chance of such symmetries will be extremely low to detect the watermark pattern at an arbitrary position in an image frame the entire image could be filtered with the kernel of the correlation detector and attempt to detect the pattern at every pixel position in the frame as the watermark pattern is complex and the frame may be large a naive implementation of this would be extremely slow nevertheless the idea is practical because the filtering can be done very fast with fast filtering methods based on fast fourier transforms it would be advantageous to map the images video frames and or watermark patterns to a special domain in which this computation is not required also it would be convenient if the search for a pattern over ranges of rotation and resizing appeared like the search for a watermark pattern with arbitrary spatial position as described above the preferred domain is the log polar coordinate system it is a modification of the polar coordinate system in polar coordinates a point is represented by its distance r from a fixed central point and an angle u03b8 representing the direction of the vector from the central point to the point in question usually the angle is measured between the vector and a horizontal vector pointing to the right in the log polar coordinate system r is replaced by its logarithm l log r note that in both coordinate systems the angle u03b8 dimension is periodic with period 2 u03c0 or 360 degrees consider a point in cartesian coordinates x.