Many significant features of images are represented in their Fourier transform. The spectral phase of an image can often be measured more precisely than magnitude for frequencies of up to a few GHz. However, spectral magnitude is the only measurable data in many imaging applications. In this paper, the reconstruction of complex-valued images from either the phases or magnitudes of their Fourier transform is addressed. Conditions for unique representation of a complex-valued image by its spectral magnitude combined with additional spatial information is investigated and presented. Reconstruction algorithms of complex-valued images are developed and introduced. Three types of reconstruction algorithms are presented. (1) Algorithms that reconstruct a complex-valued image from the magnitude of its discrete Fourier transform and part of its spatial samples based on the autocorrelation function. (2) Iterative algorithms based on the Gerchberg and Saxton approach. (3) Algorithms that reconstruct a complex-valued image from its localized Fourier transform magnitude. The advantages of the proposed algorithms over the presently available approaches are presented and discussed.