This paper deals with the development of a fast and smart acquisition technique of Optical Coherence Tomography (OCT) data that has the capability to reconstruct missing data of OCT image. The main objective is to reduce the acquisition time (i.e., increase the frame rate) of an OCT-scan system by choosing a trajectory that covers entirely the image but that does not take all the measurements. The reconstruction of the missing data is achieved by applying an updated Fast Iterative Soft-Thresholding Algorithm (FISTA) on a sparse representation of the image. Several sparse representations have been tested and the shearlets-based approach seems to outperform the other ones (e.g. wavelets, curvelets and by applying bilinear interpolation). The targeted application is a fast OCT imaging solution allowing an efficient compensation of the artefacts induced by the patient physiological motions for diagnostic purpose through optical biopsies (3D micrometric resolution optical images). The obtained results seem promising in terms of low time processing and improvement of the quality of the reconstructed image compared to the traditional sparse acquisition.