$C^1$ &gt;splines for interpolation of rapidly varying data

Šalkauskas, K.

doi:10.1216/rmj-1984-14-1-239

Cited by 70 publications

(14 citation statements)

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“…The weights w i = 1 correspond to the Holladay functional while for w i = 1/(1 + τ 2 i ) 5/2 , the geometric curvature is approximated; see [2,16,21] and other papers. In our present tests, we have used functional (4.2) exclusively; see the final section.…”

Section: Convex C 2 Spline Interpolants With Minimal Curvaturementioning

confidence: 99%

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An always successful method in univariate convex $C^2$ interpolation

Schmidt

Heß

1995

Numerische Mathematik

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We consider convex interpolation with cubic C 2 splines on grids built by adding two knots in each subinterval of neighbouring data sites. The additional knots have to be variable in order to get a chance to always retain convexity. By means of the staircase algorithm we provide computable intervals for the added knots such that all knots from these intervals allow convexity preserving spline interpolation of C 2 continuity. (1991): 65D07, 41A15 Mathematics Subject Classification

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Section: Convex C 2 Spline Interpolants With Minimal Curvaturementioning

confidence: 99%

“…There is some freedom in selecting the first and second derivatives at the original knots. Again we propose to compute these by curvature minimization subject to the convexity constraints; compare with [2,16,21] and several other papers. Now we also consider some extensions concerning the so-called convexconcave interpolation.…”

Section: Introductionmentioning

confidence: 99%

An always successful method in univariate convex $C^2$ interpolation

Schmidt

Heß

1995

Numerische Mathematik

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“…The choice w i --1 corresponds to the well-known Holladay functional [15], while wi --1/(1 + z~) 3 or wi = 1/(1 + z~) 5/2 was used in [17], [25] and other papers.…”

Section: Objective Functionalsmentioning

confidence: 99%

Powell-Sabin-Splines bei der restringierten Interpolation unregelmäßig verteilter Daten

Mulansky

Schmidt

1994

Computing

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--ZusammenfassungPowelI-Sabin Splines in Range Restricted Interpolation of Scattered Data. The construction of range restricted bivariate C 1 interpolants to scattered data is considered. In particular, we deal with quadratic spline interpolation on a Powell-Sabin refinement of a triangulation of the data sites subject to piecewise constant lower and upper bounds on the values of the interpolant. The derived sufficient conditions for the fulfillment of the range restrictions result in a solvable system of linear inequalities for the gradients as parameters, which is separated with respect to the data sites. Since there exists an infinite number of spline interpolants meeting the constraints, the selection of a visually pleasant solution is based on the minimum norm modification of a suitable initial interpolant or on the minimization of the thin plate functional. While the first proposal reduces to the solution of independent local quadratic programs, the second proposal results in a global quadratic optimization problem. AMS Subject Classifications: 65D05, 65D07, 41A15Key words: Shape preserving interpolation of scattered data, range restrictions, Powell-Sabin splines, minimum norm modification, thin plate functional. Powell-Sabin-Splines bei der restringierten Interpolation unregelm[iflig verteilter

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“…Note that we could associate separate weights ~ko with each segment e+j. Particularly, different values ~oo for the coefficient of d "+ aF/d~f t might be useful for shape control (see the analogous univariate approaches due to Salkauskas [26] and Foley [8]). For the C 2 GMNN, we usually set ~ = 1~ ---1.…”

Section: Remarkmentioning

confidence: 99%

Scattered data interpolation based upon generalized minimum norm networks

Pottmann

1991

Constr. Approx

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A generalization of G. M. Nielson's method for bivariate scattered data interpolation based upon a minimum norm network is presented. The essential part of the new method is the use of a variational principle for definition of function values as well as cross-boundary derivatives over the edges of a triangulation of the data points. We mainly discuss the case of C 2 interpolants and present some examples including quality control with systems of isophotes.

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$C^1$ >splines for interpolation of rapidly varying data

Cited by 70 publications

References 0 publications

An always successful method in univariate convex $C^2$ interpolation

An always successful method in univariate convex $C^2$ interpolation

Powell-Sabin-Splines bei der restringierten Interpolation unregelmäßig verteilter Daten

Scattered data interpolation based upon generalized minimum norm networks

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