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126
A Fast Algorithm for Deblurring Models with Neumann Boundary Conditions
, 1999
"... Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for 1dimensional problems and blockToeplitz Toeplitzblock matrices for 2dim ..."
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Cited by 103 (23 self)
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Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for 1dimensional problems and blockToeplitz Toeplitzblock matrices for 2dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitzplus Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also s...
Superfast solution of real positive definite Toeplitz systems
 SIAM J. Matrix Anal. Appl
, 1988
"... Abstract. We describe an implementation of the generalized Schur algorithm for the superfast solution of real positive definite Toeplitz systems of order n + 1, where n = 2ν. Our implementation uses the splitradix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain the ..."
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Cited by 71 (1 self)
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Abstract. We describe an implementation of the generalized Schur algorithm for the superfast solution of real positive definite Toeplitz systems of order n + 1, where n = 2ν. Our implementation uses the splitradix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain the nth Szegő polynomial using fewer than 8n log2 2 n real arithmetic operations without explicit use of the bitreversal permutation. Since Levinson’s algorithm requires slightly more than 2n2 operations to obtain this polynomial, we achieve crossover with Levinson’s algorithm at n = 256. Key words. Toeplitz matrix, Schur’s algorithm, splitradix Fast Fourier Transform
The interpretation of short climate records, with comments on the North Atlantic and
, 1999
"... This pedagogical note reminds the reader that the interpretation of climate records is dependent upon understanding the behavior of stochastic processes. In particular, before concluding that one is seeing evidence for trends, shifts in the mean, or changes in oscillation periods, one must rule out ..."
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Cited by 60 (6 self)
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This pedagogical note reminds the reader that the interpretation of climate records is dependent upon understanding the behavior of stochastic processes. In particular, before concluding that one is seeing evidence for trends, shifts in the mean, or changes in oscillation periods, one must rule out the purely random fluctuations expected from stationary time series. The example of the North Atlantic oscillation (NAO) is mainly used here: the spectral density is nearly white (frequency power law ≈ s −0.2) with slight broadband features near 8 and 2.5 yr. By generating synthetic but stationary time series, one can see exhibited many of the features sometimes exciting attention as being of causal climate significance. Such a display does not disprove the hypothesis of climate change, but it provides a simple null hypothesis for what is seen. In addition, it is shown that the linear predictive skill for the NAO index must be very slight (less than 3% of the variance). A brief comparison with the Southern Oscillation shows a different spectral distribution, but again a simulation has long periods of apparent systematic sign and trends. Application of thresholdcrossing statistics (Ricean) shows no contradiction to the assumption that the Darwin pressure record is statistically stationary. 1.
Toeplitz equations by conjugate gradients with circulant preconditioner
 SIAM J. Sci. Stat. Comput
, 1989
"... Abstract. This paper studies the solution of symmetric positive definite Toeplitz systems Ax b by the preconditioned conjugate gradient method. The preconditioner is a circulant matrix C that copies the middle diagonals of A, and each iteration uses the Fast Fourier Transform. Convergence is governe ..."
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Cited by 59 (8 self)
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Abstract. This paper studies the solution of symmetric positive definite Toeplitz systems Ax b by the preconditioned conjugate gradient method. The preconditioner is a circulant matrix C that copies the middle diagonals of A, and each iteration uses the Fast Fourier Transform. Convergence is governed by the eigenvalues of CAa Toeplitzcirculant eigenvalue problemnand it is fast if those eigenvalues are clustered. The limiting behavior of the eigenvalues is found as the dimension increases, and it is proved that they cluster around A 1. For a wide class of problems the error after q conjugate gradient steps decreases as q2. Key words. Toeplitz, circulant, conjugate gradient, Hankel
OPUC on one foot
 Bull. Amer. Math. Soc
, 2005
"... Abstract. We present an expository introduction to orthogonal polynomials on the unit circle (OPUC). 1. ..."
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Cited by 47 (10 self)
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Abstract. We present an expository introduction to orthogonal polynomials on the unit circle (OPUC). 1.
Perturbations of orthogonal polynomials with periodic recursion coefficients
, 2007
"... We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well ada ..."
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Cited by 47 (17 self)
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We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.
Magnetospheric impulse response for many levels of geomagnetic activity
 J. Geophys. Res
, 1985
"... The temporal relationship between the solar wind and magnetospheric activity has been studied using 34 intervals of high time resolution IMP 8 solar wind data and the corresponding AL auroral activity index. The median values of the AL index for each interval were utilized to rank the intervals acco ..."
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Cited by 41 (5 self)
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The temporal relationship between the solar wind and magnetospheric activity has been studied using 34 intervals of high time resolution IMP 8 solar wind data and the corresponding AL auroral activity index. The median values of the AL index for each interval were utilized to rank the intervals according to geomagnetic activity level. The linear prediction filtering technique was then applied to model magnetospheric response as measured by the AL index to the solar wind input function VB s. The linear prediction filtering routine produces a filter of timelagged response coefficients which estimates the most general linear relationship between the chosen input and output parameters of the magnetospheric system. It is found that the filters are composed of two response pulses speaking at time lags of 20 and 60 min. The amplitude of the 60min pulse is the larger for moderate activity levels, while the 20min pulse is the larger for strong activity levels. A possible interpretation is that the 20min pulse represents magnetospheric activity driven directly by solar wind coupling and that the 60min pulse represents magnetospheric activity driven by the release of energy previously stored in the magnetotail. If this interpretation is correct, the linear filtering results suggest that both the driven and the unloading models of magnetospheric response are inportant facets of a more comprehensive response model.
Generalized Stochastic Subdivision
 ACM Transactions on Graphics
, 1987
"... This paper describes the basis for techniques such as stochastic subdivision in the theory of random processes and estimation theory. The popular stochastic subdivision construction is then generalized to provide control of the autocorrelation and spectral properties of the synthesized random functi ..."
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Cited by 39 (2 self)
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This paper describes the basis for techniques such as stochastic subdivision in the theory of random processes and estimation theory. The popular stochastic subdivision construction is then generalized to provide control of the autocorrelation and spectral properties of the synthesized random functions. The generalized construction is suitable for generating a variety of perceptually distinct highquality random functions, including those with nonfractal spectra and directional or oscillatory characteristics. It is argued that a spectral modeling approach provides a more powerful and somewhat more intuitive perceptual characterization of random processes than does the fractal model. Synthetic textures and terrains are presented as a means of visually evaluating the generalized subdivision technique. Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/Image Generation; I.3.7 [Computer Graphics]: Three Dimensional Graphics and Realism <F11.