On the reproduction properties of non-stationary subdivision schemes
For researchers in geometric modeling and computer graphics, this provides a theoretical foundation for designing subdivision schemes that reproduce exponential polynomials, which are useful for representing conic sections and spirals.
This paper derives algebraic conditions for the symbols of convergent non-stationary subdivision schemes to reproduce exponential polynomials, extending polynomial reproduction theory. The results enable construction of new schemes with desired reproduction properties.
We present an accurate investigation of the algebraic conditions that the symbols of a convergent, univariate, binary, non-stationary subdivision scheme should fulfill in order to reproduce spaces of exponential polynomials. A subdivision scheme is said to possess the property of reproducing exponential polynomials if, for any initial data uniformly sampled from some exponential polynomial function, the scheme yields the same function in the limit. The importance of this property is due to the fact that several functions obtained as combinations of exponential polynomials (such as conic sections, spirals or special trigonometric and hyperbolic functions) are of great interest in graphical and engineering applications. Since the space of exponential polynomials trivially includes standard polynomials, the results in this work extend the recently developed theory on polynomial reproduction to the non-stationary context. A significant application of the derived algebraic conditions on the subdivision symbols is the construction of new non-stationary subdivision schemes with specified reproduction properties.