Transformation Properties

The following definitions will be used throughout the paper. Let $T_i$ for $i \in Q = \{A, B, C, \ldots \}$ denote a set of homogeneous, topologically-equivalent anatomical images defined on the coordinate system or domain $\Omega = [0,1]^3$. For example, $T_A$, $T_B$, etc., may correspond to a set of 3D MRI brain images collected from age and sex matched normals or abnormals or some other suitable classification criteria. Let $h_{AB}$ represent the transformation from the coordinate system of image $T_A$ to that of image $T_B$ in terms of the coordinate system of image $T_B$ as shown in Figure 1. Let the linear transformation $x = h_{AB}(y)$ deform image $T_A(x)$ into a new image $\tilde{T}(y) = T_A(h_{AB}(y))$ that resembles the shape of image $T_B(y)$ by transforming the coordinate system of image $T_A$ to that of image $T_B$. Define $H$ as the set of all transformations $h_{AB}(x)$ for $A,B \in Q$ and $x \in \Omega$. Let $\vert\vert x \vert\vert = \sqrt{x_1^2 + x_2^2 +x_3^2}$ denote the standard 2-norm.

Figure 1: Notation used to describe transformations from one coordinate system to another.

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