Optimal filtering in fractional fourier domains

For time-invariant degradation models and stationary signals and noise, the classical Fourier domain Wiener filter, which can be implemented in O(NlogN) time, gives the minimum mean-square-error estimate of the original undistorted signal. For time-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N2) time for implementation. We consider filtering in fractional Fourier domains, which enables significant reduction of the error compared with ordinary Fourier domain filtering for certain types of degradation and noise (especially of chirped nature), while requiring only O(N\og N) implementation time. Thus, improved performance is achieved at no additional cost. Expressions for the optimal filter functions in fractional domains are derived, and several illustrative examples are given in which significant reduction of the error (by a factor of 50) is obtained. © 1997 IEEE

Digital computation of the fractional fourier transform

An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(JVlogjY) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed. © 1996 IEEE

Power-Law Shot Noise Model For The Ultrasound Rf Echo

We model the radio-frequency (RF) ultrasound echo by a shot noise process with narrow-band power-law filter function. As a consequence, the in-phase and quadrature components of the return signal are shown to exhibit 1=f fi type spectral behavior. The envelope also exhibits this type of spectral behavior but with a different exponent. The model parameters, namely the rate of the point process and the power-law exponent, are related to tissue density and attenuation, respectively. Since these tissue characteristics change due to disease, estimates of the model parameters are investigated as potential tissue characterization features. We validate our claims based on clinical ultrasound images. 1. INTRODUCTION Ultrasound is a widely used medical imaging technique because of its low cost, relative safety, and versatility. Our goal here is to process the ultrasound RF echo, rather than the recorded ultrasound image, in order to derive tissue characterization features that are observer-in..

Optimal image restoration with the fractional Fourier transform

The classical Wiener filter, which can be implemented in O(N log N) time, is suited best for space-invariant degradation models and space-invariant signal and noise characteristics. For space-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N 2) time for implementation. Optimal filtering in fractional Fourier domains permits reduction of the error compared with ordinary Fourier domain Wiener filtering for certain types of degradation and noise while requiring only O(N log N) implementation time. The amount of reduction in error depends on the signal and noise statistics as well as on the degradation model. The largest improvements are typically obtained for chirplike degradations and noise, but other types of degradation and noise may also benefit substantially from the method (e.g., nonconstant velocity motion blur and degradation by inhomegeneous atmospheric turbulence). In any event, these reductions are achieved at no additional cost

The Discrete Fractional Fourier Transform

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

Nonseparable two-dimensional fractional Fourier transform

Cataloged from PDF version of article.Previous generalizations of the fractional Fourier transform to two dimensions assumed separable kernels. We present a nonseparable definition for the two-dimensional fractional Fourier transform that includes the separable definition as a special case. Its digital and optical implementations are presented. The usefulness of the nonseparable transform is justified with an image-restoration example. (C) 1998 Optical Society of America

The discrete fractional fourier transform

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. © 2000 IEEE