Can I Calculate the Fourier Transform of an Image by Hand?

[u0362565]

Hi all,

I have a somewhat qualitative understanding of image Fourier transforms and what they represent which for the most part is sufficient for me. However i am interested to know how when i use an image analysis program to produce the Fourier transform of a real image, what is actually going on? Would it be possible by hand to calculate the Fourier transform of an image and if i wanted to do this what would be the process? Ultimately all you have in the real image is pixels which represent changes in intensity over 2 dimensional space? What values from the real image are used in the Fourier transform equation?

Thanks for the help!

 

[Orodruin]

Do you know how Fourier series work mathematically? What the image program does is essentially to make a Fourier expansion of the intensity functions for each channel (RGB). The result is that the middle pixel of the Fourier transform represents the overall average intensity of the image and the other pixels how the colors vary over the image. A high intensity of one color near the center means that color displays large variations on large scales of the image and the further you get from the center the smaller the scale becomes. All of the values from the real image are used in creating the transform and no information is lost.

Technically you could compute the Fourier transform by hand, but it would be extremely tedious and you would likely end up making errors just because of moving a lot of numbers around. A computer is much more suited to do this.

 

[enorbet]

Thankfully Joseph didn't mind a little tedium ;) The degree of tedium though does depend upon just how much information you wish to encode and study. By the late 1960s hand-held, real time instruments were available that would handle FFTs. I owned a few that cost considerably less than $200 USD at the time. It would be 20 years before real microprocessing with even 1MB Ram would exist.

 

[u0362565]

Hi guys,

What i know is that to produce the Fourier transform you're essentially breaking the image up into its component sinusoids (variations in intensity in space). These are then mapped to 2d axes. Where as you say the origin represents the zero frequency and increasing spatial frequencies are further from the origin. An intuitive understanding of what's going on is not so difficult to comprehend but when i see the Fourier formula and say i have an image, I wouldn't know what values from the image are actually being plugged into the formula. Is it an operation that works on each pixel in turn? But a pixel only tells you two things, its position and its intensity, where is this utilised in the formula?

Thanks

 

[Orodruin]

Essentially all of the pixels are used for all of the Fourier modes (although some may be multiplied by zero depending on the sinusoid). The operation transforms all of the information in the picture into a different way of displaying the same information.

 

[u0362565]

Ok thanks, i think that's all i need to know.