Showing papers on "Discrete-time Fourier transform published in 2011"
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 results show that the property of a Boolean function having a concise Fourier representation is locally testable and an “implicit learning” algorithm is given that lets us test any subproperty of Fourier concision.
Abstract: We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable. We give the first efficient algorithms for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients) and for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of $\mathbb{F}_2^n$. In both cases we also prove lower bounds showing that any testing algorithm—even an adaptive one—must have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Building on these results, we give an “implicit learning” algorithm that lets us test any subproperty of Fourier concision. We also present some applications of these results to exact learning and decoding. Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [V. Feldman, P. Gopalan, S. Khot, and A. Ponnuswami, Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2006, pp. 563-576].
85 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
Book•
14 Jun 2011
TL;DR: The Finite Fourier Transform (FTT) as discussed by the authors is an extension of the Fourier transform for linear operators and has been used in many applications, e.g., time-frequency analysis.
Abstract: Preface.- The Finite Fourier Transform.- Translation-Invariant Linear Operators.- Circulant Matrices.- Convolution Operators.- Fourier Multipliers.- Eigenvalues and Eigenfunctions.- The Fast Fourier Transform.- Time-Frequency Analysis.- Time-Frequency Localized Bases.- Wavelet Transforms and Filter Banks.- Haar Wavelets.- Daubechies Wavelets.- The Trace.- Hilbert Spaces.- Bounded Linear Operators.- Self-Adjoint Operators.- Compact Operators.- The Spectral Theorem.- Schatten-von Neumann Classes.- Fourier Series.- Fourier Multipliers on S1.- Pseudo-Differential Operators on S1.- Pseudo-Differential Operators on Z.- Bibliography.- 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
TL;DR: A novel multi-image encryption and decryption algorithm based on Fourier transform and fractional Fourier transforms that has features of enhancement in decryption accuracy and high optical efficiency is presented.
47 citations
TL;DR: A novel decoupling of the least-squares problem is demonstrated which results in two systems of equations, one of which may be solved quickly by means of fast Fourier transforms (FFTs) and another that is demonstrated to be well approximated by a low-rank system.
Abstract: A new algorithm is presented which provides a fast method for the computation of recently developed Fourier continuations (a particular type of Fourier extension method) that yield superalgebraically convergent Fourier series approximations of nonperiodic functions. Previously, the coefficients of an approximating Fourier series have been obtained by means of a regularized singular value decomposition (SVD)-based least-squares solution to an overdetermined linear system of equations. These SVD methods are effective when the size of the system does not become too large, but they quickly become unwieldy as the number of unknowns in the system grows. We demonstrate a novel decoupling of the least-squares problem which results in two systems of equations, one of which may be solved quickly by means of fast Fourier transforms (FFTs) and another that is demonstrated to be well approximated by a low-rank system. Utilizing randomized algorithms, the low-rank system is reduced to a significantly smaller system of equations. This new system is then efficiently solved with drastically reduced computational cost and memory requirements while still benefiting from the advantages of using a regularized SVD. The computational cost of the new algorithm in on the order of the cost of a single FFT multiplied by a slowly increasing factor that grows only logarithmically with the size of the system.
46 citations
TL;DR: In this article, the authors define the quadratic algebra which is a one-parameter deformation of the Lie algebra extended by a parity operator, and investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra.
Abstract: We define the quadratic algebra which is a one-parameter deformation of the Lie algebra extended by a parity operator. The odd-dimensional representations of (with representation label j, a positive integer) can be extended to representations of . We investigate a model of the finite one-dimensional harmonic oscillator based upon this algebra . It turns out that in this model the spectrum of the position and momentum operator can be computed explicitly, and that the corresponding (discrete) wavefunctions can be determined in terms of Hahn polynomials. The operation mapping position wavefunctions into momentum wavefunctions is studied, and this so-called discrete Fourier–Hahn transform is computed explicitly. The matrix of this discrete Fourier–Hahn transform has many interesting properties, similar to those of the traditional discrete Fourier transform.
45 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: In this article, a method which determines a distribution from the knowledge of its q-Fourier transform and some supplementary information is presented, which conveniently extends the inverse of the standard Fourier transform, and is therefore expected to be very useful in many complex systems.
Book•
01 Apr 2011
TL;DR: In this article, the Fourier transform of a generalized function is defined as a series of generalized functions, and the asymptotic expression of a function with a finite number of singularities is derived.
Abstract: Contents 1 Introduction Preliminary remarks Introductory remarks on Fourier series Half-range Fourier series Verification of conjecture Verification of conjecture Verification of conjecture Construction of an odd periodic function Theoretical development of Fourier transforms Half-range Fourier sine and cosine integrals Introduction to the first generalized functions Heaviside unit step function and its relation with Dirac's delta function Exercises References 2 Generalized functions and their Fourier transforms Introduction Definitions of good functions and fairly good functions Generalized functions. The delta function and its derivatives Ordinary functions as generalized functions Equality of a generalized function and an ordinary function in an interval Simple definition of even and odd generalized functions Rigorous definition of even and odd generalized functions Exercises References 3 Fourier transforms of particular generalized functions Introduction Non-integral powers Non-integral powers multiplied by logarithms Integral powers of an algebraic function Integral powers multiplied by logarithms The Fourier transform of xn ln|x| The Fourier transform of x-m ln|x| The Fourier transform of x-m ln|x| sgn(x) Summary of results of Fourier transforms Exercises References 4 Asymptotic estimation of Fourier transforms Introduction The Riemann-Lebesgue lemma Generalization of the Riemann-Lebesgue lemma The asymptotic expression of the Fourier transform of a function with a finite number of singularities Exercises References 5 Fourier series as series of generalized functions Introduction Convergence and uniqueness of a trigonometric series Determination of the coefficients in a trigonometric series Existence of Fourier series representation for any periodic generalized function Some practical examples: Poisson's summation formula Asymptotic behaviour of the coefficients in a Fourier series Exercises References 6 The fast Fourier transform (FFT) Introduction Some preliminaries leading to the fast Fourier transforms The discrete Fourier transform The fast Fourier transform An observation of the discrete Fourier transforms Mathematical aspects of FFT Reviews of some works on FFT algorithms Cooley-Tukey algorithms Application of FFT to wave energy spectral density Exercises References Appendix A: Table of Fourier transforms Fourier transforms g(y)=F{f(x)}=oA - A f(x)e?2?ixydx Appendix B: Properties of impulse function (?(x)) at a glance Introduction Impulse function definition Properties of impulse function Sifting property Scaling property Product of a ?-function by an ordinary function Convolution property ?-Function as generalized limits Time convolution Frequency convolution Appendix C: Bibliography
TL;DR: In this paper, the appropriateness of the Fast Fourier Transform for decomposition and reconstruction of wave records taken at fixed locations and transposed to a different temporal and spatial point is examined.
TL;DR: A new convolution structure is proposed for the LCT, which states that a modified ordinary convolution in the time domain is equivalent to a simple multiplication operation for LCT and Fourier transform (FT) and the convolution theorem in FT domain is shown to be a special case of this structure.
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: In this article, an uncertainty principle for the representation of a vector in two bases has been shown to be applicable to the discrete version of the short time Fourier transform (SFTF).
TL;DR: In this article, the limiting empirical singular value distribution for discrete Fourier transform (DFT) matrices when a random set of columns and rows is removed is determined. But the limiting singular value distributions for DFT matrices are not known.
Abstract: We determine the limiting empirical singular value distribution for discrete Fourier transform (DFT) matrices when a random set of columns and rows is removed.
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: New sufficient conditions for the representation of a function via an absolutely convergent Fourier integral via anabsolutely convergent Fry's inequality are obtained in the paper.
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: For functions that are best described in terms of polar coordinates, a polar version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, and so on as discussed by the authors.
Abstract: For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform can be written in terms of polar coordinates as a combination of Hankel transforms and Fourier series—even if the function does not have circular symmetry. However, to be as useful as its Cartesian counterpart, a polar version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, and so on. This chapter derives the requisite polar version of the standard Fourier operations. In particular, convolution—two-dimensional, circular, and radial one-dimensional—is discussed in detail. It is shown that standard multiplication/convolution rules do apply as long as the correct definition of convolution is applied.
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.
13 Oct 2011
TL;DR: The number of possible algorithms for 2n-point FFTs with radix-2 butterfly operation is determined and a simple method to determine the twiddle factor indices for each algorithm based on the binary tree representation is proposed.
Abstract: In this work a systematic method to generate all possible fast Fourier transform (FFT) algorithms is proposed based on the relation to binary trees. The binary tree is used to represent the decomposition of a discrete Fourier transform (DFT) into sub-DFTs. The radix is adaptively changed according to compute sub-DFTs in proposed decomposition. In this work we determine the number of possible algorithms for 2n-point FFTs with radix-2 butterfly operation and propose a simple method to determine the twiddle factor indices for each algorithm based on the binary tree representation.
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.
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.