An Algorithm for Precise Aperture Photometry of Critically Sampled Images

We present an algorithm for performing precise aperture photometry on critically sampled astrophysical images. The method is intended to overcome the small-aperture limitations imposed by point-sampling. Aperture fluxes are numerically integrated over the desired aperture, with sinc-interpolation used to reconstruct values between pixel centers. Direct integration over the aperture is computationally intensive, but the integrals in question are shown to be convolution integrals and can be computed ~10000x faster as products in the wave-number domain. The method works equally well for annular and elliptical apertures and could be adapted for any geometry. A sample of code is provided to demonstrate the method.Comment: Accepted MNRA

Significance of Weighted-Type Fractional Fourier Transform in FIR Filters

The desired frequency response of a filter is periodic in frequency and can be expanded in Fourier series. One possible way of obtaining FIR filter is to truncate the infinite Fourier series. But abrupt truncation of the Fourier series results in oscillation in the pass band and stop band. These oscillations are due to slow convergence of the Fourier series by the Gibb's phenomenon. To reduce these oscillations the Fourier coefficients of the filter are modified by multiplying the infinite impulse response with a finite weighing sequence called a window. The Fourier transform (FT) of a window consists of a central lobe and side lobes. The central lobe contains most of the energy of the window. To get an FIR filter, the desired impulse response and window function are multiplied, which results to give finite length non-causal sequence. Since Fractional Fourier Transform (FrFT) is generalization of FT. Here an attempt is to implement filters using window by using Weighted Type Fractional Fourier Transform (WFrFt), differentiator and integrator using weighted FrFt is also present

Efficient detection of a CW signal with a linear frequency drift

An efficient method is presented for the detection of a continuous wave (CW) signal with a frequency drift that is linear in time. Signals of this type occur in transmissions between any two locations that are accelerating relative to one another, e.g., transmissions from the Voyager spacecraft. We assume that both the frequency and the drift are unknown. We also assume that the signal is weak compared to the Gaussian noise. The signal is partitioned into subsequences whose discrete Fourier transforms provide a sequence of instantaneous spectra at equal time intervals. These spectra are then accumulated with a shift that is proportional to time. When the shift is equal to the frequency drift, the signal to noise ratio increases and detection occurs. Here, we show how to compute these accumulations for many shifts in an efficient manner using a variety of Fast Fourier Transformations (FFT). Computing time is proportional to L log L where L is the length of the time series

On the spectrum of the Thue-Morse quasicrystal and the rarefaction phenomenon

The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated, and characterized up to a measure zero set, by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of another conjecture that we call Aubry-Godr\`eche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner; Drmota and Skalba,...), namely the fractality of sum-of-digits functions, play a fundamental role in the description of the singular continous part of the spectrum, combined with some classical results on Riesz products of Peyri\`ere and M. Queff\'elec. The dominant scaling of the sequences of approximant measures on a part of the singular component is controlled by certain inequalities in which are involved the class number and the regulator of real quadratic fields.Comment: 35 pages In honor of the 60-th birthday of Henri Cohe

Significance of Weighted-Type Fractional Fourier Transform in FIR Filters

The desired frequency response of a filter is periodic in frequency and can be expanded in Fourier series. One possible way of obtaining FIR filter is to truncate the infinite Fourier series. But abrupt truncation of the Fourier series results in oscillation in the pass band and stop band. These oscillations are due to slow convergence of the Fourier series by the Gibb’s phenomenon. To reduce these oscillations the Fourier coefficients of the filter are modified by multiplying the infinite impulse response with a finite weighing sequence called a window. The Fourier transform (FT) of a window consists of a central lobe and side lobes. The central lobe contains most of the energy of the window. To get an FIR filter, the desired impulse response and window function are multiplied, which results to give finite length non-causal sequence. Since Fractional Fourier Transform (FrFT) is generalization of FT. Here an attempt is to implement filters using window by using  Weighted Type Fractional Fourier Transform (WFrFt),  differentiator and integrator using weighted FrFt is also present

A Review on Encryption and Decryption of Image using Canonical Transforms & Scrambling Technique

Data security is a prime objective of various researchers & organizations. Because we have to send the data from one end to another end so it is very much important for the sender that the information will reach to the authorized receiver & with minimum loss in the original data. Data security is required in various fields like banking, defence, medical etc. So our objective here is that how to secure the data. So for this purpose we have to use encryption schemes. Encryption is basically used to secure the data or information which we have to transmit or to store. Various methods for the encryption are provided by various researchers. Some of the methods are based on the random keys & some are based on the scrambling scheme. Chaotic map, logistic map, Fourier transform & Fractional Fourier transform etc. are widely used for the encryption process. Now day’s image encryption method is very popular for the encryption scheme. The information is encrypted in the form of image. The encryption is done in a format so no one can read that image. Only the person who are authenticated or have authentication keys can only read that data or information. So this work is based on the same fundamental concept. Here we use Linear Canonical Transform for the encryption process

ShearLab: A Rational Design of a Digital Parabolic Scaling Algorithm

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is three-fold: We firstly develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implicates that shearlet theory provides a unified treatment of both the continuum and digital realm. Secondly, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying Matlab toolbox called ShearLab is provided. And, thirdly, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.Comment: submitted to SIAM J. Multiscale Model. Simu