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
(12b)where and is the number of orientations and determines the filter’s angular bandwidth. By modulating and and convolving with the 2-D image, we get a set of responses at each point as . The local amplitude at scale and orientation is Display Formula
(13a)and the local energy along orientation is Display Formula
(13b)where Display Formula
(13c)The 2-D PCM at is defined as Display Formula
(13d)where is a small positive constant. It should be noted that is a real number within [0,1]. The PCM of an image is defined as Display Formula
(13e)where is the size of the image. The range of PCM is [0,1].