A novel phase retrieval algorithm for broadband hyperspectral phase imaging from noisy intensity observations is proposed. It utilizes advantages of the Fourier transform spectroscopy in the self-referencing optical setup and provides additional, beyond spectral intensity distribution, reconstruction of the investigated object's phase. The noise amplification Fellgett's disadvantage is relaxed by the application of a sparse wavefront noise filtering embedded in the proposed algorithm. The algorithm reliability is proved by simulation tests and by results of physical experiments for transparent objects. These tests demonstrate precise phase imaging and object depth (profile) reconstruction.