How Do Steerable Pyramids Work?
We give a precise description of the pyramid algorithm in this paper: http://www.ipol.im/pub/art/2014/79/

It's a wavelet transform.

If you like to use the exact filters you can use Fourier transform but the filters are usually well approximated in spatial domain on a small support (8 x 8 px^2). Note that the direct convolution is faster than the Fourier implementation when the filters have few pixels (less than 16 x 16 px).