Novel Schemes of Trivariate Linear and One-Dimensional Quadratic B-Spline Interpolation Functions Based on the Sub-pixel Efficacy Region

A practical approach for the improvement of the interpolation error is presented and is applied to trivariate linear and one-dimensional quadratic B-Spline interpolation functions. The departing point of this work is that of incorporating the intensity-curvature distribution of the given interpolation function into the mathematical formulation named Intensity-Curvature Functional (∆E). While the intensity is determined by the sequel of discrete samples and also by the values established through interpolation, the curvature is expressed by the sum of second order partial derivatives of the interpolation function. The study continues by finding the solution of the polynomial consisting of the partial derivatives of ∆E with respect to each of the dimensional variables. Such a solution reveals a spatial domain of intra-pixel points called Sub-pixel Efficacy Region (SRE), which is used to improve the approximation characteristics of the interpolator. Thus, for a given re-sampling location and also for given intensity values at the pixel to re-sample and the neighbourhood, the Sub-pixel Efficacy Region is used to determine a novel re-sampling location to calculate the interpolation function. For a signal or an image, the novel re-sampling location varies pixel by pixel depending on (i) the local distribution of pixel intensity across the neighbourhood and (ii) the local curvature of the interpolation function. Benefits and limitations of the application of the SRE are studied in space domain by analysis of root mean square errors, and in frequency domain by analysis of spectral power distributions. Novel schemes of trivariate linear and onedimensional quadratic B-Spline functions are determined with improved approximation capabilities.