it is interesting.. GFD is a rotation invariant. In GFD the orginal image is transformed to polar cordinate system. This provides the rotation invariant ability. I will explain how this happens. When we map normal image in cartesian coodinate system to polar coordinates, a rotation in Cartesian coordinate will cause translation/shifting in polar system. Remeber in polar system r, and theta are the axis. So when you rotate original image, R remain same so in effect the coresponding image is shifted(to left/right according to the direction of rotation) in polar system.

Still you may be wondering even if that is the case, why Fourier transform output remains same ? because we are operating on different data.For each rotation , the image is shifted in polar coordinate system.

If you thought like that, you are thinking.. Good.

I will explain the answer to the above puzzle.

If you remember one dimensional Fourier transform you can see that we are finding the dot product of our Image data with a number of Cos,Sin vectors (N diamensional, same as image size). The sin and cos are orthonormal , so the magnitude remains same. Please read my previous post http://cgmath.blogspot.com/2011/12/image-recognition-using-phase-only.html to get an intuitive grasp on DFT.

that is

Magnitude Of DFT[ {1,2,3,4,5,6}] is equal to Magnitude of DFT[ {4,5,6,1,2,3}]

Magnitude is Sqrt( i*i + j*j ) of the complex number.

Thus we get rotation in-variance. Now the next thing is Fourier descriptor. I will post about that after some days. It is simple( Really ? )

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