Results 1  10
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116
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 191 (14 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.
Height and gradient from shading
 International Journal of Computer Vision
, 1990
"... Abstract: The method described here for recovering the shape of a surface from a shaded image can deal with complex, wrinkled surfaces. Integrability can be enforced easily because both surface height and gradient are represented (A gradient field is integrable if it is the gradient of some surface ..."
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Cited by 107 (1 self)
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Abstract: The method described here for recovering the shape of a surface from a shaded image can deal with complex, wrinkled surfaces. Integrability can be enforced easily because both surface height and gradient are represented (A gradient field is integrable if it is the gradient of some surface height function). The robustness of the method stems in part from linearization of the reflectance map about the current estimate of the surface orientation at each picture cell (The reflectance map gives the dependence of scene radiance on surface orientation). The new scheme can find an exact solution of a given shapefromshading problem even though a regularizing term is included. The reason is that the penalty term is needed only to stabilize the iterative scheme when it is far from the correct solution; it can be turned off as the solution is approached. This is a reflection of the fact that shapefromshading problems are not illposed when boundary conditions are available, or when the image contains singular points. This paper includes a review of previous work on shape from shading and photoclinometry. Novel features of the new scheme are introduced one at a time to make it easier to see what each contributes. Included is a discussion of implementation details that are important if exact algebraic solutions of synthetic shapefromshading problems are to be obtained. The hope is that better performance on synthetic data will lead to better performance on real data.
On the Gibbs phenomenon and its resolution
 SIAM Rev
, 1997
"... Abstract. The nonuniform convergence of the Fourier series for discontinuous functions, and in particular the oscillatory behavior of the finite sum, was already analyzed by Wilbraham in 1848. This was later named the Gibbs phenomenon. This article is a review of the Gibbs phenomenon from a differen ..."
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Cited by 85 (2 self)
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Abstract. The nonuniform convergence of the Fourier series for discontinuous functions, and in particular the oscillatory behavior of the finite sum, was already analyzed by Wilbraham in 1848. This was later named the Gibbs phenomenon. This article is a review of the Gibbs phenomenon from a different perspective. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a function from its expansion coefficients. Alternatively it can be viewed as the possibility of the recovery of local information from global information. The main theme here is not the structure of the Gibbs oscillations but the understanding and resolution of the phenomenon in a general setting. The purpose of this article is to review the Gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case. This is done by using the finite expansion series to construct a different, rapidly convergent, approximation.
Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators
, 1997
"... Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on RN. The definition and properties of these generalized Hermite systems extend naturally those of their classi ..."
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Cited by 37 (7 self)
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Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on RN. The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In case of the symmetric group SN, our setting includes the polynomial eigenfunctions of certain CalogeroSutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl’s Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive oneparameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments.
Minimal surfaces in pseudohermitian geometry and the Bernstein problem in the Heisenberg group
, 2004
"... We develop a surface theory in pseudohermitian geometry. We define a notion of (p)mean curvature and the associated (p)minimal surfaces. As a differential equation, the pminimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theo ..."
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Cited by 34 (5 self)
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We develop a surface theory in pseudohermitian geometry. We define a notion of (p)mean curvature and the associated (p)minimal surfaces. As a differential equation, the pminimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and hence solves the analogue of the Bernstein problem in the Heisenberg group H1. In H1, identified with the Euclidean space R 3, the pminimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set. We interpret the pmean curvature: as the curvature of a characteristic curve, as the tangential sublaplacian of a defining function, and as a quantity in terms of calibration geometry. We also show that there are no closed, connected, C 2 smoothly embedded constant pmean curvature or pminimal surfaces of genus greater than one in the standard S 3. This fact
A Projection Method for Locally Refined Grids
, 1996
"... A numerical method for the solution of the twodimensional Euler equations for incompressible flow on locally refined grids is presented. The method is a second order Godunovprojection method adapted from Bell, Colella, and Glaz. Second order accuracy of the numerical method in time and space is ..."
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Cited by 28 (2 self)
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A numerical method for the solution of the twodimensional Euler equations for incompressible flow on locally refined grids is presented. The method is a second order Godunovprojection method adapted from Bell, Colella, and Glaz. Second order accuracy of the numerical method in time and space is established through numerical experiments. The main contributions of this work concern the formulation and implementation of a projection for refined grids. A discussion of the adjointness relation between gradient and divergence operators for a refined grid MAC projection is presented, and a refined grid approximate projection is developed. An efficient multigrid method which exactly solves the projection is developed, and a method for casting certain approximate projections as MAC projections on refined grids is presented. Subject Classification Index Numbers: 65M06, 65M12, 76D05 Keywords: Incompressible Flow, Mesh Refinement, Projection Methods, Godunov Methods As a draft, thi...
On the Gibbs phenomenon IV: Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function
 Math. Computat
, 1995
"... We continue our investigation of overcoming Gibbs phenomenon, i.e., to obtain exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the rst N Gege ..."
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Cited by 26 (4 self)
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We continue our investigation of overcoming Gibbs phenomenon, i.e., to obtain exponential accuracy at all points (including at the discontinuities themselves), from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. We show that if we are given the rst N Gegenbauer expansion coe cients, based on the Gegenbauer polynomials C k (x) with the weight function (1 x 2) 1 2 for any constant 0, of an L 1 function f(x), we can construct an exponentially convergent approximation to the point values of f(x) inany subinterval in which the function is analytic. The proof covers the cases of Chebyshev or Legendre partial sums, which are most common in applications.
The resolution of the Gibbs phenomenon for spherical harmonics
 Mathematics of Computation
, 1997
"... Abstract. Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion ap ..."
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Cited by 25 (1 self)
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Abstract. Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable oscillations over the entire sphere. Recently, methods for removing the Gibbs phenomenon for onedimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first N expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function f(x) is enough to recover an exponentially convergent approximation to the point values of f(x) in any subinterval in which the function is analytic. Here we take a similar approach, proving that knowledge of the first N spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function f(θ, φ) in any subinterval [θ1,θ2], φ ∈ [0,2π], where the function is analytic. Thus we entirely overcome the Gibbs phenomenon. 1.
Theorems on existence and global dynamics for the Einstein equations
 Living Rev. Relativ
"... This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symme ..."
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Cited by 22 (3 self)
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This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section. 1 1
Shape from shading: Level set propagation and viscosity solutions, Int
 J. Comput. Vision
, 1995
"... ron @ tx.teclmion.ac.il ..."