Shape preserving histogram approximation
The authors develop a new method for the approximation of the density function, underlying a given histogram, by considering polynomials taken from the quadratic-like, three dimensional space span\(\{1,(1-t)^n,t^m\}\), which for large \(n\) and \(m\) tend to the characteristic function of \([0,1]\). They study the corresponding basic theory and apply such polynomials to the construction of univariate and bivariate \(C^1\) histosplines, i.e. splines satisfying the interpolatory area-matching conditions, which arise from a given histogram. The advantage of such quadratic-like, variable degree splines is the low computational cost and the shape control by a pseudo-Bézier net on which the degrees have immediate geometric meaning and possess tension properties, acting as design parameters. The classical quadratic \(C^1\) splines are just a special case of those ones. Since the histospline should preserve some shape properties of the given histogram, as positivity and local monotonicity, some shape constraints are defined and results on existence of suitable sequences of degrees such that the corresponding histospline satisfy such constraints, i.e. it has neither negative values nor extraneous inflections, are proved. The shape preserving histospline can be obtained by using an automatic selection of the degrees. The spline tends, for limit values of the degrees, to the corresponding Bézier net, with the consequence of a linearization of the constraints and the advantage of simplicity and low computational cost, especially in the bivariate case. In such a case the tensor product approach presents some intrinsic defects, therefore a new scheme based on the Boolean sum of univariate B-spline functions is proposed, where some additional parameters permit a more flexible control on the shape of the surface. A comparison between the two approaches is carried out. Finally upper bounds for the error in both one-dimensional and two-dimensional cases are obtained and some numerical and graphical test examples are provided.
scientific article; zbMATH DE number 5146136