Lens-free holographic microscopy is a promising diagnostic approach because it is cost-effective, compact, and suitable for point-of-care applications, while providing high resolution together with an ultra-large field-of-view. It has been applied to biomedical sensing, where larger targets like eukaryotic cells, bacteria, or viruses can be directly imaged without labels, and smaller targets like proteins or DNA strands can be detected via scattering labels like micro- or nano-spheres. Automated image processing routines can count objects and infer target concentrations. In these sensing applications, sensitivity and specificity are critically affected by image resolution and signal-to-noise ratio (SNR). Pixel super-resolution approaches have been shown to boost resolution and SNR by synthesizing a high-resolution image from multiple, partially redundant, low-resolution images. However, there are several computational methods that can be used to synthesize the high-resolution image, and previously, it has been unclear which methods work best for the particular case of small-particle sensing. Here, we quantify the SNR achieved in small-particle sensing using regularized gradient-descent optimization method, where the regularization is based on cardinal-neighbor differences, Bayer-pattern noise reduction, or sparsity in the image. In particular, we find that gradient-descent with sparsity-based regularization works best for small-particle sensing. These computational approaches were evaluated on images acquired using a lens-free microscope that we assembled from an off-the-shelf LED array and color image sensor. Compared to other lens-free imaging systems, our hardware integration, calibration, and sample preparation are particularly simple. We believe our results will help to enable the best performance in lens-free holographic sensing.