This chapter reports on the initial idea which gave birth to the investigation which subsequently became the unifying theory. The intuition herein illustrated consists of the realization that the following concept might have been supported by reasonable ground of truth after extensive study. The concept is that there exists a region of spatial extent within two nodes (1D), a pixel (2D), or a voxel (3D) where the interpolation function has best approximation properties. Naturally, the adjective best is to be interpreted with its relativity to the potentiality of the specific model interpolation function to determine approximation properties. Such potentiality according to the intuition resides in: (i) the sequel of discrete samples (e.g. the pixel intensities for the two-dimensional case), and (ii) the curvature of the model interpolator as expressed by its second order derivatives. The study in this chapter is initiated for the trivariate linear interpolation function and formalized through a set of definitions, an observation and a theorem.
Let V1 = (0, 0, 0), V2 = (1, 0, 0), V3 = (1, 1, 0), V4 = (0, 1, 0) be the quadruple of vertices of the rectangle α. Let α be lying on the plane π1 of equation C * z + D = 0 ║ to the XY plane of the absolute right handed coordinate system Ξ with origin in O. Let V5 = (0, 0, 1), V6 = (1, 0, 1), V7 = (1, 1, 1), V8 = (0, 1, 1) be the quadruple of vertices of the rectangle β. Let β be lying on the plane π2 of equation C * (z + ξ1) + D = 0 ║ to the same XY plane and with ξ1 being a constant. While Vi (i = 1…4) follows each other counter-clock wise on α, Vi (i = 5…8) do it on β. These eight vertices are located at the boundary surface ∑ of the parallelepiped (voxel) as shown in figure 1.
Figure 1. Top
Intuition: The Sub-pixel Efficacy Region. The Voxel (a), the hyperbolic paraboloid given by equation (1) for arbitrary values of nodes’ intensity (b) and the visualization of the Sub-Pixel Efficacy Region as seen by the intuition presented in this chapter. This picture is found in: Ciulla, C. (2002). Development and characterization of methodology and technology for the alignment of fMRI time series. Unpublished doctoral dissertation, New Jersey Institute of Technology - Newark.
Let V9 = (X, 0, 0), V10 = (1, Y, 0), V11 = (X, 1, 0), V12 = (0, Y, 0) be any points of the segments [V1, V2], [V2, V3], [V3, V4], [V4, V1] respectively and V13 = (X, 0, 1), V14 = (1, Y, 1), V15 = (X, 1, 1), V16 = (0, Y, 1) any points of the segments [V5, V6], [V6, V7], [V7, V8], [V8, V5] respectively. Also, let V17 = (1, 0, Z), V18 = (1, 1, Z) be any points of the segments [V4, V8], [V3, V7] respectively, and V19 = (0, 0, Z), V20 = (1, 0, Z) be any points of the segments [V1, V5], [V2, V6] respectively. Where X, Y and Z are in between the range [0, 1], and Vi (i = 9…20) is located at the boundary surface ∑ of the parallelepiped (voxel) as shown in figure 1.