By investigating links between Reed-Muller transform and Walsh-Hadamard spectra an exact and non-exhaustive algorithm for the generation of optimal Reed-Muller expansions directly from just few Walsh-Hadamard spectral coefficients has been developed. The algorithm makes use of the properties of Walsh-Hadamard spectra and by using only few Walsh-Hadamard coefficients the optimal Reed-Muller expansion is obtained for all Boolean functions through the provided equations in the new algorithm.