Repository for the presented work: Compressive sensing model for Hadamard-based singlepixel microscopy supported by kernel density estimators

Compressive sensing model for Hadamard-based singlepixel microscopy supported by kernel density estimators H. Tobon-Maya1,*, S. Zapata-Valencia1, M. Obando2, F. Lucka2, E. Tajahuerce1, J. Lancis1. 1. A Institute of New Imaging Technologies (INIT), Universitat Jaume I, Castelló E12071, Spain 2. Centrum Wiskunde & Informatica, Science Park 123, Amsterdam, the Netherlands, 1098 XG *Author e-mail address: [email protected] Abstract: A statistical compressive sensing model for Hadamard-based single-pixel microscopy (HSPM) is constructed using kernel density estimators on a data base of microscopy images. The model’s performance is evaluated using a HSPM experimental set-up. © 2025 The Author(s) 1. Introduction. Unlike conventional imaging systems, in structured illumination single-pixel microscopy (SPM) a scanning base is projected over the sample using a spatial light modulator (SLM) [1]. This process is often performed by employing high-speed digital micromirror devices (DMDs) due to their achievable frame rates and wide spectral range. Each pattern of the scanning basis is demagnified by the microscope setup and projected over the sample plane under study. Various deterministic sampling strategies have been proposed [2,3]. In this work, Hadamard-based SPM (HSPM) is explored. Its performance under noise and low-intensity sensing makes it suitable for microscopy applications where signals are weak or sensitive to environmental disturbances such as mechanical vibrations. In HSPM, each pattern of the Hadamard basis 𝐻𝑘(𝑥, 𝑦), corresponding to a two-dimensional rectangular function, is demagnified over the sample plane using a microscope setup [4]. The pointwise product between 𝐻𝑘 and the sample 𝑠(𝑥, 𝑦) is integrated by a bucket detector, creating an intensity value 𝑦(𝑘). This yields an intensity vector 𝑦, which can be expressed as 𝑦 = 𝐻(𝑠), where H(∙) denotes the Hadamard transform. The intensity vector contains the Hadamard frequency spectrum (HS) of the sample. However, full reconstruction requires sampling of many patterns. To address this situation, diverse compressive sensing (CS) alternatives have been developed [5–7]. In these CS approaches, a likely estimation of the complete intensity vector y is retrieved based on a limited number of selected measurements. This concept is illustrated in Fig. 1. An example image is shown in panels (a) and (b), together with its corresponding two-dimensional Hadamard spectrum presented in panels (c) and (d). A compressive ratio metric, the Approximate Compressive Ratio (ACR), is used to quantify the fraction of Hadamard coefficients required for image reconstruction. The ACR is defined as 𝐴𝐶𝑅 =∑1(|𝐻𝑘(𝑠)| > 𝑚𝑒𝑎𝑛|𝐻(𝑠)|) 𝑇 𝑘=1 /𝑇 . In panel (e) and (f), the required sampling coefficients based on the ACR are highlighted in white. These correspond to the image reconstruction shown in panels (e) and (f), using the ACR as a selection criterion. In this case no CS algorithms are applied for reconstruction, but prior knowledge of the patterns is required.. A similarity index (SSIM) of 0.84 and a signal-to-noise ratio (SNR) of 77 dB were achieved by sampling only 21% of the total Hadamard basis Fig 1. Sampling ratio (SR) in Hadamard based single-pixel imaging. The images under study and their corresponding Hadamard transform are presented in panel (a) to (d) respectively. In (e) and (f) the needed coefficients to retrieve the images shown in (g) and (h) are shown. Since the prior knowledge of the sample HS demands an initial full sampling, computing the ACR for a given sample does not constitute a practical implementation. For fast and optimal subsampling scenarios, CS sampling approaches are applied. As an alternative to those traditional compressive sensing (CS) sampling methods [7–9], this work proposes a new approach based on the analysis of ACR filters. The goal is to define a general sampling kernel that can be used as a measurement input for CS in Hadamard-based single-pixel microscopy (HSPM). CW1B.4 Optica Imaging Congress 2025 (3D, COSI, DH, IS, pcAOP, RadIT) © Optica Publishing Group 2025 This Article © 2025 The Author(s) 2. Methods and results. The similarity in the distribution patterns of ACR filters, shown in Fig. 1, motivates the search for a function capable of generalizing the sampling of relevant Hadamard spectrum (HS) coefficients. To address this, a probabilistic sampling kernel is proposed. This kernel is constructed using the kernel density estimation (KDE) algorithm. To ensure generality, the analysis is extended to a large dataset of images. This enables the generation of robust sampling kernels optimized through KDE, an unsupervised machine learning method [10]. The KDE model does not assume a predefined distribution, which makes it suitable for modeling irregular data distributions, such as those observed in ACR filters in Fig 1. In the proposed approach, a dataset of ACR filters is used to determine the probability of a given Hadamard coefficient being significant enough to be included in the ACR filter. By selecting a sufficiently representative dataset of ACR filter coefficients 𝑓 𝑥, 𝑓 𝑦, KDE generates a probability density function (PDF) by summing individual probability kernels centered around each observation [11]. While KDE offers an accurate estimation of the PDF for irregular data, its reliability depends on the quantity of input data. A small dataset may fail to capture the full variability of the HS coefficients. To train the KDE model for ACR filter estimation in HSPM, a dataset of relevant sampling coefficients was constructed using the PatchCamelyon dataset [12]. This dataset is ideal due to its combination of both highand low-frequency image content. Once the set of relevant sampling coefficients was obtained, the KDE model was trained using the scikit-learn KDE implementation. [10] To test the performance of the proposed KDE sampling approach, an HSPM system was constructed as described in reference [4]. A 20× Mitutoyo microscope objective was used in a telecentric configuration to demagnify the encoded patterns projected by a Vialux 7001 DMD onto the sample plane. A Thorlabs avalanche photodiode APD431A was implemented as the bucket detector, and a digital acquisition device NI USB-6353 from National Instruments was used to digitize the voltage signal. The KDE model was trained on an AMD Ryzen 7 5800H processor without GPU support. Once trained, the model was capable of generating sampling bases at various sampling ratios. In Fig. 2, the model’s performance is illustrated. Panels (a) and (b) present the results obtained using scrambledbased CS sampling for HSPM. The inverse problem in all cases was solved using the L1-Magic algorithm [13]. Panels (c) and (d) show the results obtained using the proposed KDE-based approach. These demonstrate that the method effectively captures high-frequency information without compromising the resolution of the reconstructed image, even for sampling ratios as low as 20%. Fig 2. In panel (a) and (b) the results of retrieving an USAF test resolution chart using random Hadamard sampling are shown using a SR of 20% and 30% respectively. In (c) and (d) the results of applying the same SR but using the proposed method using Hadamard samples from the KDEbased PDF are shown. The scale bar corresponds to 9 𝜇𝑚 3. References [1] Y. Liu, J. Suo, Y. Zhang, and Q. 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