<title>Image restoration based on the discrete fraction Fourier transform</title>
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The fractional Fourier transform is the powerful tool for time-variant signal analysis. For space-variant degradation and non-stationary processes the filtering in fractional Fourier domains permits reduction of the error compared with ordinary Fourier domain filtering. In this paper the concept of filtering in fractional Fourier domains is applied to the problem of estimating degraded images. Efficient digital implementation using discrete Hermite eigenvectors can provide similar results to match the continuous outputs. Expressions for the 2D optimal filter function in fractional domains will be given for transform domains characterized by the two rotation angle parameters of the 2D fractional Fourier transform. The proposed method is used to restore images that have several degradations in the experiments. The results show that the method presented in this paper is valid.Keywords:
Discrete-time Fourier transform
Conventional Fourier analysis has many schemes for different types of signals. They are Fourier transform (FT), Fourier series (FS), discrete-time Fourier transform (DTFT), and discrete Fourier transform (DFT). The goal of this article is to develop two absent schemes of fractional Fourier analysis methods. The proposed methods are fractional Fourier series (FRFS) and discrete-time fractional Fourier transform (DTFRFT), and they are the generalizations of Fourier series (FS) and discrete-time Fourier transform (DTFT), respectively.
Discrete-time Fourier transform
Discrete Fourier series
Fourier inversion theorem
Discrete sine transform
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This chapter helps the reader to understand the difference between periodic and nonperiodic signals and to become familiar with the Fourier transform. It explains how to apply the Fourier transform to the spectral analysis of nonperiodic signals. The chapter examines the main Fourier transform pairs and the main properties of the Fourier transform. It provides study examples of applications of the Fourier transform to the spectral analysis of various nonperiodic signals and systems. The chapter also discusses four types of signals – continuous periodic, continuous nonperiodic, discrete periodic, and discrete nonperiodic. It also discusses four types of the Fourier tools: continuous-time Fourier series, continuous-time Fourier transform, discrete Fourier transform (DFT), and discrete-time Fourier transform. The chapter also why only DFT can be employed for digital signal processing and how DFT can be applied for the spectral analysis of both discrete periodic and discrete nonperiodic signals.
Discrete-time Fourier transform
Spectral density estimation
Discrete Fourier series
Discrete sine transform
Harmonic wavelet transform
Fourier inversion theorem
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The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security.
Discrete-time Fourier transform
Discrete sine transform
Discrete Fourier series
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The continuous fractional Fourier transform (FRFT) represents a rotation of signal in time-frequency plane, and it has become an important tool for signal analysis. A discrete version of fractional Fourier transform has been developed but its results do not match those of continuous case. In this paper, we propose a new version of discrete fractional Fourier transform (DFRFT). This new DFRFT will provide similar transforms as those of continuous fractional Fourier transform and also hold the rotation properties.
Discrete-time Fourier transform
Discrete sine transform
Discrete Hartley transform
Harmonic wavelet transform
Hartley transform
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The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform.
Discrete-time Fourier transform
Discrete sine transform
Hartley transform
Harmonic wavelet transform
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Four Fourier transforms are usually defined, the Integral Fourier transform, the Discrete-Time Fourier transform (DTFT), the Discrete Fourier transform (DFT) and the Integral Fourier transform for periodic functions. However, starting from their definitions, we show that all four Fourier transforms can be reduced to actually only one Fourier transform, the Fourier transform in the distributional sense.
Discrete-time Fourier transform
Fourier inversion theorem
Harmonic wavelet transform
Hartley transform
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We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.
Discrete Hartley transform
Discrete sine transform
Discrete-time Fourier transform
Hartley transform
DFT matrix
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Discrete sine transform
Discrete-time Fourier transform
Fourier inversion theorem
Discrete Hartley transform
Discrete Fourier series
Hartley transform
Harmonic wavelet transform
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Fractional Fourier transform (FRFT) becomes an important tool for time-variant signal analysis, and performs a rotation of signals in the time-frequency plane. Because of the importance of the fractional Fourier transform, the implementation of discrete fractional Fourier transformwill be an importance issue. The 2D discrete fraction Fourier transform based on the discrete Hermite eigenvectors is used to analyze the 2D discrete signals in this paper. This DFRFT can preserve the rotation properties and provide similar results to continuous FRFT. ;;
Discrete-time Fourier transform
Discrete sine transform
Discrete frequency domain
Discrete Hartley transform
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