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Carlo Ciulla (Lane College, USA)

Source Title: Improved Signal and Image Interpolation in Biomedical Applications: The Case of Magnetic Resonance Imaging (MRI)

Copyright: © 2009
|Pages: 8
DOI: 10.4018/978-1-60566-202-2.ch002

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TopLet V_{1} = (0, 0, 0), V_{2} = (1, 0, 0), V_{3} = (1, 1, 0), V_{4} = (0, 1, 0) be the quadruple of vertices of the rectangle α. Let α be lying on the plane π_{1} of equation C * z + D = 0 ║ to the XY plane of the absolute right handed coordinate system Ξ with origin in O. Let V_{5} = (0, 0, 1), V_{6} = (1, 0, 1), V_{7} = (1, 1, 1), V_{8} = (0, 1, 1) be the quadruple of vertices of the rectangle β. Let β be lying on the plane π_{2} of equation C * (z + ξ_{1}) + D = 0 ║ to the same XY plane and with ξ_{1} being a constant. While V_{i} (i = 1…4) follows each other counter-clock wise on α, V_{i} (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.

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 Unpublished doctoral dissertation, New Jersey Institute of Technology - Newark.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 V_{9} = (X, 0, 0), V_{10} = (1, Y, 0), V_{11} = (X, 1, 0), V_{12} = (0, Y, 0) be any points of the segments [V_{1}, V_{2}], [V_{2}, V_{3}], [V_{3}, V_{4}], [V_{4}, V_{1}] respectively and V_{13} = (X, 0, 1), V_{14} = (1, Y, 1), V_{15} = (X, 1, 1), V_{16} = (0, Y, 1) any points of the segments [V_{5}, V_{6}], [V_{6}, V_{7}], [V_{7}, V_{8}], [V_{8}, V_{5}] respectively. Also, let V_{17} = (1, 0, Z), V_{18} = (1, 1, Z) be any points of the segments [V_{4}, V_{8}], [V_{3}, V_{7}] respectively, and V_{19} = (0, 0, Z), V_{20} = (1, 0, Z) be any points of the segments [V_{1}, V_{5}], [V_{2}, V_{6}] respectively. Where X, Y and Z are in between the range [0, 1], and V_{i} (i = 9…20) is located at the boundary surface ∑ of the parallelepiped (voxel) as shown in figure 1.

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