Hu [5,6], stated that
the continuous two-dimensional order Cartesian moment is defined in terms of Riemann
integrals as:
(13)
It is assumed that is a piecewise continuous, bounded function and that it can have non-zero
values only in the finite region of the plane (i.e. all values outside the image plane are zero - see the Taylor series expansion (Equation 1.6) and explanation in the previous section). If this is so, then moments of all orders exist and the following uniqueness theorem holds [6]:
Theorem 1
Uniqueness theorem : the moment sequence (Equation 1.13 - the basis ) is uniquely defined by and
conversely, is uniquely defined by .
This implies that the original image can be described and reconstructed, if sufficiently high order
moments are used.
By adapting Equation 1.5 to two dimensions, the Cartesian moments (Equation 1.13) can be expressed in terms of the moment generating function. Analysing a two-dimensional irradiance distribution :
(14)
and expanding the exponential using Taylor series produces:
(15)
where are the moments of this two dimensional distribution.