To compute the PCM of 2-D grayscale images, we can apply the 1-D analysis over several orientations and then combine the results according to some rules that optimize performance. The 1-D log-Gabor filters described above can be extended to 2-D ones by applying a Gaussian function across the filter perpendicular to its orientation.^{30}^{,}^{34}^{,}^{36}^{,}^{37} The 2-D log-Gabor function has the following transfer function Display Formula
$G2(\omega ,\theta j)=e\u2212[log(\omega /\omega 0)]22\sigma r2\xb7e\u2212(\theta \u2212\theta j)22\sigma \theta 2,$(12b)
where $\theta j=j\pi 2\u2009\u2009J$ and $j=0,1,2,\u2026,J\u22121$ is the number of orientations and $\sigma \theta $ determines the filter’s angular bandwidth. By modulating $\omega 0$ and $\theta j$ and convolving $G2$ with the 2-D image, we get a set of responses at each point $(x,y)$ as $[en,\theta j(x,y),on,\theta j(x,y)]$. The local amplitude at scale $n$ and orientation $\theta j$ is Display Formula$An,\theta j=en,\theta j2(x,y)+on,\theta j2(x,y)$(13a)
and the local energy along orientation $\theta j$ is Display Formula$E\theta j=F\theta j2(x,y)+H\theta j2(x,y),$(13b)
where Display Formula$F\theta j(x,y)=\u2211nen,\theta j(x,y),H\theta j(x,y)=\u2211non,\theta j(x,y).$(13c)
The 2-D PCM at $(x,y)$ is defined as Display Formula$PC2D(x,y)=\u2211jE\theta j(x,y)\u2211n\u2211jAn,\theta j(x,y)+\epsilon ,$(13d)
where $\epsilon $ is a small positive constant. It should be noted that $PC2D(x,y)$ is a real number within [0,1]. The PCM of an image is defined as Display Formula$PCM=1MN\u2211(x,y)PC2D(x,y)=1MN\u2211(x,y)\u2211jE\theta j(x,y)\u2211n\u2211jAn,\theta j(x,y)+\epsilon ,$(13e)
where $M\xd7N$ is the size of the image. The range of PCM is [0,1].