R. E. Barnhill, "Representation and approximation of surfaces," in Mathematical Software III, J. R. Rice, Ed., pp. 69--120, Academic, New York #1977#.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....One function is for the translation in the direction of the x axis and the other for the translation in the direction of the y axis. Scattered data interpolation problems occur frequently in scientic and engineering problems and many solutions have been proposed. The reader is referred to [3] [4] [5] 6] 7] for excellent reviews. In image warping, an additional problem to that of interpolation is that of maintaining the one to one property of the warping transformation. Eachpoint in the transformed image should correspond to only one point in the original image and vice versa. If the ....

R. Barnhill, Representation and approximation of surfaces. In Mathematical software III, J.R.Rice (Ed.), Academic Press, New York, 68-119 (1977).

....by the recursive expandable Shepard method proposed also in [14] This modi cation is based on the similar idea how one can get the Newton form of linear interpolation from the Lagrange type interpolation. Beside the application oriented investigation of Shepard s method such as [14] see also [13, 40, 86, 100, 93, 20]) an increasing interest was arisen from mathematical researchers to examine the approximation propertyofformula (3.37) 3.37) was generalized as follows S ### #f;x## f#x # ##x ; #; n ##; # : 3.38) for an arbitrary f C##; ##, where x # #k ##; n#, in general, denotes the ....

R. E. Barnhill. Representation and approximation surfaces. In: Mathematical Software III (J. R. Rice, ed.), Academic Press, New York, pp. 69120. 1977.

....recursive expandable Shepard method proposed also in [4] This modification is based on the similar idea of how one can derive the Newton form of linear interpolation from the Lagrange type interpolation. Beside the application oriented investigation of Shepard s method such as [4] see also [3,14,20,23,22,5]) an increasing interest has arisen from mathematical researchers to examine the approximation property of formula (10) Formula (10) was generalized as follows S n,# (f, x) k=0 f(x k ) x k=0 (x , # 0, n = 1, 2 . 11) for an arbitrary f C[0, 1] where x k (k = 0, ....

R. Barnhill, Representation and approximation surfaces, in: J. Rice, ed., Mathematical Software III (Academic Press, New York, 1977), 69--120.

....Though usually simple to verify, conditions (I) and (II) are somewhat restrictive. It would be interesting to find conditions weaker than (I) even though the price to pay may be implementations of the paradigm that take more than cubic time. Listings of optimality criteria can be found in [Barn77, BeEp92, Lind83, Schu87]. Furthermore, implementations for criteria satisfying (I) and (II) that run in time o(n 3 ) and o(n 2 log n) are sought. Acknowledgment The authors thank two anonymous referees for suggestions on improving the style of this paper. ....

R. E. Barnhill. Representation and approximation of surfaces. Math. Software III, J. R. Rice, ed., Academic Press, 1977, 69--120.

....Though simple to be verified, conditions (I) and (II) are somewhat restrictive. It would be interesting to find conditions weaker than (I) even though the price to pay may be implementations of the paradigm that take more than cubic time. Listings of optimality criteria can be found in [Barn77, Lind83, Schu87]. Furthermore, implementations for criteria satisfying (I) and (II) that run in time o(n 3 ) and o(n 2 log n) are sought. ....

R. E. Barnhill. Representation and approximation of surfaces. Math. Software III, J. R. Rice, ed., Academic Press, 1977, 69--120.

.... as it signifies the importance of the vertex in computations [FrFi91] It is, however, an NPcomplete problem to decide whether a point set with constraining edges has a triangulation with vertex degree at most 7 [Jans92] No solution is known for the next problem compiled from [Schu87, page 222] [Barn77, page 84], GeSh90, page 202] and [GCR77, Lind83] Problem 6 Can a min max or a max min optimal triangulation based on any one of the following quality measures be computed efficiently: area; aspect ratio; degree; radius of inscribed circle; ratio of the area of the inscribed circle to the area of the ....

R. E. Barnhill. Representation and approximation of surfaces. Math. Software III, J. R. Rice, ed., Academic Press, 1977, 69--120.

....Delta Sum based interpolation schemes are described and discussed in [22] A recent paper on Shepard s method is [37] It surveys the properties of the method, and in particular analyzes its approximation order. Other papers on implementations, modifications, and tests of Shepard s method include [15], 17] 40] 42] 49] and [56] 4.2 Radial Interpolants The term radial is due to Rippa [68] Radial interpolants are of the form s(x) N X i=1 ff i g(kx 0 x i k) pm (x) 26) where g is a given univariate so called radial function, and pm 2 IP k m . The coefficients of s are ....

Barnhill, R. E., Representation and approximation of surfaces, in Mathematical Software III J. R. Rice (ed.), Academic Press, New York, 1977, 68--119.

.... led to the use of edge flipping (with the appropriate definitions of reversed quadrilateral) for finding triangulations that approximately optimize other criteria, such as vertex degree [90] maximum angle [104] total edge length [220] or the ratio of the areas of inscribed circle and triangle [16]. Edge flipping to improve these criteria, however, will not usually compute a global optimum. The problem is that the algorithm can get stuck in a local optimum, in which no flip improves the triangulation. A local optimum can be very far from a global optimum; for example, it may have total ....

....d. Writing d as c 1 a c 2 b c 3 c, with c 1 c 2 c 3 = 1 and c 1 ; c 2 ; c 3 0, we have f T (d) c 1 f(a) c 2 f(b) c 3 f(c) We say that f T interpolates S. The question arises: which triangulations are good for interpolation This question has been discussed in the literature [16, 56, 66, 128, 195]. Rippa [177] recently proved a surprising result. Regardless of the input elevations, the Delaunay triangulation gives an interpolating surface, or elevated triangulation, optimal in a certain least energy sense. Theorem 12 (Rippa [177] Let f T be a piecewise linear function interpolating S ....

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R.E. Barnhill. Representation and approximation of surfaces. Math. Software III, J.R. Rice, ed., Academic Press, 1977, 69--120.

....algorithms for generating surfaces on arbitrary mesh topologies, and some techniques and algorithms for generating blend surfaces. 2.1 Surfaces Defined over Triangular Regions 2.1. 1 Local Interpolation Schemes There are many schemes based on triangles for interpolating and approximating surfaces [2, 3, 39]; however most of these techniques fall into the interpolation category. Research in triangular regions first stemmed from the scattered data problem (the problem of fitting a surface to arbitrarily spaced data in the plane) Since interpolating or approximating scattered data does not fit into ....

....Gregory and involve the use of projectors and Boolean sums of such projectors. A simple example of projectors and Boolean sums is the bilinearly blended Coons patch. Let P and P be linear projectors defined to operate on a primitive bivariate function F(u,v) as 1 2 follows: See Barnhill [2, 3] for surveys of these methods. 9 P F = 1 u) F(0,v) u F(1,v) 1 P F = 1 v) F(u,0) v F(u,1) 2 The Boolean sum of these projectors is denoted P P = P P P P , 1 2 1 2 1 2 where P P is the composition, or tensor product, of the projectors. The bilinearly blended 1 2 Coons patch is ....

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Barnhill, R. E., "Representation and Approximation of Surfaces," in Mathematical Software III, John R. Rice, ed., Academic Press, New York, 1977, pp. 68-119.

....Performance results and a comparison to thin plate splines and hierarchical B spline refinement are presented in Section 7. Conclusions are given in Section 8. 2 PREVIOUS WORK There is a vast amount of literature devoted to scattered data interpolation. The reader is referred to [22] 28] 44] [2] for excellent surveys. In this section, we review several dominant approaches based on Shepard s method, radial basis functions, thin plate splines, and finite element methods. We also consider related research in image processing and geometric modeling, including multiresolution filtering, ....

....by Lawson [30] to derive optimal triangulations in which long thin triangles with small angles are avoided. Piecewise linear approximation over the triangulation is not smooth, achieving only C 0 continuity. The most common C 1 method uses the Clough Tocher triangular interpolant [7] [2], 26] A related technique was proposed in [39] Triangulation methods, however, are sensitive to data distribution, i.e. long thin triangles cannot always be avoided. Schumaker [44] proposed a two stage method that first generates a grid of data using any method for scattered data ....

R. Barnhill, "Representation and Approximation of Surfaces," J.R. Rice, ed., Mathematical Software III, pp. 68-119. New York: Academic Press, 1977.

....to optimize the triangulation in some sense. While a full blown optimization is the type of expensive problem we seek to avoid, some improvement may be obtained by moving vertices around or constructing triangulations on a given set of vertices. Some discussion of such schemes can be found in Barnhill (1977). There are a number of directions in which this work can be extended. Perhaps the most obvious is beyond two dimensions, where as already noted there is potential for triangulations to be much more efficient than rectangular grids. An obvious extension of this work (indeed the original ....

Barnhill, R. E. (1977). Representation and approximation of surfaces. In Mathematical Software III, Ed. J. Rice. Academic Press, New York.

....use a similar experience or similar experiences to form a local model are often referred to as nearest neighbor or k nearest neighbor approaches. Local models (often polynomials) have been used for many years to smooth time series [59, 60, 75, 44] and interpolate and extrapolate from limited data. [6, 54] survey the use of nearest neighbor interpolators to fit surfaces to arbitrarily spaced points. 20] surveys the use of nearest neighbor estimators in nonparametric regression. 40] refer to nearest neighbor approaches as moving least squares and survey their use in fitting surfaces to data. ....

R. E. Barnhill. Representation and approximation of surfaces. In J. R. Rice, editor, Mathematical Software III, pages 69--120. Academic Press, New York, NY, 1977.

....specifications of grid density by introducing nodal and linear sources. The contributions from nodal sources are inversely proportional to the square of the distance. A sample triangulation based on a nodal source is shown in Fig. 9. This is very similar to the Shepard method of interpolation [5] [6]. However, the contributions from the linear sources are modeled similar to the diffusion which is not consistent with the nodal sources. The linear source integral formulation in [4] becomes singular near the ends of the elements. Consequently, the spacing is more concentrated near the middle of ....

Barnhill, R. E., "Representation and Approximation of Surfaces," in: J. R. Rice, ed., Mathematical Software III, Academic Press, New York, 1977.

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R. E. Barnhill, "Representation and approximation of surfaces," in Mathematical Software III, J. R. Rice, Ed., pp. 69--120, Academic, New York #1977#.

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R. E. Barnhill. Representation and approximation of surfaces. In J. D. Rice, editor, Mathematical Software III, pages 69--120. Academic Press, 1977.

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R. E. Barnhill. Representation and approximation of surfaces. In J. D. Rice, editor, Mathematical Software III, pages 69--120. Academic Press, 1977.

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R. E. Barnhill. Representation and approximation of surfaces. In J. R. Rice, editor, Mathematical Software III. Academic Press, 1977.

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Barnhill, R. E. (1977) Representation and approximation of surfaces. Mathematical Software III. New York: Academic Press.

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Barnhill, R. E. (1977) Representation and approximation of surfaces. Mathematical Software III. New York: Academic Press.

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