  Single-pixel imaging (SPI) has the advantages of high-speed acquisition over
a broad wavelength range and system compactness, which are difficult to achieve
by conventional imaging sensors. However, a common challenge is low image
quality arising from undersampling. Deep learning (DL) is an emerging and
powerful tool in computational imaging for many applications and researchers
have applied DL in SPI to achieve higher image quality than conventional
reconstruction approaches. One outstanding challenge, however, is that the
accuracy of DL predictions in SPI cannot be assessed in practical applications
where the ground truths are unknown. Here, we propose the use of the Bayesian
convolutional neural network (BCNN) to approximate the uncertainty (coming from
finite training data and network model) of the DL predictions in SPI. Each
pixel in the predicted result from BCNN represents the parameter of a
probability distribution rather than the image intensity value. Then, the
uncertainty can be approximated with BCNN by minimizing a negative
log-likelihood loss function in the training stage and Monte Carlo dropout in
the prediction stage. The results show that the BCNN can reliably approximate
the uncertainty of the DL predictions in SPI with varying compression ratios
and noise levels. The predicted uncertainty from BCNN in SPI reveals that most
of the reconstruction errors in deep-learning-based SPI come from the edges of
the image features. The results show that the proposed BCNN can provide a
reliable tool to approximate the uncertainty of DL predictions in SPI and can
be widely used in many applications of SPI.
