Properties of Shepard's surfaces

Barnhill, Robert E.; Dube, R. P.; Little, F.F.

doi:10.1216/rmj-1983-13-2-365

Cited by 62 publications

(27 citation statements)

References 6 publications

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“…Shepard's method [1,2] is a frequently employed scattered data interpolation scheme in computer graphics. In this method the interpolated value is a weighted sum of the surrounding data points Figure 4.…”

Section: Shepard's Methodsmentioning

confidence: 99%

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Pose space deformation

Lewis¹,

Cordner²,

Fong³

2000

Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH '00

686

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Pose space deformation generalizes and improves upon both shape interpolation and common skeleton-driven deformation techniques. This deformation approach proceeds from the observation that several types of deformation can be uniformly represented as mappings from a pose space, defined by either an underlying skeleton or a more abstract system of parameters, to displacements in the object local coordinate frames. Once this uniform representation is identified, previously disparate deformation types can be accomplished within a single unified approach. The advantages of this algorithm include improved expressive power and direct manipulation of the desired shapes yet the performance associated with traditional shape interpolation is achievable. Appropriate applications include animation of facial and body deformation for entertainment, telepresence, computer gaming, and other applications where direct sculpting of deformations is desired or where real-time synthesis of a deforming model is required.

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Section: Shepard's Methodsmentioning

confidence: 99%

“…By writing the delta shape form as (1 − 1 w k )S0 + 1 w k S k it is clear that the space of achievable shapes is identical in both variations. 1 An attractive feature of shape interpolation is that the desired expressions can be directly specified by sculpting.…”

Section: Shape Interpolationmentioning

confidence: 99%

Pose space deformation

Lewis¹,

Cordner²,

Fong³

2000

Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH '00

686

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show abstract

“…where g i : → , i = 1, , n, are cardinal basis functions, that is, satisfy (2). Note that for a given z ∈ , each g i (z) takes a scalar value, while f (z i ) ∈ Y , and so F (z) ∈ Y .…”

Section: Cardinal Basis Interpolation In Banach Spacesmentioning

confidence: 99%

“…We choose = 2 in Shepard-type cardinal basis functions because, in general, a small value of avoids "flat spots" near the nodes (see, e.g., [2]). The cardinal basis interpolant, in the barycentric form, is…”

Section: Downloaded By [Miami University Libraries] At 14:04 12 Octobmentioning

confidence: 99%

Lagrange Interpolation on Arbitrarily Distributed Data in Banach Spaces

Allasia

Bracco

2011

Numerical Functional Analysis and Optimization

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The Lagrange interpolation problem in Banach spaces is approached by cardinal basis interpolation. Some error estimates are given and the results of several numerical tests are reported in order to show the approximation performances of the proposed interpolants. A comparison between some examples of interpolants is presented in the noteworthy case ofHilbert spaces, with some considerations about the possible localization of the formulas. Finally, some remarks about the cardinal basis interpolation framework are made from the application point of view.

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“…These cusps turn into corners when u = 1 and atten out when u > 1. 3 Another useful property of Shepard's interpolation is that the surface stays within the extrema of the data points. 1 That is, min(F j ) f(x; y) max(F j ) for j = 1; 2; :::;N. Figure 1 illustrates the behavior of Shepard's interpolation using di erent distance power values.…”

Section: Environmental Datamentioning

confidence: 99%

<title>Data quality issues in visualization</title>

1994

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Recent e orts in visualization have concentrated on high volume data sets from numerical simulations and medical imaging. There is another large class of data, characterized by their spatial sparsity with noisy and possibly missing data points, that also need to be visualized. Two places where these type of data sets can be found are in oceanographic and atmospheric science studies. In such cases, it is not uncommon to have on the order of one percent of sampled data available within a space volume. Techniques that attempt to deal with the problem of lling-in-the-holes range in complexity from simple linear interpolation to more sophisticated multiquadric and optimal interpolation techniques. These techniques will generally produce results that do not fully agree with each other. To avoid misleading the users, it is important to highlight these di erences and make sure the users are aware of the idiosyncrasies of the di erent methods. This paper compares some of these interpolation techniques on sparse data sets and also discusses how other parameters such as con dence levels and drop-o rates may be incorporated into the visual display.

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Properties of Shepard's surfaces

Cited by 62 publications

References 6 publications

Pose space deformation

Pose space deformation

Lagrange Interpolation on Arbitrarily Distributed Data in Banach Spaces

<title>Data quality issues in visualization</title>

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