When a parallel beam of light undergoes a discontinuous change in phase or in phase gradient in the central part of the beam, two wavefronts with a common border are formed. In this paper, we formulate the interference of two such wavefronts and show that conventional diffraction and interference are special cases of a generalized interference with amplitudes that are determined by the distances of the observation point from the wavefronts and their boundaries. This formulation permits us to reconstruct the interfering wavefronts at their original locations. Also, we modulate a phase function on the interference fringes near the common boundary, where the diffraction effect is significant, and show this kind of modulation provides a basis for quantitative imaging of phase objects in small scales. Furthermore, we evaluate the theoretical results by relevant experiments and elaborate on features and application potential of the interferometry introduced.