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239
Determining Optical Flow
 ARTIFICIAL INTELLIGENCE
, 1981
"... Optical flow cannot be computed locally, since only one independent measurement is available from the image sequence at a point, while the flow velocity has two components. A second constraint is needed. A method for finding the optical flow pattern is presented which assumes that the apparent veloc ..."
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Cited by 1883 (7 self)
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Optical flow cannot be computed locally, since only one independent measurement is available from the image sequence at a point, while the flow velocity has two components. A second constraint is needed. A method for finding the optical flow pattern is presented which assumes that the apparent velocity of the brightness pattern varies smoothly almost everywhere in the image. An iterative implementation is shown which successfully computes the optical flow for a number of synthetic image sequences. The algorithm is robust in that it can handle image sequences that are quantized rather coarsely in space and time. It is also insensitive to quantization of brightness levels and additive noise. Examples are included where the assumption of smoothness is violated at singular points or along lines in the image.
Numerical Shape from Shading and Occluding Boundaries
 Artifical Intelligence
, 1981
"... An iterative method for computing shape from shading using occluding boundary information is proposed. Some applications of this method are shown. We employ the stereographic plane to express the orientations of surface patches, rather than the more commonly.used gradient space. Use of the stereogra ..."
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Cited by 206 (15 self)
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An iterative method for computing shape from shading using occluding boundary information is proposed. Some applications of this method are shown. We employ the stereographic plane to express the orientations of surface patches, rather than the more commonly.used gradient space. Use of the stereographic plane makes it possible to incorporate occluding boundary information, but forces us to employ a smoothness constraint different from the one previously proposed. The new constraint follows directly from a particular definition of surface smoothness. We solve the set of equations arising from the smoothness constraints and the imageirradiance equation iteratively, using occluding boundary information to supply boundary conditions. Good initial values are found at certain points to help reduce the number of iterations required to reach a reasonable solution. Numerical experiments show that the method is effective and robust. Finally, we analyze scanning electron microscope (SEM) pictures using this method. Other applications are also proposed. 1.
Pseudospectra of linear operators
 SIAM Rev
, 1997
"... Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A ..."
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Cited by 124 (9 self)
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Abstract. If a matrix or linear operator A is far from normal, its eigenvalues or, more generally, its spectrum may have little to do with its behavior as measured by quantities such as ‖An ‖ or ‖exp(tA)‖. More may be learned by examining the sets in the complex plane known as the pseudospectra of A, defined by level curves of the norm of the resolvent, ‖(zI − A) −1‖. Five years ago, the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications. Since that time, analogous computations have been carried out for differential and integral operators. This paper, a companion to the earlier one, presents ten examples, each chosen to illustrate one or more mathematical or physical principles.
A Banach space of test functions for Gabor analysis
 IN &QUOT;GABOR ANALYSIS AND ALGORITHMS: THEORY AND APPLICATIONS
, 1998
"... We introduce the Banach space S 0 # L which has a variety of properties making it a useful tool in Gabor analysis. S 0 can be characterized as the smallest timefrequency homogeneous Banach space of (continuous) functions. We also present other characterizations of S 0 turning it into a very ..."
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Cited by 54 (15 self)
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We introduce the Banach space S 0 # L which has a variety of properties making it a useful tool in Gabor analysis. S 0 can be characterized as the smallest timefrequency homogeneous Banach space of (continuous) functions. We also present other characterizations of S 0 turning it into a very flexible tool for Gabor analysis and allowing for simplifications of various proofs. A careful
Suboptimal Schemes for Atmospheric Data Assimilation Based on the Kalman Filter
, 1994
"... This work is directed toward approximating the evolution of forecast error covariances for data assimilation. We study the performance of different algorithms based on simplification of the standard Kalman filter (KF). These are suboptimal schemes (SOS's) when compared to the KF, which is optim ..."
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Cited by 42 (9 self)
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This work is directed toward approximating the evolution of forecast error covariances for data assimilation. We study the performance of different algorithms based on simplification of the standard Kalman filter (KF). These are suboptimal schemes (SOS's) when compared to the KF, which is optimal for linear problems with known statistics. The SOS's considered here are several versions of optimal interpolation (OI), a scheme for height error variance advection, and a simplified KF in which the full height error covariance is advected. In order to employ a methodology for exact comparison among these schemes we maintain a linear environment, choosing a betaplane shallow water model linearized about a constant zonal flow for the testbed dynamics. Our results show that constructing dynamicallybalanced forecast error covariances, rather than using conventional geostrophicallybalanced ones, is essential for successful performance of any SOS. A posteriori initialization of SOS's to comp...
Robust Numerical Methods for Contingent Claims under Jump Diffusion Processes
 IMA Journal of Numerical Analysis
, 2003
"... An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models wit ..."
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Cited by 37 (13 self)
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An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models with uncertain volatility or transaction costs. Proofs of timestepping stability and convergence of a fixed point iteration scheme are presented. For typical model parameters, it is shown that the fixed point iteration reduces the error by two orders of magnitude at each iteration. The correlation integral is computed using a fast Fourier transform (FFT) method. Techniques are developed for avoiding wraparound effects. Numerical tests of convergence for a variety of options are presented.
The Convergence Rate Of Approximate Solutions For Nonlinear Scalar Conservation Laws
, 1992
"... . Let fv " (x; t)g "?0 be a family of approximate solutions for the nonlinear scalar conservation law u t + f(u)x = 0 with C 1 0 initial data. Assume that fv " (x; t)g are Lip + stable in the sense that they satisfy Oleinik's Eentropy condition. It is shown that if the ..."
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Cited by 35 (14 self)
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. Let fv " (x; t)g "?0 be a family of approximate solutions for the nonlinear scalar conservation law u t + f(u)x = 0 with C 1 0 initial data. Assume that fv " (x; t)g are Lip + stable in the sense that they satisfy Oleinik's Eentropy condition. It is shown that if these approximate solutions are Lip 0 consistent, i.e., if kv " (\Delta; 0) \Gamma u(\Delta; 0)k Lip 0 (x) + kv " t + f(v " )x k Lip 0 (x;t) = O("), then they converge to the entropy solution, and the convergence rate estimate kv " (\Delta; t) \Gamma u(\Delta; t)k Lip 0 (x) = O(") holds. Consequently, the familiar L p type and new pointwise error estimates are derived. These convergence rate results are demonstrated in the context of entropy satisfying finitedifference and Glimm's schemes. Key Words. Conservation laws, entropy stability, weak consistency, error estimates,postprocessing, finitedifference approximations, Glimm scheme AMS(MOS) subject classification. 35L65, 65M10,65M15. 1. Intro...
Numerical Methods for Neuronal Modeling
 In Methods in Neuronal Modeling
, 1989
"... Introduction In this chapter we will discuss some practical and technical aspects of numerical methods that can be used to solve the equations that neuronal modelers frequently encounter. We will consider numerical methods for ordinary differential equations (ODEs) and for partial differential equa ..."
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Cited by 28 (1 self)
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Introduction In this chapter we will discuss some practical and technical aspects of numerical methods that can be used to solve the equations that neuronal modelers frequently encounter. We will consider numerical methods for ordinary differential equations (ODEs) and for partial differential equations (PDEs) through examples. A typical case where ODEs arise in neuronal modeling is when one uses a single lumpedsoma compartmental model to describe a neuron. Arguably the most famous PDE system in neuronal modeling is the phenomenological model of the squid giant axon due to Hodgkin and Huxley. The difference between ODEs and PDEs is that ODEs are equations in which the rate of change of an unknown function of a single variable is prescribed, usually the derivative with respect to time. In contrast, PDEs involve the rates of change of the solution with respect to two or more independent variables, such as time and space. The numerical methods we will discuss for both ODEs and
Approximation of dual Gabor frames, window decay, and wireless communications
"... wireless communications ..."
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An Analytical Comparison Of The Acoustic Analogy And Kirchhoff Formulation For Moving Surfaces
 AIAA Journal
, 1997
"... The Lighthill acoustic analogy, as embodied in the Ffowcs WilliamsHawkings (FWH) equation, is compared with the Kirchhoff formulation for moving surfaces. A comparison of the two governing equations reveals that the main Kirchhoff advantage (namely nonlinear flow effects are included in the surf ..."
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Cited by 25 (6 self)
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The Lighthill acoustic analogy, as embodied in the Ffowcs WilliamsHawkings (FWH) equation, is compared with the Kirchhoff formulation for moving surfaces. A comparison of the two governing equations reveals that the main Kirchhoff advantage (namely nonlinear flow effects are included in the surface integration) is also available to the FWH method if the integration surface used in the FWH equation is not assumed impenetrable. The FWH equation is analytically superior for aeroacoustics because it is based upon the conservation laws of fluid mechanics rather than the wave equation. This means that the FWH equation is valid even if the integration surface is in the nonlinear region. This is demonstrated numerically in the paper. The Kirchhoff approach can lead to substantial errors if the integration surface is not positioned in the linear region. These errors may be hard to identify. Finally, new metrics based on the Sobolev norm are introduced which may be used to compare in...