Topic
Non-uniform discrete Fourier transform
About: Non-uniform discrete Fourier transform is a research topic. Over the lifetime, 4067 publications have been published within this topic receiving 123952 citations.
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TL;DR: This paper presents a brief review of the algebraic problem of the uniqueness of the solution for both discrete and continuous phase retrieval models and considers the discrete phase retrieval problem as a special case of a more general problem of recovering a real-valued signal x from the magnitude of the output of a linear distortion.
Abstract: In this paper we deal with the problem of retrieving a finite-extent signal from the magnitude of its Fourier transform. We will present a brief review of the algebraic problem of the uniqueness of the solution for both discrete and continuous phase retrieval models. Several important issues which are yet unresolved will be pointed out and discussed. We will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real-valued signal x from the magnitude of the output of a linear distortion: |Hx|(j), j = 1, ..., n . An important result concerning the conditioning of this problem will be obtained for this general setting by means of algebraic-geometric techniques. In particular, the problems of the existence of a solution for phase retrieval, conditioning of the problem and stability of the (essentially) unique solution will be addressed.
20 citations
TL;DR: It has been observed that full hybrid wavelet transform obtained by combining Real Fourier Transform and DCT gives best performance of all, and is compared with DCT Full Wavelet Transform.
Abstract: This paper proposes new image compression technique that uses Real Fourier Transform. Discrete Fourier Transform (DFT) contains complex exponentials. It contains both cosine and sine functions. It gives complex values in the output of Fourier Transform. To avoid these complex values in the output, complex terms in Fourier Transform are eliminated. This can be done by using coefficients of Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST). DCT as well as DST are orthogonal even after sampling and both are equivalent to FFT of data sequence of twice the length. DCT uses real and even functions and DST uses real and odd functions which are equivalent to imaginary part in Fourier Transform. Since coefficients of both DCT and DST contain only real values, Fourier Transform obtained using DCT and DST coefficients also contain only real values. This transform called Real Fourier Transform is applied on colour images. RMSE values are computed for column, Row and Full Real Fourier Transform. Wavelet transform of size N2xN2 is generated using NxN Real Fourier Transform. Also Hybrid Wavelet Transform is generated by combining Real Fourier transform with Discrete Cosine Transform. Performance of these three transforms is compared using RMSE as a performance measure. It has been observed that full hybrid wavelet transform obtained by combining Real Fourier Transform and DCT gives best performance of all. It is compared with DCT Full Wavelet Transform. It beats the performance of Full DCT Wavelet transform. Reconstructed image quality obtained in Real Fourier-DCT Full Hybrid Wavelet Transform is superior to one obtained in DCT, DCT Wavelet and DCT Hybrid Wavelet Transform.
20 citations
TL;DR: An interpretation of the Frei-Chen masks in terms of eight-dimensional Fourier transform coefficient vectors is introduced and a modified set of eight orthogonal masks based on the frequency space analysis is proposed.
20 citations
TL;DR: In this article, a new interpretation of the self-imaging phenomenon using the Fourier plane of periodical objects is proposed, in which all properties of self-images may be described, in the Fresnel approximation, by the quadratic phase corrections of the object Fourier transform.
Abstract: A new interpretation of the self-imaging phenomenon using the Fourier plane of periodical objects is proposed. All properties of the self-images may be described, in the Fresnel approximation, by the quadratic phase corrections of the object Fourier transform. The angular dimensions of the self-images, as well as the notions of the constant of periodical field configuration and the self-image vergence, are introduced. They allow the characterization, in a uniform manner, of the field distribution in the whole space independently of the chosen self-image plane. The equivalency between the self-imaging phenomenon and the image defocusing by an optical system are considered. The general formulae for the harmonics analysis of the intensity distribution are derived.
20 citations
TL;DR: A radix-7, decimation-in-space fast Fourier transform (FFT) for images defined on hexagonal aggregates, expressed in terms of the p-product, a generalization of matrix multiplication.
Abstract: Hexagonal aggregates are hierarchical arrangements of hexagonal cells These hexagonal cells may be efficiently addressed using a scheme known as generalized balanced ternary for dimension 2, or GBT_2 The objects of interest in this paper are digital images whose domains are hexagonal aggregates We define a discrete Fourier transform (DFT) for such images The main result of this paper is a radix-7, decimation-in-space fast Fourier transform (FFT) for images defined on hexagonal aggregates The algorithm has complexity N log_7 N It is expressed in terms of the p-product, a generalization of matrix multiplication Data reordering (also known as shuffle permutations) is generally associated with FFT algorithms However, use of the p-product makes data reordering unnecessary
20 citations