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24
Boundary Finding with Parametrically Deformable Models
, 1992
"... Introduction This work describes an approach to finding objects in images based on deformable shape models. Boundary finding in two and three dimensional images is enhanced both by considering the bounding contour or surface as a whole and by using modelbased shape information. Boundary finding u ..."
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Cited by 278 (6 self)
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Introduction This work describes an approach to finding objects in images based on deformable shape models. Boundary finding in two and three dimensional images is enhanced both by considering the bounding contour or surface as a whole and by using modelbased shape information. Boundary finding using only local information has often been frustrated by poorcontrast boundary regions due to occluding and occluded objects, adverse viewing conditions and noise. Imperfect image data can be augmented with the extrinsic information that a geometric shape model provides. In order to exploit modelbased information to the fullest extent, it should be incorporated explicitly, specifically, and early in the analysis. In addition, the bounding curve or surface can be profitably considered as a whole, rather than as curve or surface segments, because it tends to result in a more consistent solution overall. These models are best suited for objects whose diversity and irregularity of shape make
The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remar ..."
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Cited by 153 (51 self)
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This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourierseries method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...
AN INTRODUCTION TO NUMERICAL TRANSFORM INVERSION AND ITS APPLICATION TO PROBABILITY MODELS
, 1999
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Wavelets Associated with Periodic Basis Functions
"... In this paper, we investigate a class of nonstationary, orthogonal, periodic scaling functions and wavelets generated by continuously differentiable periodic functions with positive Fourier coefficients; such functions are termed periodic basis functions. For this class of wavelets, the decompositio ..."
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Cited by 30 (3 self)
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In this paper, we investigate a class of nonstationary, orthogonal, periodic scaling functions and wavelets generated by continuously differentiable periodic functions with positive Fourier coefficients; such functions are termed periodic basis functions. For this class of wavelets, the decomposition and reconstruction coefficients can be computed in terms of the discrete Fourier transform, so that FFT methods apply for their evaluation. In addition, decomposition at the n th level only involves 2 terms from the higher level. Similar remarks apply for reconstruction. We apply a periodic "uncertainty principle" to obtain an angle/frequency uncertainty "window" for these wavelets, and we show that for many wavelets in this class the angle/frequency localization is good.
Detection of Discontinuities in Scattered Data Approximation
 Numerical Algorithms
, 1997
"... . A Detection Algorithm for the localisation of unknown fault lines of a surface from scattered data is given. The method is based on a local approximation scheme using thin plate splines, and we show that this yields approximation of second order accuracy instead of first order as in the global cas ..."
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Cited by 11 (4 self)
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. A Detection Algorithm for the localisation of unknown fault lines of a surface from scattered data is given. The method is based on a local approximation scheme using thin plate splines, and we show that this yields approximation of second order accuracy instead of first order as in the global case. Furthermore, the Detection Algorithm works with triangulation methods, and we show their utility for the approximation of the fault lines. The output of our method provides polygonal curves which can be used for the purpose of constrained surface approximation. 1 Introduction Feature recognition has become an attractive field for research especially within industrial applications. A feature of a surface f : R 2 ! R typically reflects characteristic properties of f , such as discontinuities across planar curves. In geophysical sciences these discontinuities are referred to as fault lines [1], [20], [9]. The motivation for this work is given by applications from oil industry where method...
Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization
 J. Math. Phys
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Time series aggregation, disaggregation and long memory ∗
, 2008
"... Abstract The paper studies the aggregation/disaggregation problem of random parameter AR(1) processes and its relation to the long memory phenomenon. We give a characterization of a subclass of aggregated processes which can be obtained from simpler, ”elementary”, cases. In particular cases of the m ..."
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Cited by 4 (1 self)
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Abstract The paper studies the aggregation/disaggregation problem of random parameter AR(1) processes and its relation to the long memory phenomenon. We give a characterization of a subclass of aggregated processes which can be obtained from simpler, ”elementary”, cases. In particular cases of the mixture densities, the structure (moving average representation) of the aggregated process is investigated. AMS classification: 62M10; 91B84 Keywords: random coefficient AR(1), long memory, aggregation, disaggregation
A General NearZone Light Source Model and its Application to Computer Automated Reflector Design
 SPIE Optical Engineering
, 1996
"... The goal of this work is to develop a general light source model that is physically accurate, intuitively descriptive, computationally convenient, and applicable to real sources. This paper solves two major problems in illumination optics: nearzone general extended light source characterization for ..."
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Cited by 4 (0 self)
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The goal of this work is to develop a general light source model that is physically accurate, intuitively descriptive, computationally convenient, and applicable to real sources. This paper solves two major problems in illumination optics: nearzone general extended light source characterization for accurate image rendering and raytracing, and computerautomated reflector design using the resulting nearzone model. The main approach is to combine measurements with Fourier analysis, using judiciously chosen coordinate systems and orthogonal fitting functions. This approach has several advantages over standard raytracing: it provides for natural data compression and interpolation, it bypasses the problem of computing the radiance distribution of a real source by using actual pinhole CCD camera measurements, and it eliminates the computationally intensive rayfilament intersection problem by transforming the source into an equivalent nonuniform spherical radiator. A method for treating th...
On Asymptotic Normality when the Number of Regressors Increases and the Minimum Eigenvalue of X'X/n Decreases
, 1990
"... Conditions under which a regression estimate based on p regressors is asymptotically normally distributed when the minimum eigenvalue of X 0 X=n decreases with p are obtained. The results are relevant to the regressions on truncated series expansions that arise in neural networks, demand analysis, ..."
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Cited by 3 (2 self)
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Conditions under which a regression estimate based on p regressors is asymptotically normally distributed when the minimum eigenvalue of X 0 X=n decreases with p are obtained. The results are relevant to the regressions on truncated series expansions that arise in neural networks, demand analysis, and asset pricing applications. 0 1 Introduction The p leading terms of a series expansion fOE j g 1 j=1 are often used in regression analysis to either represent or approximate the conditional expectation with respect to x of a dependent variable y. Either explicitly or implicitly, p usually grows with the sample size n in these applications. The growth may follow some deterministic rule fp n g or it may be adaptive with p increased when a ttest rejects or some model selection rule such as Schwarz's (1978) criterion, Mallow's (1973) C p ; or cross validation suggests an increase. The most familiar examples of expansions viewed as representations are experimental designs, which are Ha...
Double tori solution to an equation of mean curvature and Newtonian potential. preprint
"... Studies of near periodic patterns in many selforganizing physical and biological systems give rise to a nonlocal geometric problem in the entire space involving the mean curvature and the Newtonian potential. One looks for a set in space of the prescribed volume such that on the boundary of the set ..."
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Cited by 1 (1 self)
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Studies of near periodic patterns in many selforganizing physical and biological systems give rise to a nonlocal geometric problem in the entire space involving the mean curvature and the Newtonian potential. One looks for a set in space of the prescribed volume such that on the boundary of the set the sum of the mean curvature of the boundary and the Newtonian potential of the set, multiplied by a parameter, is constant. Despite its simple form, the problem has a rich set of solutions and its corresponding energy functional has a complex landscape. When the parameter is sufficiently large, there exists a solution that consists of two tori: a larger torus and a smaller torus. Due to the axisymmetry, the problem is formulated on a half plane. A variant of the LyapunovSchmidt procedure is developed to reduce the problem to minimizing the energy of the set of two exact tori, as an approximate solution, with respect to their radii. A reparameterization argument shows that the double tori so obtained indeed solves the equation of mean curvature and Newtonian potential. One also obtains the asymptotic formulae for the radii of the tori in terms of the parameter. This double tori set is the first known disconnected solution.