupler so it is in-band and co-propagating with the probe. The SNR is defined from measured photon counts with/without ASE and dark counts. Patterns and acquisition: Walsh 2D masks are uploaded to the DMD. For each digit, 300 masks are used by default; two white patterns are appended for target demarcation. The DMD pattern change time is ~100 µs, and photon-counting events during switching are discarded. For illustrative acquisition settings: DD path often records 10 events per pattern at 100 µs each, averaging over 8 events (first and last dropped); QPMS path often records 25 events per pattern at 40 µs each, averaging over 20 events (first three and last two dropped), giving 0.8 ms effective integration per pattern. Integration time and probe power are varied to control photon counts and study shot-noise effects, while avoiding SPD saturation. Deep neural network model: A 7-layer DNN is used for classification with ReLU activations in hidden layers and a log-softmax before the output layer. Dataset comprises 1000 samples (100 images per digit) with about 300 mean photon counts per sample. Train/test split is 75%/25%. Training epochs vary by experiment (e.g., DD typically 300 epochs; QPMS 200 epochs). MATLAB is used for post-processing. Characterization: Dark counts versus integration time are measured for both SPDs. Representative dark count levels translate to ~234 counts per 800 µs for DD and ~40 counts per 200 µs for QPMS. QPMS also exhibits Raman scattering noise in the conversion module, which is quantified separately. NEDC (noise equivalent of dark counts) is computed as dark counts divided by detection efficiency for comparable integration windows.

- Without external in-band noise: Classification accuracy strongly depends on detected photon counts per mask and detector noise. - DD (InGaAs SPD): With 600 µs/mask and probe ~14 nW, average 398.6 counts/mask yields 82.8% accuracy. Increasing integration to 800 µs at same power gives 551.2 counts/mask and 90.0% accuracy. With 800 µs but lower probe (~8.5 nW), 391.5 counts/mask yields 80.4% accuracy. Reducing probe to ~4 nW (800 µs) yields ~299.7 counts/mask and only 31.6% accuracy. DD accuracy degrades rapidly below ~300 counts/mask, consistent with shot-noise limits and higher dark count fraction. - QPMS (Si-SPD after upconversion): With 200 µs/mask and probe ~14 nW, average 485.8 counts/mask yields 98% accuracy—significantly higher than DD at similar or even higher counts, due to higher total efficiency (12.0% vs 4.2%) and lower dark counts. - Accuracy versus average counts and mask number: For QPMS, at ~119.7 counts/mask, accuracy is 97.2% (300 masks) and 93.2% (100 masks); with ~500 counts/mask, accuracy saturates near 99.2% for both. For DD, at ~59.5 counts/mask, accuracy is 42.4% (300 masks) and 26.4% (100 masks); at ~550 counts/mask, accuracy is ~90% (300 masks) and 82.8% (100 masks). DD was limited by InGaAs SPD saturation preventing higher counts. - Dark counts and baseline noise: Measured dark counts scale with integration time. Example effective baselines: ~234 counts per 800 µs for DD versus ~40 counts per 200 µs for QPMS. NEDC per 40 µs window is 281 for DD (11.8 dark counts, 4.2% efficiency) versus 71 for QPMS (8.5 dark counts, 12.0% efficiency), indicating much lower noise-equivalent baseline for QPMS. - With in-band ASE noise: QPMS shows strong robustness, while DD degrades severely. - DD: With SNR 0 dB (ASE power equal to signal at combining point), counts per pattern ~67–115, accuracy 22.0% (300 masks, 300 epochs). At SNR 3 dB, accuracy improves to 51.6%. Attempting SNR −3 dB led to SPD saturation. - QPMS: Maintains >98% accuracy when the signal is 100× weaker than noise (SNR −20 dB). At SNR −27 dB (signal 500× weaker), accuracy is still 94%. - Noise rejection: QPMS rejects >99.9% of in-band noise so that detected noise is near the Si-SPD dark count level, preserving high contrast in photon-count features across DMD masks. - Cross-SNR generalization: Training at one ASE level and testing at another maintains high accuracy with QPMS (e.g., training at 0 dB and testing without noise; training at −10 dB and testing at −17 dB), achieving ≥98% with 300 masks. - Practical insight: Reducing SPD integration time increases the relative dark count contribution; QPMS benefits from higher dynamic range and lower dark counts, yielding more pronounced features for the DNN to classify.

The findings directly address the core challenge of noise in single-photon compressive sensing. DD performance is constrained by higher dark counts, lower detection efficiency, and saturation limits of InGaAs SPDs, causing the baseline to occupy a larger fraction of total counts and obscuring feature variations across DMD masks. In contrast, QPMS provides spatiotemporal mode selectivity that preferentially converts the targeted return mode, effectively rejecting >99.9% of in-band background photons even when they are co-temporal and co-spatial. Combined with wavelength translation to the visible for detection by low-dark-count, high-dynamic-range Si-SPDs, QPMS reduces the noise-equivalent baseline (lower NEDC) and increases usable photon-count contrast, enabling near-saturation classification accuracy at modest photon counts per mask and robustness under extreme in-band noise conditions (down to SNR ≈ −27 dB). The observed accuracy improvements with increasing photon counts reflect shot-noise behavior; QPMS’s higher total detection efficiency (12.0%) accelerates entry into the high-SNR regime compared to DD (4.2%). The study also suggests an additional potential advantage of QPMS via picosecond-scale timing resolution approaching the inter-pixel time-of-flight variations introduced by DMD/SLM geometry, possibly encoding additional spatial distribution information in the counts. Although not verified here, this could further enhance feature separability for classification. Overall, the results imply that QPMS-based single-pixel systems can maintain high recognition performance in realistic environments dominated by sunlight or ambient noise, with fewer masks and shorter integration times than DD.

The work demonstrates a noise-resilient single-pixel compressive sensing and classification system by integrating mode-selective upconversion (QPMS) with single-photon counting and machine learning. Using MNIST digits, the system achieves 98–99% accuracy without added noise at moderate photon counts per mask and maintains 94% accuracy even when the signal is 500× weaker than in-band ASE noise (SNR −27 dB), far outperforming direct detection. Improvements stem from QPMS’s strong mode selectivity, wavelength translation to low-noise Si-SPDs, higher total detection efficiency, and lower noise-equivalent baselines. The approach offers a practical pathway to robust single-photon sensing and classification in challenging ambient conditions such as daylight and multi-scattering environments. Future work could validate and exploit potential timing-resolution advantages of QPMS for encoding spatial information, explore higher compression (fewer masks) with advanced learning models, optimize upconversion efficiency and losses, and extend to other sensing tasks (e.g., LiDAR, 3D imaging) and more complex datasets.

- Detector trade-offs: InGaAs SPD performance requires balancing quantum efficiency, dark counts, and saturation; lowering dark counts by reducing QE also reduces saturation counts and dynamic range. Si-SPD offers lower dark counts and higher saturation but requires upconversion. - Saturation constraints: In the DD channel, saturation limited the maximum usable photon counts, restricting achievable accuracy improvements. - Unverified advantages: The hypothesized benefit from picosecond timing resolution (potential encoding of spatial distribution via time-of-flight differences) was not verified in this study. - Setup constraints: The DMD+SLM configuration was chosen due to equipment availability; other configurations may perform differently. Coupling and component losses impact total detection efficiencies. - Dataset and protocol: Experiments used a reduced MNIST subset (100 images per digit) and specific training regimes; generalization to larger, more diverse datasets or different tasks may require additional validation. - Noise model: ASE was used to emulate in-band background; real-world ambient noise may include spectral/temporal/spatial characteristics not fully captured by ASE.