Abstract
In this Chapter and in the next Chap. 7, we deal with continuous rather than discrete and discrete-time splines. In these and only these chapters, we abandon the assumption that the grid, on which the splines are constructed, is uniform and consider splines on arbitrary grids. Two types of local cubic and quadratic splines on non-uniform grids are described: 1. The simplest variation-diminishing splines and 2. The quasi-interpolating splines. The splines are computed by simple fast computational algorithms that utilize relations between the splines and interpolation polynomials. In addition, these relations provide sharp estimations of splines’ approximation accuracy. These splines can serve as an efficient tool for real-time signal processing. As an input, they use either clean or noised arbitrarily-spaced samples. On the other hand, the capability to adapt the grid to the structure of an object and minimal requirements to the operating memory are great advantages for off-line processing of signals and multidimensional data arrays, especially in the 2D case. The contents of this chapter, as well as Chap. 7, is based on (Averbuch, Neittaanmäki, Shefi and Zheludev, Adv. Comput. Math. 43(4), 733–758, 1917) [2].
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Notes
- 1.
This means preservation of monotonicity and convexity of a given data.
- 2.
If a function f(t) is continuous on the interval [a, b] and \(\alpha \) and \(\beta \) are real numbers with the same sign, then there exists a point \(c\in [a,b]\) such that \( \alpha f(a)+ \beta f(b)= (\alpha + \beta ) f(c).\)
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Averbuch, A.Z., Neittaanmäki, P., Zheludev, V.A. (2019). Local Splines on Non-uniform Grid. In: Spline and Spline Wavelet Methods with Applications to Signal and Image Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-92123-5_6
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DOI: https://doi.org/10.1007/978-3-319-92123-5_6
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