HP Labs Technical Reports

Systematic Derivation of Spline Bases

Seroussi, Gadiel; Lempel, Abraham



Abstract: In this paper we present an explicit derivation of general spline bases over function spaces closed under differentiation. We determine the necessary and sufficient conditions for the existence of B-splines in such a function space and prove that whenever such a basis exist it is essentially unique. Rather than following the common practice of presenting a mathematical definition of B-splines and then proceeding to prove some of their desired properties, we begin by stipulating a set of design objecti ves and then proceed to derive functions that meet these objectives. We stipulate the following standard design objectives: a given degree of continuity, least feasible order for the given continuity requirement, and shape invariance under translation. As it turns out, these objectives form a complete set in the sense that no other requirement can be imposed without it being already implied by the ones listed, IN other words, when the listed objectives are translated into algebraic constraints, the resulting equations have an essentially unique solution with no remaining degrees of freedom. Clearly, when other desirable objectives such as convexity and the variation diminishing property are attainable, they follow as a by-product by virtue of uniqueness.

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