Snapshot spectral imaging methods acquire compressed 2D projections of the entire data cube, which reduces the cost of storage, transmission and data processing compared with the traditional scanning methods. For instance, the single pixel imaging (SPI) encoded the scene and then uses a spectrometer to sense these projections. After the acquisition, SPI recovers the hyperspectral image solving an optimization problem for the complete image which represents high costs in time and storage. For this reason, this work proposes to recover the image in SPI by formulating a problem for each band, because the same coding is applied individually to each band. This decreases the amount of data to process and facilitating its parallelization. In addition, using the result of the previous band as the initialization of the current band the proposed formulation converge faster. Results show that the proposed method outperforms in 4 dB of PSNR the method of solving the full data cube and the reconstruction times and the computation resource are reduced up to 4 times and 86% respectively.