Non-orthogonal moments

Hu [5,6], stated that the continuous two-dimensional $(p+q)^{th}$ order Cartesian moment is defined in terms of Riemann integrals as:

$$m_{pq} = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} x^{p}y^{q}~f(x,y)~dx~dy$$ (13)

It is assumed that $f(x,y)$ is a piecewise continuous, bounded function and that it can have non-zero values only in the finite region of the $x-y$ 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 $m_{pq}$ (Equation 1.13 - the basis $x^{p}y^{q}$) is uniquely defined by $f(x,y)$ and conversely, $f(x,y)$ is uniquely defined by $m_{pq}$.

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 $f(x,y)$:

$$M^{xy}(u,v) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp(ux + vy)f(x,y)~dx~dy$$ (14)

and expanding the exponential using Taylor series produces:

$$M^{xy}(u,v) = \sum_{p=0}^{\infty} \sum_{q=0}^{\infty} \frac{u^p}{p!}\frac{v^q}{q!}m_{pq}$$ (15)

where $m_{pq}$ are the moments of this two dimensional distribution.