In the present paper, the authors will introduce our research on spectral reconstruction of Fourier transform computed tomography imaging spectrometer by means of the algebraic reconstruction technology. A simulation experiment was carried out to demonstrate the algorithm. The spatial similarities and spectral similarities were evaluated using the normalized correlation coefficient. The performance of ART was evaluated when the quantity of projection is 45. In that case, filter back projection can't work well. Actual spectral slices were reconstructed by using ART in the last part of this paper.