Signal-to-noise ratio analysis of single-pixel detection multiplexing under photon-noise. Cases of Hadamard and Cosine positive modulation

In typical single-pixel detection multiplexing, an unknown object is sequentially illuminated with intensity patterns: the total signal is summed into a single-pixel detector and is then demultiplexed to retrieve the object. Because of measurement noise, the retrieved object differs from the ground truth by some error quantified by the signal-to-noise ratio (SNR). In situations where the noise only arises from the photon counting process, it has not been made clear if single-pixel detection multiplexing leads to a better SNR than simply scanning the object with a focused intensity spot - a modality known as raster scanning. This study theoretically assesses the SNR associated with certain types of single-pixel detection multiplexing, and compares it with raster scanning. In particular, we show that, under photon noise, when the positive intensity modulation is based on Hadamard or Cosine patterns, single-pixel detection multiplexing does not systematically improve the SNR as compared to raster scanning. Instead, it only improves the SNR on object pixels at least $k$ times brighter than the object mean signal $\bar{x}$, where $k$ is a constant that depends on the modulation scheme.