To date, no fringe analysis technique has the capability to provide simultaneous and direct estimation of the continuous distributions corresponding to the interference phase and its first and second-order derivatives within the framework of a single interferometric configuration. Achieving this task would provide a significant advancement in the field of optical metrology as it allows for the measurement of displacement, strain, and curvature of a deformed object and avoids the necessity of using filtering and unwrapping procedures, multiple analysis techniques, and multiple interferometric configurations. Developing such a spatial fringe analysis method with the added advantage of having less computational complexity would open up avenues for making real-time measurements such as in the study of temporal evolution of deformation and/or strain. This thesis presents a novel approach based on piecewise polynomial phase approximation as an elegant all-in-one solution to the problems mentioned above. This approach has given birth to several advanced fringe analysis methods such as discrete-chirp-Fourier transform method, high-order instantaneous moments method, and cubic-phase function method. Significant advancements brought in the field by these methods are made evident by both theoretical analysis (simulation results) and by experimental demonstrations such as the measurement of displacement, strain and curvature in digital holographic interferometry and the measurement of 3D shape, temporal evolution of deformation and/or strain in fringe projection techniques.