To solve the problem of metamerism in the color reproduction process, various spectral reflectance reconstruction methods combined with neural network have been proposed in recent years. However, these methods are generally sensitive to initial values and can easily converge to local optimal solutions, especially on small data sets. In this paper, we propose a spectral reflectance reconstruction algorithm based on the Back Propagation Neural Network (BPNN) and an improved Sparrow Search Algorithm (SSA). In this algorithm, to solve the problem that BPNN is sensitive to initial values, we propose to use SSA to initialize BPNN, and we use the sine chaotic mapping to further improve the stability of the algorithm. In the experiment, we tested the proposed algorithm on the X-Rite ColorChecker Classic Mini Chart which contains 24 colors, the results show that the proposed algorithm has significantly better performance compared to other algorithms and moreover it can meet the needs of spectral reflectance reconstruction on small data sets. Code is avaible at https://github.com/LuraZhang/spectral-reflectance-reconsctuction.