A number of methods have recently been proposed in the literature for the encryption of two-dimensional information by use of optical systems based on the fractional Fourier transform. Typically, these methods require random phase screen keys for decrypting the data, which must be stored at the receiver and must be carefully aligned with the received encrypted data. A new technique based on a random shifting, or jigsaw, algorithm is proposed. This method does not require the use of phase keys. The image is encrypted by juxtaposition of sections of the image in fractional Fourier domains. The new method has been compared with existing methods and shows comparable or superior robustness to blind decryption. Optical implementation is discussed, and the sensitivity of the various encryption keys to blind decryption is examined.

The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an $N\log N$ algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.

By use of matrix-based techniques it is shown how the space–bandwidth product (SBP) of a signal, as indicated by the location of the signal energy in the Wigner distribution function, can be tracked through any quadratic-phase optical system whose operation is described by the linear canonical transform. Then, applying the regular uniform sampling criteria imposed by the SBP and linking the criteria explicitly to a decomposition of the optical matrix of the system, it is shown how numerical algorithms (employing interpolation and decimation), which exhibit both invertibility and additivity, can be implemented. Algorithms appearing in the literature for a variety of transforms (Fresnel, fractional Fourier) are shown to be special cases of our general approach. The method is shown to allow the existing algorithms to be optimized and is also shown to permit the invention of many new algorithms.

Raman spectroscopy has been demonstrated to have diagnostic potential in areas such as urine and cervical cytology, whereby different disease groups can be classified based on subtle differences in the cell or tissue spectra using various multi-variate statistical classification tools. However, Raman scattering is an inherently weak process, which often results in low signal to noise ratios, thus limiting the method's diagnostic capabilities under certain conditions. A common approach for reducing the experimental noise is Savitzky–Golay smoothing. While this method is effective in reducing the noise signal, it has the undesirable effect of smoothing the underlying Raman features, compromising their discriminative utility. Maximum likelihood estimation is a method for estimating the parameters of a statistical model given an available dataset and a priori knowledge of the model type. In this paper, we demonstrate how Savitzky–Golay smoothing may be enhanced with maximum likelihood estimation in order to prevent significant deviation from the ‘true’ Raman signal yet retain the robust smoothing properties of the Savitzky–Golay filter. The algorithm presented here is demonstrated to have a lower impact on Raman spectral features at known spectral peaks while providing superior denoising capabilities, when compared with established smoothing algorithms; artificially noised databases and experimental data are used to evaluate and compare the performance of the algorithms in terms of the signal to noise ratio. The proposed method is demonstrated to typically provide at least a 50% increase in the signal to noise ratio when compared to the raw data, and consistently out-performs two alternative smoothing filters.

Raman spectroscopy is an optical technique based on the inelastic scattering of monochromatic light that can be used to identify the biomolecular composition of biological cells and tissues. It can be used as both an aid for understanding the etiology of disease and for accurate clinical diagnostics when combined with multivariate statistical algorithms. This method is non-destructive, potentially non-invasive and can be applied in vitro or in vivo directly or via a fiber optic probe. However, there exists a high degree of variability across experimental protocols, some of which result in large background signals that can often overpower the weak Raman signals being emitted. These protocols need to be standardised before the technique can provide reliable and reproducible experimental results in an everyday clinical environment. The objective of this study is to investigate the impact of different experimental parameters involved in the analysis of biological specimen. We investigate the Raman signals generated from healthy human cheek cells using different source laser wavelengths; 473 nm, 532 nm, 660 nm, 785 nm and 830 nm, and different sample substrates; Raman-grade calcium fluoride, IR polished calcium fluoride, magnesium fluoride, aluminium (100 nm and 1500 nm thin films on glass), glass, fused silica, potassium bromide, sodium chloride and zinc selenide, whilst maintaining all other experimental parameters constant throughout the study insofar as possible.