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This article considers the problem of recovering edge-illumination x-ray phase contrast (EIXPC) images from a set of potentially Poisson noisy projection measurements. The authors cast a recovery as a sparse regularization problem based on Anscombe multiscale variance stabilizing transform (MS-VST) with fast discrete curvelet transform which was applied to simulated edge-illumination x-ray phase contrast images. For accurate modelling, the noise characteristics of the EIXPCi data are used to determine the relative importance of each projection. Two implementations of curvelet sparse regularization transforms were applied, including the unequally-spaced fast Fourier transform and the wrapping-based transform. The algorithms were evaluated in terms of contrast improvement, quality of image restoration, object perceptibility, and peak signal-to-noise ratio. The methods provide nearly optimal solution without excessive memory and recovery time requirement. The performance of the proposed algorithms is demonstrated through a series of complex numerical geometric and anthropomorphic phantom studies. The results of numerical simulations demonstrate that the discrete curvelet transform with MS-VST is fast and robust, and it can effectively improve image quality, preserve and enhance edges and restore lost information while significantly reducing the noise. Additionally, both sparse sampling and decreasing x-ray tube current (i.e. noisy data) lead to the reduction of radiation dose in the x-ray imaging.