[eliteraspberrie]

The faster way of doing this:

    def fourier_transform(signal, period, tt):
        """ See http://en.wikipedia.org/wiki/Fourier_transform
        How come Numpy and Scipy don't implement this ??? """
        f = lambda func : (signal*func(2*pi*tt/period)).sum()
        return f(cos)+ 1j*f(sin)

is using the FFT.

What you want is the power spectral density in the discrete case, called the power spectrum. It can be calculated by multiplying the discrete Fourier transform (FFT) with its conjugate, and shifting. NumPy can do it. Here is an example: http://stackoverflow.com/questions/15382076/plotting-power-s...

[zulko]

I knew I was going to have this remark :) Now correct me if I am wrong, but I think the FFT (which computes the discrete Fourier transform) cannot replace the continous fourier transform in my case, because the optimal periods I find are non-integer values. In the first case, the holes are separated by 7.5 pixels. The FFT could only have told me that they are separated by 7 or 8 pixels, which is not precise enough. Same thing for the tempo, a beat corresponds to 7.1 frames of the video, and a FFT would have told me 7.

If someone knows a way to use the FFT to get non-integer periods (apart from oversampling the signal) I'll gladly change the code.

[peterwoo]

The maximum frequency you can detect is limited by your sampling rate, but there's not a limit on the precision with which you can break those frequencies up.

It's controlled by a parameter NFFT -- the PSD will compute (NFFT/2+1) values evenly spaced between 0 and the Nyquist frequency.

So say the frame rate is 15Hz and you compute with NFFT=2048, then PSD[970] contains the amplitude at 7.09Hz.

This was a really cool project by the way!

[Serow225]

Also, it's not as widely known as the FFT, but if you know roughly the frequency of interest you can use the Goertzel algorithm to calculate a chosen number of bins around that specific freq and then pick the max of them to find the freq of interest, instead of when using the FFT having to calculate a bunch of bins using a large nFFT in order to get enough freq resolution and then discarding 99% of the results. Going further, compared to the original Goertzel, the Generalized Goertzel algorithm does the same thing but allows you to query non-integer multiples of the fundamental frequency: http://asp.eurasipjournals.com/content/2012/1/56

[zulko]

Thanks, I learned something. I will try it and amend the blog when I have time.

[Serow225]

Forgot to say, great post! :)

[GFK_of_xmaspast]

There are a lot of parametric (as opposed to the nonparametric FFT) methods for tracking frequency, I'm not totally convinced they're applicable to this case, but I think they might be fun to try out. Maybe start here: http://en.wikipedia.org/wiki/Multiple_signal_classification