Reconstruction of a vibrating object from its time-averaged image intensities by the use of exponential filtering

We consider the reconstruction of a complex-valued object that vibrates in some out-of-plane modes. The reconstruction is based on the phase-retrieval method with the use of two intensity measurements: the two time-averaged image intensities of the object illuminated coherently, which are modulated in two Fourier-transform planes of the object by the use of two filters with exponentially decaying transmittances that complement each other. We discuss the necessary condition of the vibration for the reconstruction method. Computer-simulated examples of retrieving the phases of one-dimensional objects demonstrate that the reconstruction of a sinusoidal-vibrating and a Gaussian random-vibrating object can be treated by this method.