Fringe projection profilometry (FPP) is a non-contact, high-precision technique for measuring three-dimensional (3D) shapes. An essential step of FPP is to recover the phase distribution from the deformed fringe patterns. In real applications, the captured fringe patterns often suffer from noises, which results in degradation of the performance of phase retrieval and shape reconstruction. Fringe denoising can be applied to suppress the influence of noise in FPP. This paper introduces a novel fringe denoising method based on robust principal component analysis (RPCA). The proposed method makes use of the low-rankness of the clean fringe patterns and the sparsity of the strong impulsive fringe noise. RPCA is then applied to effectively mitigate the strong impulsive fringe noise and suppress the random additive noise. The proposed method features 2D processing of the fringe patterns and is easy to implement. Its effectiveness is demonstrated via numerical simulations.