A Review of Some Recent Work in the Area of Imaging and Optical Signal Processing <s> Fast Algorithms, Sampling and Aliasing <s> A signal may have compact support, be band-limited (i.e., its Fourier transform has compact support), or neither ("unbounded"). We determine conditions for the linear canonical transform of a signal having these properties. We examine the significance of these conditions for special cases of the linear canonical transform and consider the physical significance of our results. <s> BIB001 </s> A Review of Some Recent Work in the Area of Imaging and Optical Signal Processing <s> Fast Algorithms, Sampling and Aliasing <s> The linear canonical transform describes the effect of first-order quadratic phase optical systems on a wave field. Several recent papers have developed sampling rules for the numerical approximation of the transform. However, sampling an analog function according to existing rules will not generally permit the reconstruction of the analog linear canonical transform of that function from its samples. To achieve this, an additional sampling criterion has been developed for sampling both the input and the output wave fields. <s> BIB002 </s> A Review of Some Recent Work in the Area of Imaging and Optical Signal Processing <s> Fast Algorithms, Sampling and Aliasing <s> The numerical approximation of the linear canonical transform (LCT) is of importance in modeling first order optical systems and many signal processing applications. We have considered an approach based on discretizing the continuous LCT, making careful consideration of the consequences for the range and resolution of the output. <s> BIB003 </s> A Review of Some Recent Work in the Area of Imaging and Optical Signal Processing <s> Fast Algorithms, Sampling and Aliasing <s> The linear canonical transform provides a mathematical model of paraxial propagation though quadratic phase systems. We review the literature on numerical approximation of this transform, including discretization, sampling, and fast algorithms, and identify key results. We then propose a frequency-division fast linear canonical transform algorithm comparable to the Sande–Tukey fast Fourier transform. Results calculated with an implementation of this algorithm are presented and compared with the corresponding analytic functions. © 2010 Optical Society of America OCIS codes: 070.4560, 080.2730, 100.2000, 200.2610, 200.3050, 200.4560, 200.4740. <s> BIB004 </s> A Review of Some Recent Work in the Area of Imaging and Optical Signal Processing <s> Fast Algorithms, Sampling and Aliasing <s> Sampling a function periodically replicates its spectrum. As a bilinear function of the signal, the associated Wigner distribution function contains cross terms between the replicas. Often neglected, these cross terms affect numerical simulations of paraxial optical systems. We develop expressions for these cross terms and show their effect on an example calculation. <s> BIB005
The Fast Fourier Transform (FFT) is a way of numerically calculate in NlogN time the Discrete Fourier Transform (DFT), which requires a calculation time proportional to N 2 (where N is the number of samples being processed). The Fourier Transform (FT) is a very special case of the LCT, therefore is seems reasonable to ask whether a Fast LCT algorithm (FLCT) exists. It does and we have recently provided a very detailed description of such an algorithm in BIB004 . Given such a fast algorithm the selection of the number of samples N then become critical. It is well known that the Space Bandwidth Product (SBP), i.e., the product of spatial extent of a signal by its spatial frequency bandwidth, can be used to estimate the number of regular samples (uniform sampling rate) necessary to meet the Shannon sampling criterion (i.e. the Nyquist rate). We note that once this rate of sampling is available the analogue signal can be extracted from the digitally processed representation. However real signals cannot be finite in both the space and spatial frequency domains and therefore replication and aliasing (overlap) occurs. These effects are well understood in Fourier Optics but are more complex to examine when using PSO. The effects of compactness (finite extent) in various domains on sampling, and thus performing calculations, is explored in BIB001 , while the rates of sampling used are further explored in BIB002 BIB003 . Furthermore while the FLCT discussed above provides one approach to rapidly calculating the LCT of a signal, returning to the matrix description of such systems and decomposing the system ABCD matrix into the product of several sub-system matrices, other fast algorithms can be identified. The Fresnel Transform, (FST), is another special case of the LCT, and describes paraxial propagation in free space. For obvious reasons the FST has received a great deal of attention in the literature. At the moment it is of particular interest because of its' use in processing Digital Holographic image data. Our approach has allowed us to re-evaluate one of the most popular decomposition based algorithms, the Direct Method of calculating the FST  . We have also been able to extend the PSO analysis in such a way as to allow us to re-interpret aliasing as related to the WDF cross terms produced by sampling BIB005 .