In this paper, we make a proposal to obtain the Hilbert-transform for each entry of the projection data leaving the slice of a thin phase object. These modified projections are stacked in such a way that they form a modified sinogram called Hilbert-sinogram. We prove that the inverse Radon-transform of this sinogram is the directional Hilbert-transform of the slice function, and the reconstructed image is the directional edge enhancement of the distribution function on the slice. The Hilbert-transform is implemented by a 4f optical Fourier-transform correlator and a spatial filter consisting of a phase step of π radians. One important feature of this proposal is to perform a turn of 180° in the spatial filter at a certain value of the projection angle within the range [0°, 360°]. The desired direction of enhancement can be chosen by the proper selection of such turning angle. We present both the mathematical modeling and numerical results.