Spline interpolation

Résumé

In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all of the values at once, spline interpolation fits low-degree polynomials to small subsets of the values, for example, fitting nine cubic polynomials between each of the pairs of ten points, instead of fitting a single degree-ten polynomial to all of them. Spline interpolation is often preferred over polynomial interpolation because the interpolation error can be made small even when using low-degree polynomials for the spline. Spline interpolation also avoids the problem of Runge's phenomenon, in which oscillation can occur between points when interpolating using high-degree polynomials. Introduction Originally, spline was a term for elastic rulers that were bent to pass through a number of predefined points, or knots. These were use

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https://en.wikipedia.org/wiki/Spline_interpolation

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Theoretical and experimental study of the IR spectrum of the BrHI and BrDI anions

Andreas Osterwalder

A new potential energy surface and dipole moment surface were constructed for BrHI- by a 3-D spline interpolation of ab initio data at the MRCI/aVQZ level of theory and basis set, with ECP's for the halogens. "Exact"vibrational calcns. were performed on this potential energy surface for BrH(D)I- using two very different codes. The asym. stretch and bending modes are found to be strongly coupled, and are shown to be in close 1:2 Fermi resonance. This is verified by a simple de-perturbation of the wavefunctions. The IR transition probabilities were also obtained by integrating dipole moment matrix elements over both vibrational and rotational coordinates. Excellent agreement was found with the recently measured IR spectrum of BrH(D)I- using the Ar-messenger technique by Neumark and co-workers. [on SciFinder (R)]

2004

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New anisotropic a priori error estimates

Luca Formaggia

We prove a priori anisotropic estimates for the L2 and H1 interpolation error on linear finite elements. The full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions. This new H1 error estimate does not require the "maximal angle condition". The analysis has been carried out for the 2D case, but may be extended to three dimensions. Numerical experiments have been carried out to test our theoretical results

2001

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Hermite Snakes with Control of Tangents

Julien René Fageot, Virginie Sophie Uhlmann, Michaël Unser

We introduce a new model of parametric contours defined in a continuous fashion. Our curve model relies on Hermite spline interpolation and can easily generate curves with sharp discontinuities; it also grants direct access to the tangent at each location. With these two features, the Hermite snake distinguishes itself from classical spline-snake models and allows one to address certain bioimaging problems in a more efficient way. More precisely, the Hermite snake construction allows introducing sharp corners in the snake curve and designing directional energy functionals relying on local orientation information in the input image. Using the formalism of spline theory, the model is shown to meet practical requirements such as invariance to affine transformations and good approximation properties. Finally, the dependence on initial conditions and the robustness to the noise is studied on synthetic data in order to validate our Hermite snake model, and its usefulness is illustrated on real biological images acquired using brightfield, phase-contrast, differential-interference-contrast, and scanning-electron microscopy.

IEEE2016

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