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Monotone cubic interpolation
About: Monotone cubic interpolation is a research topic. Over the lifetime, 1740 publications have been published within this topic receiving 38111 citations.
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TL;DR: In this article, an efficient algorithm for the continuous representation of a discrete signal in terms of B-splines and for interpolative signal reconstruction with an expansion factor m are described.
Abstract: Efficient algorithms for the continuous representation of a discrete signal in terms of B-splines (direct B-spline transform) and for interpolative signal reconstruction (indirect B-spline transform) with an expansion factor m are described. Expressions for the z-transforms of the sampled B-spline functions are determined and a convolution property of these kernels is established. It is shown that both the direct and indirect spline transforms involve linear operators that are space invariant and are implemented efficiently by linear filtering. Fast computational algorithms based on the recursive implementations of these filters are proposed. A B-spline interpolator can also be characterized in terms of its transfer function and its global impulse response (cardinal spline of order n). The case of the cubic spline is treated in greater detail. The present approach is compared with previous methods that are reexamined from a critical point of view. It is concluded that B-spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner. >
510 citations
462 citations
408 citations
TL;DR: Two alternative algorithms based on known interpolating functions have been implemented for area calculations in the Lagrange method and the spline method and are described with numerical examples.
Abstract: In the trapezoidal method, linear interpolation between data points tends to overestimate or underestimate the area, depending on the concavity of the curve. In some instances, area estimates can be obtained by linear interpolation of logarithmically transformed data. Two alternative algorithms based on known interpolating functions have been implemented for area calculations. In the Lagrange method, the linear interpolations are replaced by cubic polynomial interpolations. In the spline method, the cubic functions are further modified so that the fitted curves are completely smooth. This report describes their computing procedures with numerical examples.
408 citations
TL;DR: In this article, the shape-preserving interpolation method is applied to two-dimensional semi-Lagrangian advection in plane and spherical geometry, and the derivative estimates are modified to ensure that the interpolant is monotonic.
Abstract: The more attractive one dimensional, shape-preserving interpolation schemes as determined from a companion study are applied to two-dimensional semi-Lagrangian advection in plane and spherical geometry. Hermite cubic and a rational cubic are considered for the interpolation form. Both require estimates of derivatives at data points. A cubic derivative form and the derivative estimates of Hyman and Akima are considered. The derivative estimates are also modified to ensure that the interpolant is monotonic. The modification depends on the interpolation form. Three methods are used to apply the interpolators to two-dimensional semi-Lagrangian advection. The first consists of fractional time steps or time splitting. The method has noticeable displacement errors and larger diffusion than the other methods. The second consists of two-dimensional interpolants with formal definitions of a two-dimensional monotonic surface and application of a two-dimensional monotonicity constraint. This approach is exam...
335 citations