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137
Estimating differential quantities using polynomial fitting of osculating jets
"... This paper addresses the pointwise estimation of differential properties of a smooth manifold S —a curve in the plane or a surface in 3D — assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, and either interpolation ..."
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Cited by 90 (5 self)
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This paper addresses the pointwise estimation of differential properties of a smooth manifold S —a curve in the plane or a surface in 3D — assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, and either interpolation or approximation. A jet is a truncated Taylor expansion, and the incentive for using jets is that they encode all local geometric quantities —such as normal, curvatures, extrema of curvature. On the way to using jets, the question of estimating differential properties is recasted into the more general framework of multivariate interpolation / approximation, a wellstudied problem in numerical analysis. On a theoretical perspective, we prove several convergence results when the samples get denser. For curves and surfaces, these results involve asymptotic estimates with convergence rates depending upon the degree of the jet used. For the particular case of curves, an error bound is also derived. To the best of our knowledge, these results are among the first ones providing accurate estimates for differential quantities of order three and more. On the algorithmic side, we solve the interpolation/approximation problem using Vandermonde systems. Experimental results for surfaces of R 3 are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.
A chronology of interpolation: From ancient astronomy to modern signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into histo ..."
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Cited by 75 (0 self)
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This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolutionbased interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.
Simple Constrained Deformations for Geometric Modeling and Interactive Design
 ACM Transactions on Graphics
, 1994
"... Abstract. Deformations are a powerful tool for shape modeling and design. We present a new model for producing controlled spatial deformations, which we term Simple Constrained Deformations (Scodef). The user defines a set of constraint points, giving a desired displacement and radius of influence f ..."
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Cited by 47 (0 self)
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Abstract. Deformations are a powerful tool for shape modeling and design. We present a new model for producing controlled spatial deformations, which we term Simple Constrained Deformations (Scodef). The user defines a set of constraint points, giving a desired displacement and radius of influence for each. Each constraint point determines a local Bspline basis function centered at the constraint point, falling to zero for points beyond the radius. The deformed image of any point in space is a blend of these basis functions, using a projection matrix computed to satisfy the constraints. The deformation operates on the whole space regardless of the representation of the objects embedded inside the space. The constraints directly influence the final shape of the deformed objects, and this shape can be finetuned by adjusting the radius of influence of each constraint point. The computations required by the technique can be done very efficiently, and realtime interactive deformation editing on current workstations is possible.
QLearning in Continuous State and Action Spaces
 IN AUSTRALIAN JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
, 1999
"... Qlearning can be used to learn a control policy that maximises a scalar reward through interaction with the environment. Q learning is commonly applied to problems with discrete states and actions. We describe a method suitable for control tasks which require continuous actions, in response to con ..."
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Cited by 29 (5 self)
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Qlearning can be used to learn a control policy that maximises a scalar reward through interaction with the environment. Q learning is commonly applied to problems with discrete states and actions. We describe a method suitable for control tasks which require continuous actions, in response to continuous states. The system consists of a neural network coupled with a novel interpolator. Simulation results are presented for a nonholonomic control task. Advantage Learning, a variation of Qlearning, is shown enhance learning speed and reliability for this task.
Heterogeneous Material Modeling with Distance Fields
 Computer Aided Geometric Design
, 2004
"... We propose a universal approach to the problem of computer modeling of shapes with continuously varying material properties satisfying prescribed material conditions on a finite collection of material features and global constraints. The central notion is a parameterization of space by distances f ..."
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Cited by 24 (1 self)
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We propose a universal approach to the problem of computer modeling of shapes with continuously varying material properties satisfying prescribed material conditions on a finite collection of material features and global constraints. The central notion is a parameterization of space by distances from the material features  either exactly or approximately. Functions of such distances provide a systematic and intuitive means for modeling of desired material distributions as they arise in design, manufacturing, analysis and optimization of components with varying material properties.
Design optimization of hierarchically decomposed multilevel system under uncertainty
 Proceedings of the ASME 2004 Design Engineering Technical Conferences, Salt Lake City, Utah, 28 September–2 October, DETC2004/DAC57357
, 2004
"... This paper presents a methodology for design optimization of decomposed systems in the presence of uncertainties. We extend the analytical target cascading (ATC) formulation to probabilistic design by treating stochastic quantities as random variables and parameters and posing reliabilitybased desi ..."
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Cited by 21 (12 self)
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This paper presents a methodology for design optimization of decomposed systems in the presence of uncertainties. We extend the analytical target cascading (ATC) formulation to probabilistic design by treating stochastic quantities as random variables and parameters and posing reliabilitybased design constraints. We model the propagation of uncertainty throughout the multilevel hierarchy of elements that comprise the decomposed system by using the advanced mean value (AMV) method to generate the required probability distributions of nonlinear responses. We utilize appropriate metamodeling techniques for simulationbased design problems. A simple yet illustrative hierarchical bilevel engine design problem is used to demonstrate the proposed methodology. 1
Transfinite Interpolation over Implicitly Defined Sets
, 2000
"... In a general setting, the transfinite interpolation problem requires constructing a single function #### that takes on the prescribed values and/or derivatives on some collection of point sets. The sets of points may contain isolated points, bounded or unbounded curves, as well as surfaces and reg ..."
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Cited by 19 (4 self)
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In a general setting, the transfinite interpolation problem requires constructing a single function #### that takes on the prescribed values and/or derivatives on some collection of point sets. The sets of points may contain isolated points, bounded or unbounded curves, as well as surfaces and regions of arbitrary topology. All such closed semianalytic sets may be represented implicitly by real valued functions with guaranteed differential properties.
Automatic Reconstruction of Single Trees from Terrestrial Laser Scanner Data
 ISPRS XX th Congress. Istanbul. Turkey. 12.07. 23.07.2004 Pitas, I
, 2004
"... The investigation of single trees in a forest is of ecological and economical interest. One aim is to capture the geometric aspects of a tree: the length and diameter of the trunk and individual branches, the change of the radius along the branch and similar measures. These measures can be determine ..."
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Cited by 19 (2 self)
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The investigation of single trees in a forest is of ecological and economical interest. One aim is to capture the geometric aspects of a tree: the length and diameter of the trunk and individual branches, the change of the radius along the branch and similar measures. These measures can be determined automatically from terrestrial laser scanner data. The conditions for scanning in the forest, but also the irregular structure and surface of trees aggravate the reconstruction process. The branches of the trees are locally modelled by circular cylinders. With the radius, the axis direction and the axis position the main parameters of interest are captured. We describe a set of algorithms for automatically fitting and tracking cylinders along branches and reconstructing the entire tree. Especially for coniferous trees the computation of an outer hull, giving the extent in different directions and at different heights is an alternative, as the dense foliage coverage renders a distinction between branches and needles impossible. Examples for the different reconstructions of trees are presented.
Analysis of thin plates by the elementfree Galerkin method
 Computational Mechanics
, 1996
"... 1 Introduction There is a growing interest in the socalled "meshless " methods. It might be partly traced to high costs involved in meshing procedures. Modelling of adapting domain geometry, fracture, fragmentation and similar phenomena requires considerable remeshing efforts, whi ..."
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Cited by 15 (3 self)
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1 Introduction There is a growing interest in the socalled &quot;meshless &quot; methods. It might be partly traced to high costs involved in meshing procedures. Modelling of adapting domain geometry, fracture, fragmentation and similar phenomena requires considerable remeshing efforts, which can easily constitute the largest portion of analysis costs.