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Introduction
Published in Neil Collings, Fourier Optics in Image Processing, 2018
The centre of the Fourier transform is displaced to a new position and the extent of the displacement is given by -rz0k $ -\frac{rz_0}{k} $ . Conversely, translations in the aperture plane give rise to corresponding phase changes in the output plane. This property of the Fourier transform is known as the Fourier shift theorem.
Higher-order linear differential equations
Published in Alan Jeffrey, Mathematics, 2004
If F(s) exists for s > γ, then F(s – a) exists for s – a > γ. The name ‘shift theorem’ is derived from the fact that multiplication of f(t) by eat shifts the variable s in the Laplace transform to s – a.
Ultrashort Pulses
Published in Chunlei Guo, Subhash Chandra Singh, Handbook of Laser Technology and Applications, 2021
where τ is known as the group delay. In its entirety, the second term in equation (19.14) describes a linear phase ramp in frequency which corresponds (through the well-known Fourier shift theorem [139]) to a delay of the pulse in time.
Method of correcting stitching errors in reconstructing a synthetic-aperture digital hologram with seams
Published in Journal of Modern Optics, 2019
Meng Ding, Qi Fan, Yin Su, Baiyu Yang, Changhui Tian, Binke Wang, Yunfei Wang
To further satisfy the requirements of dynamic testing and to enlarge the FOV, a novel method based on phase recovery for reconstructing a synthetic-aperture digital hologram with seams was proposed (9); the method involved merging several CCD arrays into one large recording plane at the same time. However, the synthesis of a correct hologram remains a challenge when using this method. Hongzhen Jiang et al. determined the misplacements of sub-holograms by evaluating the quality of the corresponding synthetic reconstructed images (10). The exact stitching error was determined by performing a cross-correlation of images reconstructed from the sub-holograms and calculating the amount of shift from the reconstructed object (11,12). The stitching error can be divided into integer pixel dislocations and non-integer pixel dislocations. Because the latter cannot be corrected via straightforward shifting and is more common in practice, an appropriate correction method is highly relevant and must be developed, according to (11,12). In this paper, we propose a method based on the shift theorem of the Fourier transform. The sub-holograms stemming from the dislocated recording planes in the frequency domain are multiplied by a phase factor and then superposed onto the in-place sub-holograms. The entire synthesized hologram is then transformed into the spatial domain, where it provides a high-quality reconstructed image. Using simulated particle detection as an example, we find that the CC is essentially identical to that of an image without stitching errors and that the proposed method is stable against noise in most cases. These findings demonstrate that the non-integer pixel stitching error has been corrected, further verifying the feasibility of the proposed method.