On the Approximate Nature of the Bivariate Linear Interpolation Function:

This paper describes a novel theoretical approach for the improvement of the bivariate linear interpolation function. The fundamental premise of this theory consists of quantifying the effect of the interpolation function on an image’s pixel by the product of the value of the pixel intensity times the sum of non-null second order derivatives of the function. The product is called intensity-curvature term and it is calculated (i) at the grid node, and (ii) at the generic intra-pixel location. The ratio between the two terms consists of the Intensity-Curvature Functional. First order derivatives of the IntensityCurvature Functional are computed to derive a polynomial system which zeros constitute the Sub-pixel Efficacy Region (SRE). Given a re-sampling location, the SRE is used to project it onto a novel re-sampling location where the approximation properties of the interpolation function lead to error minimization. Two conceptions are thus derived from the Intensity-Curvature Functional: (i) error improvement is set dependent on pixel intensity and curvature of the interpolation function and (ii) the novel re-sampling location varies locally between pixels depending on local properties of the interpolation function as expressed by the intensitycurvature distribution at the neighbourhood. Accordingly, a novel scheme of bivariate linear interpolation is determined with improved approximation properties.