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70
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 68 (18 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 54 (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
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 37 (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.
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 33 (5 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.
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 32 (10 self)
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Abstract. We present an expository introduction to orthogonal polynomials on the unit circle (OPUC). 1.
Automated LipSync: Background and Techniques
 Journal of Visualization and Computer Animation
, 1991
"... muscle action procedures for human face animation', The Visual Computer, 3, 290297 (1988). [16] J. Lewis and F. Parke, `Automated lipsynch and speech synthesis for character animation', In Proceedings of CHI87, ACM, New York, 143147 (Toronto, 1987). To appear in: J.Visualization and Computer Ani ..."
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Cited by 31 (3 self)
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muscle action procedures for human face animation', The Visual Computer, 3, 290297 (1988). [16] J. Lewis and F. Parke, `Automated lipsynch and speech synthesis for character animation', In Proceedings of CHI87, ACM, New York, 143147 (Toronto, 1987). To appear in: J.Visualization and Computer Animation 2, 1991 REFERENCES REFERENCES ...
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 26 (15 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.
Formally Biorthogonal Polynomials and a LookAhead Levinson Algorithm for General Toeplitz Systems
 Linear Algebra Appl
, 1993
"... Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n 2 ) arithmetic operations, as compared to O(n 3 ) operations that are needed for solving general linear syst ..."
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Cited by 25 (2 self)
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Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O(n 2 ) arithmetic operations, as compared to O(n 3 ) operations that are needed for solving general linear systems. However, the Levinson algorithm in its original form requires that all leading principal submatrices are nonsingular. In this paper, an extension of the Levinson algorithm to general Toeplitz systems is presented. The algorithm uses lookahead to skip over exactly singular, as well as illconditioned leading submatrices, and, at the same time, it still fully exploits the Toeplitz structure. In our derivation of this algorithm, we make use of the intimate connection of Toeplitz matrices with formally biorthogonal polynomials. In particular, the occurrence of singular or illconditioned submatrices corresponds to The research of this author was performed at the Research Institute for A...
The Generalized Schur Algorithm for the Superfast Solution of Toeplitz Systems
 in Rational Approximation and its Applications in Mathematics and Physics
, 1987
"... We review the connections between fast, O(n ), Toeplitz solvers and the classical theory of Szego polynomials and Schur's algorithm. ..."
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Cited by 19 (4 self)
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We review the connections between fast, O(n ), Toeplitz solvers and the classical theory of Szego polynomials and Schur's algorithm.