Fast Fourier Transform (FFT) is a powerful way of analyzing (and filtering) images. In one of the presentations today at the Royal Microscopical Society Frontiers in Bioimaging, it was proposed to evaluate and compare the resolution of various superresolution techniques. In the context of the stripy controversy, there has been some confusion over the (apparently very simple) question of what is the FFT of an array of lines.
That confusion is apparent in the response of Miao Yu and Francesco Stellacci (see below), but it is particularly stunning in this 2009 comment by one of the Nature Materials referee [Nature Materials considered and then rejected Stripy Nanoparticles Revisited in 2009]:
Arrays of parallel straight ripples are clearly visible in each nanoparticle, and it is well known that the Fourier Transform of an array of parallel straight lines is again an array of parallel straight lines. [empasis mine]
I do not know how ‘well known’ it is, but as with many ‘well known’ things, it is false. Let’s first consider the array of nice and smooth lines below.
Here is its FFT:
That is a pretty boring FFT! First thing to notice is that the FFT is constant except on one horizontal line. The reason is quite simple: the original image is invariant along the vertical axis: there is no need for vertical frequency components to describe that image. Let’s look more closely at this single line then:
We can see two nice peaks corresponding to the frequency of the oscillations above. So, in short, the FFT of an array of lines is two sharp peaks. Is it that simple? Yes, but the careful reader will have noticed the ‘nice and smooth’ above. What happens if the lines are instead steps, like this?
The FFT is still zero everywhere except on a single line, but now higher harmonics are required to account for these sharp transitions, so instead of two peaks, we get lots of them:
In the response to stripy revisited, Miao Yu and Francesco Stellacci write:
As shown in Figure S1 in the Supporting Information, a series of aligned bands (with a very well-defined wavelength λ ) shows two parallel bands in the FT, questioning this interpretation of the FT images.
How is this possible? Simple, they present clipped bands, i.e. the image is not invariant anymore; the bands have horizontal sharp edges and therefore a range of high frequency modes is required to represent those sharp edges.