# Adaptive Truncation Threshold Determination for Multimode Fiber Single-Pixel Imaging
## 1 Introduction  
Multimode fibers (MMFs) are expected extensions of fiber bundles [1; 2; 3; 4; 5; 6], and examples include endoscopic probes [7; 8]. However, mode coupling and dispersion scramble transmitted images into unrecognizable speckles, rendering direct imaging impossible. Therefore, several methods have been developed for MMF imaging, such as (1) wavefront shaping-based raster scanning [9; 10; 11; 12]; (2) single-pixel imaging (SPI [13] or, equivalently, ghost imaging [14]); and (3) machine learning [15; 16; 17; 18]. Wavefront shaping requires complex interference setups to measure transmission matrices for characterization of MMF transmission and to control the output of scanned point-by-point MMF objects. Machine learning methods leverage large datasets to train models that can reconstruct images from speckle patterns and require extensive training data. In MMF-SPI [19; 20; 21; 22; 23; 24; 25; 26; 27; 28], MMF is used as a light projector to illuminate the target; then, the light intensity response of the target is recorded. With the help of SPI algorithms, the image can be recovered [29]. Compared with raster scanning and machine learning methods, SPI has advantages in terms of simplicity compared to wavefront shaping because only light intensity is required, avoiding interference setups; it also has advantages in terms of imaging speeds compared to machine learning because a few measurements are enough. Compared with conventional SPI, MMF-SPI is more susceptible to stochastic noise; thus, not all SPI algorithms work. A linear correlation method such as second-order correlation is a simple and quick SPI algorithm, relying on the orthogonality of the measurement matrix, but it often performs poorly in practical MMF-SPI [21; 24] because the measurementmatrix derived from MMF speckle patterns typically has lower orthogonality than assumed. Regularized optimization methods such as compressive sensing (CS) [19; 20; 21; 22; 23; 24] can provide better image quality through the fine tuning of regularized parameters. However, this key parameter can only be empirically specified beforehand or is gradually found via costly iterations, ruling out high-speed imaging. For machine learning, the acquisition of large-scale data, model training, and parameter fine tuning are inevitable [30]. In terms of balancing costs, performance, and robustness, the relatively old truncated singular value decomposition (TSVD) still enjoys unique advantages. Computing the pseudo-inverse of measurement matrices with a suitable truncation threshold to maintain image information and suppress noise at the same time, TSVD has become popular with MMF-SPI [25; 26; 27; 28]. However, MMF-SPI is fragile relative to disturbances [31; 32; 33; 34; 35; 36] such as fiber bending, resulting in dynamical changes in noise levels and, thus, requiring a flexible truncation threshold for recovery. When complete knowledge of the target is known, it is possible to test all potential truncation thresholds to identify the optimal one, which is known as the clairvoyant benchmark. This benchmark minimizes the recovery error and serves as a reference for truncation threshold selection. In practice, this critical truncation threshold can be chosen without prior knowledge via traditional methods such as the L-curve [37] and generalized cross-validation (GCV) [38; 39] methods. The L-curve method selects the truncation point at the curve's corner, and it is plotted as the norm of the solution against the norm of the residual. GCV selects the threshold by minimizing the cross-validation error. The computational complexity of both methods is related to the square of the pixel number, rendering them unsuitable for high-speed imaging applications, especially in disturbed conditions requiring frequent re-determination. For more efficient methods with less computation, fixed truncation and [PERSON]'s unbiased risk estimator (SURE) [40] are recommended. The former selects truncation thresholds directly via the singular value (SV) spectrum [25; 26; 27; 28].