A Review of Some Recent Work in the Area of Imaging and Optical Signal Processing <s> Measurement: Phase, Motion and Speckle <s> A reflection-based optical implementation of two simultaneous scale-invariant fractional Fourier transforms (FRTs) is used to develop a novel compact speckle photographic system. The system allows the independent determination of both surface tilting and in-plane translational motion from two sequential mixed domain images captured using a single camera. <s> BIB001 </s> A Review of Some Recent Work in the Area of Imaging and Optical Signal Processing <s> Measurement: Phase, Motion and Speckle <s> The formation of holograms is interpreted as the consequence of the bilinearity of the ambiguity function. Reconstruction can then be regarded as the manipulation of the ambiguity function. Specifically, we show that in the case of in-line holography, the reconstruction can be regarded as phase tomography. In this way we provide a unified picture for the formulation of both noninterferometric and interferometric phaseretrieval techniques. © 2007 Optical Society of America OCIS codes: 100.5070, 090.0090, 110.6960, 110.2990, 070.0070. <s> BIB002 </s> A Review of Some Recent Work in the Area of Imaging and Optical Signal Processing <s> Measurement: Phase, Motion and Speckle <s> The signal extraction method based on intensity measurements in two close fractional Fourier domains is examined by using the phase space formalism. The fractional order separation has a lower bound and an upper bound that depend on the signal at hand and the noise in the optical system used for measurement. On the basis of a theoretical analysis, it is shown that for a given optical system a judicious choice of fractional order separation requires some a priori knowledge of the signal bandwidth. We also present some experimental results in support of the analysis. <s> BIB003
A detailed overview of recent work in the area of PSO, to which we contributed, is given in  . PSO is an approximate phase space representation firmly grounded in the more physical WDF. While other such representation of signals exist and have been used to interpret optical systems, i.e., the Ambiguity function BIB002 , the WDF and the PSO offer simplicity. Furthermore, with use they can provide intuitive insights and suggest practical applications. One such insight is that the phase of a signal can be retrieved without the use of interferometric techniques by capturing projections (marginals) of the same signals' WDF in different domains. To do this in BIB003 the Fractional Fourier transform, (FRT), distributions (intensities) for different FRT orders were imaged. Thus the FRT, which is a special case of the LCT, was used and the signal phase extracted experimentally. The effects of FRT orders, noise in the system and the spatial frequency response were quantified. Based on another insight, provided by use of the WDF and LCT, it has been shown that simple Speckle Photography systems can be used to measure in-plane tilts and rotations by capturing two sequential images of the surface before and after motion BIB001  . Once again these images must be captured in different domains, i.e. the light had passed through different QPSs, undergoing different LCTs or ABCD matrix transforms. This sounds complex but can be understood very simply using the graphical techniques associated with PSO BIB001  . 