An algorithm for image reconstruction from a series of short-exposure images with the use of the self-crosscorrelation (self-cross spectrum in the Fourier domain) between an observation g(x,y) and a truncated subimage of it, p(x,y) is described. The truncated subimages are chosen to be the brightest region in the observed images and of size comparable with that of the average point-spread function. We find that this self-cross spectrum retains the first-order approximation diffraction-limited phase information of the unknown object and does not rely on the integration of the second-order phase difference or the third-order phase closure estimate. For the case in which the components of the object corresponding to the truncated subimage p(x,y) have edges, a method to extract these edges and then use them to recover the phase of the object is presented. Results from computer-simulated data and from real data show the effectiveness of the method.