The preceding chapter is to be viewed as a purely theoretical math intuition and the claim consists in that of the existence of a region within the voxel (the three dimensional pixel) where interpolation is most beneficial because it is meant to produce the least approximation of the true intensity value and this region has been named: “Sub-pixel Efficacy Region” (SRE). An energy function will be defined here as the ratio between the energy of the original image and the energy of the interpolated image. This ratio which is called the Intensity-Curvature Functional, is symbolized by the expression ?E = Eo / EIN, and it is prone to be studied to reveal its behavior within the voxel, and it is prone to determine the boundary of the Sub-pixel Efficacy Region within the voxel. In this chapter the Intensity-Curvature Functional will be treated for what concerns the trivariate liner interpolation function such to present its original conception. An application of the Intensity-Curvature Functional for the improvement of the trivariate liner interpolation function was however previously reported (Ciulla & Deek, 2005).
TopImage Energy
The formulation of the interpolation function h(x, y, z) was given in definition IV of Chapter II. The curvature consists of the second order derivatives of h(x, y, z): ¶2 (h(x, y, z))/¶x¶y, ¶2(h (x, y, z))/¶y¶z, and ¶2(h (x, y, z))/¶z¶x. For a voxel, let the energy of the interpolated image be defined as:EIN = EIN(x, y, z) = EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz) (1) where:x/2 y/2 z/2EIN (Ψxy) = ∫ ∫ ∫ h(x, y, z) (¶2 (h(x, y, z)) /¶x¶y) dx dy dz (2)-x/2 -y/2 -z/2x/2 y/2 z/2EIN (Ψzx) = ∫ ∫ ∫ h(x, y, z) (¶2 (h(x, y, z)) /¶z¶x) dx dy dz (3)-x/2 -y/2 -z/2x/2 y/2 z/2EIN (Ψyz) = ∫ ∫ ∫ h(x, y, z) (¶2 (h(x, y, z)) /¶y¶z) dx dy dz (4)-x/2 -y/2 -z/2For the original image, let the energy be defined as:
E
o = E
o(x, y, z) = E
o (Ψ
xy) + E
o (Ψ
zx) + E
o (Ψ
yz)
(5) where:
x/2 y/2 z/2
E
o (Ψ
xy) = ∫ ∫ ∫ f(0, 0, 0) (¶
2 I /¶x¶y) dx dy dz
(6)-x/2 -y/2 -z/2x/2 y/2 z/2
E
o (Ψ
zx) = ∫ ∫ ∫ f(0, 0, 0) (¶
2 I /¶z¶x) dx dy dz
(7)-x/2 -y/2 -z/2x/2 y/2 z/2
E
o (Ψ
yz) = ∫ ∫ ∫ f(0, 0, 0) (¶
2 I /¶y¶z) dx dy dz
(8)