© 2019 SPIE. Phase unwrapping is one of the key steps for fringe projection profilometry (FPP)-based 3D shape measurements. Conventional spatial phase unwrapping schemes are sensitive to noise and discontinuities, which may suffer from low accuracies. Temporal phase unwrapping is able to improve the reliability but often requires the acquisition of additional patterns, increasing the measurement time or hardware costs. This paper introduces a novel phase unwrapping scheme that utilizes composite patterns consisting of the superposition of standard sinusoidal patterns and randomly generated speckles. The low-rankness of the deformed sinusoidal patterns is studied. This is exploited together with the sparse nature of the speckle patterns and a robust principal component analysis (RPCA) framework is then deployed to separate the deformed fringe and speckle patterns. The cleaned fringe patterns are used for generating the wrapped phase maps using the standard procedures of phase shift profilometry (PSP) or Fourier Transform profilometry (FTP). Phase unwrapping is then achieved by matching the deformed speckle patterns that encode the phase order information. In order to correct the impulsive fringe order errors, a recently proposed postprocessing step is integrated into the proposed scheme to refine the phase unwrapping results. The analysis and simulation results demonstrate that the proposed scheme can improve the accuracy of FPP-based 3D shape measurements by effectively separating the fringe and speckle patterns.