Lens-free holographic microscopy offers sub-micron resolution over an ultra-large field-of-view >20 mm2, making it suitable for bio-sensing applications that require the detection of small targets at low concentrations. Various pixel super-resolution techniques have been shown to enhance resolution and boost signal-to-noise ratio (SNR) by combining multiple partially-redundant low-resolution frames. However, it has been unclear which technique performs best for small-target sensing. Here, we quantitatively compare SNR and resolution in experiments using no regularization, cardinal-neighbor regularization, and a novel implementation of sparsity-promoting regularization that uses analytically-calculated gradients from Bayer-pattern image sensors. We find that sparsity-promoting regularization enhances the SNR by ~8 dB compared to the other methods when imaging micron-scale beads with surface coverages up to ~4%.