Potential Energy Transform

An invertible linear transform which transforms an image into an energy field by treating the pixels as an array of particles that act as the source of a Gaussian potential energy field. It is assumed that there is a spherically symmetrical potential energy field generated by each pixel, so that $E({\bf{r}}_j )$ is the total potential energy imparted to a pixel of unit intensity at the pixel location with position vector ${\bf{r}}_j$ by the energy fields of remote pixels with position vectors ${\bf{r}}_i$ and pixel intensities $P({\bf{r}}_i )$, and is given by the scalar summation,

$$ E({\bf{r}}_j ) = \sum\limits_i {\left\{ \eqalign{ {{P({\bf{r}}_i )} \over {\left| {{\bf{r}}_i - {\bf{r}}_j } \right|}}\forall i \ne j \cr 0\forall i = j \cr} \right\}}.$$

((1))

To calculate the energy field for the entire image, Eq. 10 should be applied at every pixel position. For efficiency this is calculated in the frequency domain using Eq. 11 where $\Im$ stands for FFT and \( \Im...