Transformation Functions

In principle, the value of each of a transformation's output variables may depend on the values supplied to all of its input variables. Consequently, a general transformation's forward mapping may only be specified in full by giving a complete set of transformation functions which define the precise form of this dependence for each of the output variables. The same consideration also applies to the inverse mapping. Thus, in general, a transformation's two mappings may be decomposed into a set of $n$ forward transformation functions (denoted $F_1,\ldots F_n$) and a set of $m$ inverse transformation functions (denoted $I_1,\ldots I_m$) which act upon the input and output variables, as follows:

$$\begin{array}{cc} \mbox{Forward } \left\{ \begin{array}{ccc}... ...m & = & I_m(y_1, y_2,\ldots y_n) \end{array}\right. \end{array}$$ (1)