Transfinite interpolation over implicitly defined sets

Rvachev, V. L.; Шейко, Т. И.; Shapiro, Vadim; Tsukanov, Igor

doi:10.1016/s0167-8396(01)00015-2

Cited by 102 publications

(81 citation statements)

References 9 publications

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“…Rvachev et al [30] showed that such a function representation of lower dimensional cells was quite useful in solving interpolation and boundary value problems.…”

Section: Implicit Complexesmentioning

confidence: 99%

Function-Based Shape Modeling: Mathematical Framework and Specialized Language

Pasko

Adzhiev

2004

Automated Deduction in Geometry

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Abstract. In this survey, we describe the following different aspects of modeling multidimensional point sets (shapes) using real-valued functions of several variables: algebraic system as a formal framework; representation of shapes, operations, and relations using real-valued functions, internal representation of the modeling system; specialized language for function-based modeling, and model extension to point sets with attributes (hypervolumes).

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“…Rvachev et al [30] showed that such a function representation of lower dimensional cells was quite useful in solving interpolation and boundary value problems.…”

Section: Implicit Complexesmentioning

confidence: 99%

Function-Based Shape Modeling: Mathematical Framework and Specialized Language

Pasko

Adzhiev

2004

Automated Deduction in Geometry

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“…So, the entire point set is represented by a single realvalued function that allows for combining constructive solid modeling with scalar field techniques based on the theory of Rfunctions [31,26,[30][31][32][33]. For k-dimensional objects in nD space with k<n, the main idea is that the function F will be zero only at the points of this object and negative everywhere else.…”

Section: Function Representation Of Geometrymentioning

confidence: 99%

“…Rvachev et al [30] show that such a function representation of dimensionally heterogeneous objects is very useful in solving interpolation and boundary value problems. See also applications in [33].…”

Section: Function Representation Of Geometrymentioning

confidence: 99%

Cellular-functional modeling of heterogeneous objects

Adzhiev

Kartasheva

Kunii

et al. 2002

Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications

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The paper presents an approach to modeling heterogeneous objects as multidimensional point sets with multiple attributes (hypervolumes). A theoretical framework is based on a hybrid model of hypervolumes combining a cellular representation and a constructive representation using real-valued functions. This model allows for independent but unifying representation of geometry and attributes, and makes it possible to represent dimensionally non-homogeneous entities and their cellular decompositions. Hypervolume model components such as objects, operations and relations are introduced and outlined. The framework's inherent multidimensionality allowing, in particular, to deal naturally with time dependence promises to model complex dynamic objects composed of different materials with constructive building of their geometry and attributes. Attributes given at each point can represent properties of arbitrary nature (material, photometric, physical, statistical, etc.). To demonstrate a particular application of the proposed framework, we present an example of multimaterial modeling -a multilayer geological structure with cavities and wells. Another example illustrating the treatment of attributes other than material distributions is concerned with time-dependent adaptive mesh generation where function representation is used to describe object geometry and density of elements in the cellular model of the mesh. The examples have been implemented by using a specialized modeling language and software tools being developed by the authors.

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“…Then, eight corresponding intersection points between those rays and their closest adjacent contour lines are identified, and the elevation value F(n) is computed from these intersection points using the inverse distance weight method (Figure 1) based on these eight crossing points (C1, C2… C8). However, this approach cannot reflect the topography of mountaintops and depressions with sufficient resolution for most purposes (Figure 2) (Rvachev et al, 2001;Dinis et al, 2007;Watson, 1999). An example to illustrate the drawbacks of the weighted moving average technique: it is hard to determine whether P is in a valley or on top of a hill.…”

Section: Introductionmentioning

confidence: 99%

An algorithm for generation of DEMs from contour lines considering geomorphic features

Rui¹,

Yu²,

Lu³

et al. 2016

Earth sci. res. j.

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Geomorphic information is omitted from many existing methods of generating gridded digital elevation models (DEMs) from contour lines, resulting in significant errors during interpolation. Here, we present an advanced schema for improvement of the comprehensive regionalized method of linear interpolation. This approach uses a moving fitting method for an interpolated point and selects elevation points that are representative of geomorphic features as a whole to improve interpolation quality. A total of 16 points are selected, according to certain criteria, in eight directions surrounding the interpolated point; thus, there are two points in each direction, which is sufficient to provide an accurate representation of the geomorphic features of the DEM. Our method introduces virtual control points to prevent sudden changes in the interpolation results, which helps to overcome problems related to the distortion of the local geospatial distribution in areas where feature geomorphic information is inadequate. We construct the spline interpolation function using intersection points and virtual control points, all of which are applied to compute the point elevation. Moreover, we index all elevation values and spatial points of linear features using the R-tree method to ensure that points related to an interpolated position can be retrieved as quickly as possible. Finally, we test our method using a coal mine elevation dataset. The results confirm that our proposed method can generate DEMs smoothly and, in particular, avoid problems related to local distortion.La información geomórfica se omite en muchos de los métodos de generación de Modelos Digitales de Elevación (DEM, en inglés) que se elaboran a partir de líneas de contorno, lo que resulta en errores significativos durante la interpolación. En este trabajo se presenta un esquema avanzado para el mejoramiento del método comprensivo regionalizado de interpolación lineal. Esta aproximación utiliza un método de ajuste movil para un punto interpolado y selecciona puntos de elevación representativos o características geomórficas como un todo para mejorar la calidad de la interpolación. Se seleccionaron 16 puntos de acuerdo con ciertos criterios, en ocho direcciones alrededor del punto interpolado; por lo tanto, hay dos puntos en cada dirección, lo que es suficiente para proveer una representación precisa de las características geomórficas del DEM. El método propuesto consta de puntos virtuales de control para prevenir cambios repentinos en los resultados de la interpolación, lo que ayuda a vencer los problemas relacionados a la distorsión de la distribución geoespacial local en áreas donde la información de las características geomórficas es inadecuada. Se utilizó una función de interpolación spline con puntos de intersección y puntos de control virtual, que fueron utilizados para calcular el punto de elevación. Además, se indexaron todos los valores de elevación y los puntos espaciales de las características lineales con el método de árbol-R para asegurar que los puntos...

show abstract

Transfinite interpolation over implicitly defined sets

Cited by 102 publications

References 9 publications

Function-Based Shape Modeling: Mathematical Framework and Specialized Language

Function-Based Shape Modeling: Mathematical Framework and Specialized Language

Cellular-functional modeling of heterogeneous objects

An algorithm for generation of DEMs from contour lines considering geomorphic features

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