  Single-pixel imaging (SPI) is very popular in subsampling applications, but
the random measurement matrices it typically uses will lead to measurement
blindness as well as difficulties in calculation and storage, and will also
limit the further reduction in sampling rate. The deterministic Hadamard basis
has become an alternative choice due to its orthogonality and structural
characteristics. There is evidence that sorting the Hadamard basis is
beneficial to further reduce the sampling rate, thus many orderings have
emerged, but their relations remain unclear and lack a unified theory. Given
this, here we specially propose a concept named selection history, which can
record the Hadamard spatial folding process, and build a model based on it to
reveal the formation mechanisms of different orderings and to deduce the mutual
conversion relationship among them. Then, a weight ordering of the Hadamard
basis is proposed. Both numerical simulation and experimental results have
demonstrated that with this weight sort technique, the sampling rate,
reconstruction time and matrix memory consumption are greatly reduced in
comparison to traditional sorting methods. Therefore, we believe that this
method may pave the way for real-time single-pixel imaging.
