python - Phase correlation

Question

How can rotation angle be determined by phase correlation(using fft) of 2 images? The algorithm given in http://en.wikipedia.org/wiki/Phase_correlation returns linear shift, not angular. It also mentions images have to be converted to log-polar coordinates to compute rotation. How is this conversion achieved in python? And post conversion do the same steps of the algorithm hold?

Answer 1

Log polar transformation is actually rotation and scale invariant.. Rotation corresponds to shift in y axis and scaling corresponds to shift in x axis in log polar transformation

So simple steps are as follows for finding an image x in image y:

1. Find image x in image y (use phase correlation in cartesian coordinates)

2. Compute log polar transforms of both x and y (this is a whole other problem, see references below), make sure to center on the same feature in both images.

3. Find FFT of x and y, say F(X) and F(y)

4. Find phase correlation of F(x) and F(y), call it R

5. Find the IFFT (inverse FFT) of R. The peak value of R corresponds to the rotation deviation in the Y Axis and to the Scaling deviation in the X Axis from the original Image.

References:

1. http://etd.lsu.edu/docs/available/etd-07072005-113808/unrestricted/Thunuguntla_thesis.pdf