Hadamard coding spectral imaging technology is a computational spectral imaging technology, which modulates the target's spectral information and recovers the original spectrum by inverse transformation. Because it has the advantage of multichannel detection, it is being studied by more researchers. For the engineering realization of push-broom coding spectral imaging instrument, it will inevitably be subjected to push-broom error, template error and detection noise, the redundant sampling problem caused by detector. Therefore, three restoration methods are presented in this paper: firstly, the one is the least squares solution, the two is the zero-filling inverse solution by extending the coding matrix in the redundant coding state to a complete higher order Hadamard matrix, the three is sparse method. Secondly, the numerical and principle analysis shows that the inverse solution of zero-compensation has better robustness and is more suitable for engineering application; its conditional number, error expectation and covariance are better and more stable because it directly uses Hadamard matrix, which has good generalized orthogonality. Then, a real-time spectral reconstruction method is presented, which is based on inverse solution of zero-compensation. Finally, simulation analysis shows that spectral data could be destructed relative accuracy in the error condition; however, the effect of template noise and push error on reconstruction is much greater than that of detection error. Therefore, in addition to reducing the detection noise as much as possible, lower template noise and more accurate push controlling should be guaranteed specifically in engineering realization.