Showing papers on "Non-uniform discrete Fourier transform published in 2011"
Posted Content•
TL;DR: In this paper, a generalized discrete cosine transform with three parameters was proposed and its orthogonality was proved for some new cases, and a new type of DCT was also proposed.
Abstract: The discrete cosine transform (DCT), introduced by Ahmed, Natarajan and Rao, has been used in many applications of digital signal processing, data compression and information hiding. There are four types of the discrete cosine transform. In simulating the discrete cosine transform, we propose a generalized discrete cosine transform with three parameters, and prove its orthogonality for some new cases. A new type of discrete cosine transform is proposed and its orthogonality is proved. Finally, we propose a generalized discrete W transform with three parameters, and prove its orthogonality for some new cases.
1,096 citations
TL;DR: A watermarking method, which minimizes the impact of the watermark implementation on the overall quality of an image, is developed using a peak signal-to-noise ratio to evaluate quality degradation.
Abstract: In this paper, we evaluate the degradation of an image due to the implementation of a watermark in the frequency domain of the image. As a result, a watermarking method, which minimizes the impact of the watermark implementation on the overall quality of an image, is developed. The watermark is embedded in magnitudes of the Fourier transform. A peak signal-to-noise ratio is used to evaluate quality degradation. The obtained results were used to develop a watermarking strategy that chooses the optimal radius of the implementation to minimize quality degradation. The robustness of the proposed method was evaluated on the dataset of 1000 images. Detection rates and receiver operating characteristic performance showed considerable robustness against the print-scan process, print-cam process, amplitude modulated, halftoning, and attacks from the StirMark benchmark software.
115 citations
TL;DR: The aim of this monograph is to clarify the role of Fourier Transforms in the development of Functions of Complex Numbers and to propose a procedure called the Radon Transform, which is based on the straightforward transformation of the Tournaisian transform.
Abstract: Series Editor s Preface. Preface. 1 Introduction. 1.1 Signals, Operators, and Imaging Systems. 1.2 The Three Imaging Tasks. 1.3 Examples of Optical Imaging. 1.4 ImagingTasks inMedical Imaging. 2 Operators and Functions. 2.1 Classes of Imaging Operators. 2.2 Continuous and Discrete Functions. Problems. 3 Vectors with Real-Valued Components. 3.1 Scalar Products. 3.2 Matrices. 3.3 Vector Spaces. Problems. 4 Complex Numbers and Functions. 4.1 Arithmetic of Complex Numbers. 4.2 Graphical Representation of Complex Numbers. 4.3 Complex Functions. 4.4 Generalized Spatial Frequency Negative Frequencies. 4.5 Argand Diagrams of Complex-Valued Functions. Problems. 5 Complex-Valued Matrices and Systems. 5.1 Vectors with Complex-Valued Components. 5.2 Matrix Analogues of Shift-Invariant Systems. 5.3 Matrix Formulation of ImagingTasks. 5.4 Continuous Analogues of Vector Operations. Problems. 6 1-D Special Functions. 6.1 Definitions of 1-D Special Functions. 6.2 1-D Dirac Delta Function. 6.3 1-D Complex-Valued Special Functions. 6.4 1-D Stochastic Functions Noise. 6.5 Appendix A: Area of SINC[x] and SINC2[x]. 6.6 Appendix B: Series Solutions for Bessel Functions J0[x] and J1[x]. Problems. 7 2-D Special Functions. 7.1 2-D Separable Functions. 7.2 Definitions of 2-D Special Functions. 7.3 2-D Dirac Delta Function and its Relatives. 7.4 2-D Functions with Circular Symmetry. 7.5 Complex-Valued 2-D Functions. 7.6 Special Functions of Three (orMore) Variables. Problems. 8 Linear Operators. 8.1 Linear Operators. 8.2 Shift-Invariant.Operators. 8.3 Linear Shift-Invariant (LSI) Operators. 8.4 Calculating Convolutions. 8.5 Properties of Convolutions. 8.6 Autocorrelation. 8.7 Crosscorrelation. 8.8 2-DLSIOperations. 8.9 Crosscorrelations of 2-D Functions. 8.10 Autocorrelations of 2-D.Functions. Problems. 9 Fourier Transforms of 1-D Functions. 9.1 Transforms of Continuous-Domain Functions. 9.2 Linear Combinations of Reference Functions. 9.3 Complex-Valued Reference Functions. 9.4 Transforms of Complex-Valued Functions. 9.5 Fourier Analysis of Dirac Delta Functions. 9.6 Inverse Fourier Transform. 9.7 Fourier Transforms of 1-D Special Functions. 9.8 Theorems of the Fourier Transform. 9.9 Appendix: Spectrum of Gaussian via Path Integral. Problems. 10 Multidimensional Fourier Transforms. 10.1 2-D Fourier Transforms. 10.2 Spectra of Separable 2-D Functions. 10.3 Theorems of 2-D Fourier Transforms. Problems. 11 Spectra of Circular Functions. 11.1 The Hankel Transform. 11.2 Inverse Hankel Transform. 11.3 Theorems of Hankel Transforms. 11.4 Hankel Transforms of Special Functions. 11.5 Appendix: Derivations of Equations (11.12) and (11.14). Problems. 12 The Radon Transform. 12.1 Line-Integral Projections onto Radial Axes. 12.2 Radon Transforms of Special Functions. 12.3 Theorems of the Radon Transform. 12.4 Inverse Radon Transform. 12.5 Central-Slice Transform. 12.6 Three Transforms of Four Functions. 12.7 Fourier and Radon Transforms of Images. Problems. 13 Approximations to Fourier Transforms. 13.1 Moment Theorem. 13.2 1-D Spectra via Method of Stationary Phase. 13.3 Central-Limit Theorem. 13.4 Width Metrics and Uncertainty Relations. Problems. 14 Discrete Systems, Sampling, and Quantization. 14.1 Ideal Sampling. 14.2 Ideal Sampling of Special Functions. 14.3 Interpolation of Sampled Functions. 14.4 Whittaker Shannon Sampling Theorem. 14.5 Aliasingand Interpolation. 14.6 Prefiltering to Prevent Aliasing. 14.7 Realistic Sampling. 14.8 Realistic Interpolation. 14.9 Quantization. 14.10 Discrete Convolution. Problems. 15 Discrete Fourier Transforms. 15.1 Inverse of the Infinite-Support DFT. 15.2 DFT over Finite Interval. 15.3 Fourier Series Derived from Fourier Transform. 15.4 Efficient Evaluation of the Finite DFT. 15.5 Practical Considerations for DFT and FFT. 15.6 FFTs of 2-D Arrays. 15.7 Discrete Cosine Transform. Problems. 16 Magnitude Filtering. 16.1 Classes of Filters. 16.2 Eigenfunctions of Convolution. 16.3 Power Transmission of Filters. 16.4 Lowpass Filters. 16.5 Highpass Filters. 16.6 Bandpass Filters. 16.7 Fourier Transform as a Bandpass Filter. 16.8 Bandboost and Bandstop Filters. 16.9 Wavelet Transform. Problems. 17 Allpass (Phase) Filters. 17.1 Power-Series Expansion for Allpass Filters. 17.2 Constant-Phase Allpass Filter. 17.3 Linear-Phase Allpass Filter. 17.4 Quadratic-Phase Filter. 17.5 Allpass Filters with Higher-Order Phase. 17.6 Allpass Random-Phase Filter. 17.7 Relative Importance of Magnitude and Phase. 17.8 Imaging of Phase Objects. 17.9 Chirp Fourier Transform. Problems. 18 Magnitude Phase Filters. 18.1 Transfer Functions of Three Operations. 18.2 Fourier Transform of Ramp Function. 18.3 Causal Filters. 18.4 Damped Harmonic Oscillator. 18.5 Mixed Filters with Linear or Random Phase. 18.6 Mixed Filter with Quadratic Phase. Problems. 19 Applications of Linear Filters. 19.1 Linear Filters for the Imaging Tasks. 19.2 Deconvolution Inverse Filtering . 19.3 Optimum Estimators for Signals in Noise. 19.4 Detection of Known Signals Matched Filter. 19.5 Analogies of Inverse and Matched Filters. 19.6 Approximations to Reciprocal Filters. 19.7 Inverse Filtering of Shift-Variant Blur. Problems. 20 Filtering in Discrete Systems. 20.1 Translation, Leakage, and Interpolation. 20.2 Averaging Operators Lowpass Filters. 20.3 Differencing Operators Highpass Filters. 20.4 Discrete Sharpening Operators. 20.5 2-DGradient. 20.6 Pattern Matching. 20.7 Approximate Discrete Reciprocal Filters. Problems. 21 Optical Imaging in Monochromatic Light. 21.1 Imaging Systems Based on Ray Optics Model. 21.2 Mathematical Model of Light Propagation. 21.3 Fraunhofer Diffraction. 21.4 Imaging System based on Fraunhofer Diffraction. 21.5 Transmissive Optical Elements. 21.6 Monochromatic Optical Systems. 21.7 Shift-Variant Imaging Systems. Problems. 22 Incoherent Optical Imaging Systems. 22.1 Coherence. 22.2 Polychromatic Source Temporal Coherence. 22.3 Imaging in Incoherent Light. 22.4 System Function in Incoherent Light. Problems. 23 Holography. 23.1 Fraunhofer Holography. 23.2 Holography in Fresnel Diffraction Region. 23.3 Computer-Generated Holography. 23.4 Matched Filtering with Cell-Type CGH. 23.5 Synthetic-Aperture Radar (SAR). Problems. References. Index.
80 citations
TL;DR: This work shows that the fast Fourier transform, so called hyperbolic cross FFT, suffers from an increase of its condition number for both increasing refinement and increasing spatial dimension.
Abstract: A straightforward discretisation of problems in high dimensions often leads to an exponential growth in the number of degrees of freedom. Sparse grid approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivatives and the fast Fourier transform (FFT) has been adapted to this thin discretisation. We show that this so called hyperbolic cross FFT suffers from an increase of its condition number for both increasing refinement and increasing spatial dimension.
73 citations
TL;DR: Fast Fourier transform (FFT) algorithm can be introduced into the calculation of convolution format of gyrator transform in the discrete case by using convolution operation.
65 citations
28 Jan 2011
TL;DR: When compared, Discrete Cosine Transform and Fast Fourier Transform give better compression ratio, while Discrete Wavelet Transform yields good fidelity parameters with comparable compression ratio.
Abstract: In this paper, a transform based methodology is presented for compression of electrocardiogram (ECG) signal. The methodology employs different transforms such as Discrete Wavelet Transform (DWT), Fast Fourier Transform (FFT) and Discrete Cosine Transform (DCT). A comparative study of performance of different transforms for ECG signal is made in terms of Compression ratio (CR), Percent root mean square difference (PRD), Mean square error (MSE), Maximum error (ME) and Signal-to-noise ratio (SNR). The simulation results included illustrate the effectiveness of these transforms in biomedical signal processing. When compared, Discrete Cosine Transform and Fast Fourier Transform give better compression ratio, while Discrete Wavelet Transform yields good fidelity parameters with comparable compression ratio.
58 citations
Patent•
27 Dec 2011TL;DR: In this paper, a video encoder may transform residual data by using a transform selected from a group of transforms, which is applied to the residual data to create a two-dimensional array of transform coefficients.
Abstract: A video encoder may transform residual data by using a transform selected from a group of transforms. The transform is applied to the residual data to create a two-dimensional array of transform coefficients. A scanning mode is selected to scan the transform coefficients in the two-dimensional array into a one-dimensional array of transform coefficients. The combination of transform and scanning mode may be selected from a subset of combinations that is based on an intra-prediction mode. The scanning mode may also be selected based on the transform used to create the two-dimensional array. The transforms and/or scanning modes used may be signaled to a video decoder.
54 citations
TL;DR: In this chapter, the fundamentals of uniform and nonuniform sampling methods in one- and multidimensional NMR are reviewed.
Abstract: Beginning with the introduction of Fourier Transform NMR by Ernst and Anderson in 1966, time domain measurement of the impulse response (free induction decay) consisted of sampling the signal at a series of discrete intervals. For compatibility with the discrete Fourier transform, the intervals are kept uniform, and the Nyquist theorem dictates the largest value of the interval sufficient to avoid aliasing. With the proposal by Jeener of parametric sampling along an indirect time dimension, extension to multidimensional experiments employed the same sampling techniques used in one dimension, similarly subject to the Nyquist condition and suitable for processing via the discrete Fourier transform. The challenges of obtaining high-resolution spectral estimates from short data records were already well understood, and despite techniques such as linear prediction extrapolation, the achievable resolution in the indirect dimensions is limited by practical constraints on measuring time. The advent of methods of spectrum analysis capable of processing nonuniformly sampled data has led to an explosion in the development of novel sampling strategies that avoid the limits on resolution and measurement time imposed by uniform sampling. In this chapter we review the fundamentals of uniform and nonuniform sampling methods in one- and multidimensional NMR.
44 citations
TL;DR: The problem of estimating the frequency of a complex single tone is considered and two iterative Fourier interpolation algorithms are generalized by introducing an additional parameter to allow for selection of the Fouriers interpolation coefficients relative to the true frequency.
42 citations
TL;DR: The proposed algorithm has significantly lower arithmetic complexity, shorter delays and simpler indexing schemes than existing algorithms based on the concatenation of the WHT and FFT, and saves about 70%-36% in computer run-time for transform lengths of 16-4096.
Abstract: An efficient fast Walsh-Hadamard-Fourier transform algorithm which combines the calculation of the Walsh-Hadamard transform (WHT) and the discrete Fourier transform (DFT) is introduced. This can be used in Walsh-Hadamard precoded orthogonal frequency division multiplexing systems (WHT-OFDM) to increase speed and reduce the implementation cost. The algorithm is developed through the sparse matrices factorization method using the Kronecker product technique, and implemented in an integrated butterfly structure. The proposed algorithm has significantly lower arithmetic complexity, shorter delays and simpler indexing schemes than existing algorithms based on the concatenation of the WHT and FFT, and saves about 70%-36% in computer run-time for transform lengths of 16-4096.
40 citations
TL;DR: It is shown that this DFrFT definition based on the eigentransforms of the DFT matrix mimics the properties of continuous fractional Fourier transform (FrFT) by approximating the samples of the continuous FrFT.
TL;DR: In this article, the interpolated discrete Fourier transform (IpDFT) with maximum sidelobe decay windows is investigated for machinery fault feature identification, which combines the idea of local frequency band zooming-in with the IpDFTs and demonstrates high accuracy and frequency resolution in signal parameter estimation when different characteristic frequencies are very close.
Abstract: Complex systems can significantly benefit from condition monitoring and diagnosis to optimize operational availability and safety. However, for most complex systems, multi-fault diagnosis is a challenging issue, as fault-related components are often too close in the frequency domain to be easily identified. In this paper, the interpolated discrete Fourier transform (IpDFT) with maximum sidelobe decay windows is investigated for machinery fault feature identification. A novel identification method called the zoom IpDFT is proposed, which combines the idea of local frequency band zooming-in with the IpDFT and demonstrates high accuracy and frequency resolution in signal parameter estimation when different characteristic frequencies are very close. Simulation and a case study on rolling element bearing vibration data indicate that the proposed zoom IpDFT based on multiple modulations has better capability to identify characteristic components than do traditional methods, including fast Fourier transform (FFT) and zoom FFT.
TL;DR: The computer simulation results show that the proposed image encryption algorithm is feasible, secure and robust to noise attack and occlusion.
TL;DR: This chapter presents the development and applications of non-uniform Fourier transform, which provides the possibility to acquire NMR spectra of ultra-high dimensionality and/or resolution which allow easy resonance assignment and precise determination of spectral parameters.
Abstract: Fourier transform can be effectively used for processing of sparsely sampled multidimensional data sets. It provides the possibility to acquire NMR spectra of ultra-high dimensionality and/or resolution which allow easy resonance assignment and precise determination of spectral parameters, e.g., coupling constants. In this chapter, the development and applications of non-uniform Fourier transform is presented.
TL;DR: An O(NlogN) algorithm to compute the LCT is obtained by using a chirp-FFT-chirp transformation yielded by a convergent quadrature formula for the fractional Fourier transform to give a unitary discrete LCT in closed form.
01 Jan 2011
TL;DR: A good starting point is the finite Fourier transform that underpins the contents of the first thirteen chapters of the book as mentioned in this paper, which is the basis for the present paper's analysis.
Abstract: A good starting point is the finite Fourier transform that underpins the contents of the first thirteen chapters of the book.
TL;DR: In this paper, a DCT-based speech enhancement system with pitch synchronous analysis is proposed to overcome the drawbacks of fixed window shift and the amount of shift in the analysis window is now based on the pitch period, thus increasing the interframe similarities.
Abstract: Discrete cosine transform (DCT) has been proven to be a good approximation to the Karhunen-Loeve Transform (KLT) and has similar properties to the discrete Fourier transform (DFT). It also possesses a better energy compaction capability which is advantageous for speech enhancement. However, frame to frame variations of DCT coefficients even for a perfectly stationary signal can be observed. Therefore a DCT-based speech enhancement system with pitch synchronous analysis is proposed to overcome this problem. It reduces the drawbacks of fixed window shift and the amount of shift in the analysis window is now based on the pitch period, thus increasing the inter-frame similarities. Furthermore, a Wiener filter using the a priori signal-to-noise ratio (SNR) with an adaptive parameter is also derived and implemented as an advanced noise reduction filter. This proposed speech enhancement system is evaluated in terms of several objective measures and the experimental results demonstrate the good performance of the proposed system.
TL;DR: A method to optimize the reconstruction of a hologram when the storage device has a limited dynamic range and a minimum grain size is demonstrated.
Abstract: We demonstrate a method to optimize the reconstruction of a hologram when the storage device has a limited dynamic range and a minimum grain size. The optimal solution at the recording plane occurs when the object wave has propagated an intermediate distance between the near and far fields. This distance corresponds to an optimal order and magnification of the fractional Fourier transform of the object.
Patent•
07 Nov 2011TL;DR: In this paper, an optical image processing system is used to calculate a product of a measured magnitude of a Fourier transform of a complex transmission function of an object or optical image.
Abstract: A method utilizes an optical image processing system. The method includes calculating a product of (i) a measured magnitude of a Fourier transform of a complex transmission function of an object or optical image and (ii) an estimated phase term of the Fourier transform of the complex transmission function. The method further includes calculating an inverse Fourier transform of the product, wherein the inverse Fourier transform is a spatial function. The method further includes calculating an estimated complex transmission function by applying at least one constraint to the inverse Fourier transform.
TL;DR: A Fourier-based regularized method for reconstructing the wavefront from multiple directional derivatives is presented, which is robust to noise, and specially suited for deflectometry measurement.
Abstract: We present a Fourier-based regularized method for reconstructing the wavefront from multiple directional derivatives. This method is robust to noise, and is specially suited for deflectometry measurement.
10 Nov 2011
TL;DR: When compared, Discrete Wavelet Transform gives higher compression respect to Discrete Cosine Transform and Fast Fourier Transform in terms of compression ratio, and DWT as well as good fidelity parameters also.
Abstract: Speech Compression is a field of digital signal processing that focuses on reducing bit-rate of speech signals to enhance transmission speed and storage requirement of fast developing multimedia. This paper explores a transform based methodology for compression of the speech signal. In this methodology, different transforms such as Discrete Wavelet Transform (DWT), fast Fourier Transform (FFT) and Discrete Cosine Transform (DCT) are exploited. A comparative study of performance of different transforms is made in terms of Signal-to-noise ratio (SNR), Peak signal-to-noise ratio (PSNR) and Normalized root-mean square error (NRMSE). The simulation results included illustrate the effectiveness of these transforms in the field of data compression. When compared, Discrete Wavelet Transform gives higher compression respect to Discrete Cosine Transform and Fast Fourier Transform in terms of compression ratio, and DWT as well as good fidelity parameters also.
TL;DR: It is shown that by judiciously choosing sample points on these curved reference surfaces, it is possible to represent the diffracted signals in a nonredundant manner and provide a simple and robust basis for accurate and efficient computation.
Abstract: Fresnel integrals corresponding to different distances can be interpreted as scaled fractional Fourier transformations observed on spherical reference surfaces. We show that by judiciously choosing sample points on these curved reference surfaces, it is possible to represent the diffracted signals in a nonredundant manner. The change in sample spacing with distance reflects the structure of Fresnel diffraction. This sampling grid also provides a simple and robust basis for accurate and efficient computation, which naturally handles the challenges of sampling chirplike kernels.
TL;DR: A twofold generalization of the optical schemes that perform the discrete Fourier transform (DFT) is given: new passive planar architectures are presented where the 2 × 2 3 dB couplers are replaced by M × M hybrids, reducing the number of required connections and phase shifters.
Abstract: A twofold generalization of the optical schemes that perform the discrete Fourier transform (DFT) is given: new passive planar architectures are presented where the 2 × 2 3 dB couplers are replaced by M × M hybrids, reducing the number of required connections and phase shifters. Furthermore, the planar implementation of the discrete fractional Fourier transform (DFrFT) is also described, with a waveguide grating router (WGR) configuration and a properly modified slab coupler.
TL;DR: A fast Fourier transform on regular d-dimensional lattices is introduced, which can be used to perform a fast multivariate wavelet decomposition, where the wavelets are given as trigonometric polynomials and the preferred directions of the decomposition itself can be characterized.
Abstract: We introduce a fast Fourier transform on regular d-dimensional lattices. We investigate properties of congruence class representants, i.e. their ordering, to classify directions and derive a Cooley-Tukey-Algorithm. Despite the fast Fourier techniques itself, there is also the advantage of this transform to be parallelized efficiently, yielding faster versions than the one-dimensional Fourier transform. These properties of the lattice can further be used to perform a fast multivariate wavelet decomposition, where the wavelets are given as trigonometric polynomials. Furthermore the preferred directions of the decomposition itself can be characterised.
TL;DR: This paper introduces a method for determining the locations of jump discontinuities, or edges, in a one-dimensional periodic piecewise-smooth function from nonuniform Fourier coefficients.
Abstract: Edge detection is important in a variety of applications. While there are many algorithms available for detecting edges from pixelated images or equispaced Fourier data, much less attention has been given to determining edges from nonuniform Fourier data. There are applications in sensing (e.g. MRI) where the data is given in this way, however. This paper introduces a method for determining the locations of jump discontinuities, or edges, in a one-dimensional periodic piecewise-smooth function from nonuniform Fourier coefficients. The technique employs the use of Fourier frames. Numerical examples are provided.
TL;DR: In this article, the authors prove a sampling theorem for holomorphic functions on the hyperboloid (or its stereographic projection onto the open unit disk), seen as a homogeneous space of the pseudo-unitary group SU(1,1), and provide a reconstruction formula for bandlimited functions, through a sinc-type kernel, and a discrete Fourier transform from N samples properly chosen.
Abstract: Using Coherent-State (CS) techniques, we prove a sampling theorem for holomorphic functions on the hyperboloid (or its stereographic projection onto the open unit disk $\mathbb{D}_{1}$
), seen as a homogeneous space of the pseudo-unitary group SU(1,1). We provide a reconstruction formula for bandlimited functions, through a sinc-type kernel, and a discrete Fourier transform from N samples properly chosen. We also study the case of undersampling of band-unlimited functions and the conditions under which a partial reconstruction from N samples is still possible and the accuracy of the approximation, which tends to be exact in the limit N→∞.
TL;DR: A generalized interpolated Fourier transform, hereafter called GIFT, is proposed to speed up the Radon transform, and a methodology that can detect straight lines from a gray scale image without any pre-processing is implemented.
01 Mar 2011
TL;DR: This implementation of a two-phase implementation of the filters used in the computation of the fractional Fourier transform speeds up the classical code by an average factor from 2 to 4.
Abstract: We describe the implementation of a two-phase implementation of the filters used in the computation of the fractional Fourier transform. This implementation speeds up the classical code by an average factor from 2 to 4.
12 Dec 2011
TL;DR: In this article, it was shown that the linear canonical transform (LCT) is a variation of the standard Fourier transform and, as such, many of its properties, such as its inversion formula, sampling theorems, convolution theorem and Hilbert transform can be deduced from those of the Fourier Transform by a simple change of variable.
Abstract: Linear canonical transform (LCT) is a four-parameter (a,b,c,d) class of linear integral transform. It has been the focus of many research papers. In this paper, we show that the linear canonical transform is nothing more than a variation of the standard Fourier transform and, as such, many of its properties, such as its inversion formula, sampling theorems, convolution theorems and Hilbert transform can be deduced from those of the Fourier transform by a simple change of variable. Finally, An example of the application of the LCT is also given.