The Extension of the Theory to the Trivariate Linear Interpolation Function

Methodology

To capture the variation across the voxel of both intensity and curvature of the function, the intensity-curvature term is defined as the product of the interpolation function times the sum of its second order partial derivatives: ∂² (h (x, y, z))/δxδy, ∂²(h (x, y, z))/δxδz, and ∂²(h (x, y, z))/δyδz respectively. The intensity-curvature term is computed at the grid node and named E_o, and at the generic intra-pixel location and named E_IN. The ratio E_o(x, y, z) / E_IN(x, y, z) constitutes the Intensity-Curvature Functional: ΔE = ΔE(x, y, z). For ΔE, the first order partial derivatives are calculated with respect to x, y, and z. A polynomial system constituted by the first order partial derivatives of ΔE is thus obtained and its solution determines the set of points that constitute the Sub-pixel Efficacy Region. These sets of points are of the type (x_sre, y_sre, z_sre) and they initiate the process of re-calculation of the interpolation function at each voxel.

Given a misplacement (x₀, y_0, z₀), the misplacement used for signal interpolation is calculated based on the Sub-pixel Efficacy Region in order to improve the interpolation error and it consists of the novel re-sampling location (x^r0, y^r0_, z^r0). These novel re-sampling locations are derived through equation (1), which recalls equation (21) of Chapter VII.