# Why are there vertical lines in the power spectrum?

I took an image, did an fft2 to it in Matlab and then squared the abs of the fourier transform to get the power spectrum.

Something I notice about the power spectrum however is that there are vertical lines in it like this:

This is the original image which the power spectrum was calculated from, for reference:

What are these lines and why do I see such phenomenon from the transformed image? Do the lines imply anything about the original image and the fourier transform?

• Do you have the original image ? It looks like you have a $sinc$ in the horizontal dimension in your FFT, i.e., some rectangular shapes in your original image. – sansuiso Mar 13 '13 at 8:02
• I have an idea, it's about the nature of the blur that was used to generate the image. But here is THE question: is it for some school assignment of any sort? I don't want to spoil any homework. – sansuiso Mar 13 '13 at 8:53
• – Maurits Mar 13 '13 at 9:16
• I don't think so, I believe it is related to the shape of the blur kernel. But thanks for the link, I didn't know this paper by L. Moisan. – sansuiso Mar 13 '13 at 9:34

Let's interpret what we are seeing: the spectrum seems to be made of the pointwise multiplication of 2 known Fourier patterns:

• a first one is the usual messy image Fourier transform
• a second pattern that modulates the image spectrum and that has a $sinc$ shape.

Now, let's see what the theory provides us:

• multiplication in the spectral domain is a convolution in the spatial domain => we are observing the convolution of an image by a blur kernel
• the inverse Fourier transform of a $sinc$ is a box => the blur kernel has an horizontal box shape (we can infer that from the orientation of the zero bands).

Finally, let's summarize: the image was blurred by a linear (horizontal) kernel. This is a typical example of motion blur, where the motion is horizontal. This is confirmed by looking at the original image.

And as a bonus, here is a paper where special shapes of motion blurs are designed, where the blur shapes do not have zeros that will disturb the deconvolution process: http://web.media.mit.edu/~raskar/deblur/CodedExpousreLowres.pdf

• Thanks for your really informative answer! +1! One thing I'm not entirely sure of, though, is how do I observe that the blur kernel has an horizontal box shape? You mentioned that we can infer from the orientation of the zero bands. I'm not quite sure what zero bands are and I looked up on that and I got the definition as $x\cdot y=x$. But I don't understand how I can understand the shape from $x\cdot y=x$? – xenon Mar 14 '13 at 3:29
• I'm talking about the bands in your figure where the modulation function is zero, i.e., the vertical black lines, and not of some analytical line equations.They correspond to horizontal frequencies that are shut down, i.e., a horizontal sinus cardial, i.e. in space domain an horzontal box blur. – sansuiso Mar 14 '13 at 8:34