Images in the frequency domain

[ndesk1900]

Hi,

I've been trying to understand more on frequency plots of images.

1. Assuming a 1d example of just a strip of an image; in the frequency map there are 'same' no. of frequencies as there are pixels. What is the reason for this? Does this have something to do with the Nyquist theorom.

2. What you see in the frequency domain is just a plot of all the spatial frequencies and their amplitude corresponds to the coefficient of that frequency? Is this correct?

3. Now if in the 1d example, i see a one recurring pattern...
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

How would this map to the frequency domain? So for example, i see the sequence '0 0 0 1' 4 times. What can i make out of this?

4. In 2d all the points near the center are the lower frequencies and away from the center are higher frequencies. What can be said about the image if we get higher amplitudes in the lower freqencies as compared to the higher frequencies. What does this mean? Can we say that simpler (or more complicated) images will not peak at the center like this? [PS:- i do know that lower frequencies capture the general aspects of the image whereas the higher frequencies capture the more 'detailed' aspects (e.g. edges)]

Thanks
Neville

 

[MATLABdude]

Howdy, and welcome to Physics Forums.

My first inclination was that this might be more appropriate in the Computer Science forum, but then again, we do signal processing in EE, as well. And many of us have taken undergraduate optics whereas some CS types haven't taken either of the above.

So, I found this website when I googled image frequency domain:
http://local.wasp.uwa.edu.au/~pbourke/miscellaneous/imagefilter/

1) IIRC, if you had more frequencies than pixels, the contribution of those frequencies (higher than the number of pixels you had) would not contribute anything to said image.

2) Yes, that's the definition of a Fourier transform.

3) That's where it gets a little tricky. Your lowest frequency would be one or two 1s in the center. The next highest frequency would be two 1s separating three equal-sized blocks of 0s (I might be wrong about the relative spacing here, but you get the idea). There's probably a frequency that corresponds to what you have (it would have an imaginary or complex coefficient because the ones are shifted off to one side).

4) Given the above discussion in point 3, generally speaking, if you have sharp (and lots of) transitions (i.e. edges), you'll have lots of high frequency components. If you look at the link above, they start with a sharp box, and by filtering out high frequency components, they get one with fuzzy edges.

 

[rrogers]

1) You actually end up with only n/2 frequencies that are distinct. The extra degrees (n/2) of freedom (lost pixel information) ends up in the phase information; or sine/cos if you transform the mag/phase to cartesian coordinates. Remember that this is just a transform or alternately a description of the original data set; there are others Mellin/Wavelets. If the original data had n degrees of freedom, the description must have the same number of coefficients in order to reconstruct the orginal data.
2) Yes
3) This is the repeating impulse pattern. It looks similar in the frequency domain; but inverted subtly. A single data point "1" produces a uniform set of ones in the frequency domain amplitude; but with phases determined by the position. When more impulses are added (repeating in your case) the phases start interacting to produce interference. When the pattern repeats you end up with regular impulses in the frequency domain. Let's say you have two impulses neatly spaced the fundamental frequency will end up at 2/n and have uniform harmonics from there. You can actually watch this during transformation because the integral/sum for each frequency becomes a simple summation and the transform kernel is sampled at the particular points where the input is non-zero. Thus in the continuous case you get e^-jw*ts+e^jw*ts; where ts is the position of the none-zero time. If you walk through these simple cases you get to see the time/picture data move over to the frequency description. In the simple case of impulses it's just high school complex numbers in action; in the case of rectangles it's just a little more complicated but there is a trick using Dirac delta functionals that works very neatly; and provides insight. As usual try to get inside the operations looking out; instead of outside watching things happen. Impulse are the easy way to see what is happening.
4) What was said by Matlabdude is a good description. An alternate way of looking at the center frequency is that it measures the total signal amplitude/energy of the picture; averaged/summed. The next harmonic is the average against one frequency cycle fit into the picture, etc ... So just pick the term you want put down a sine/cos wave on the picture of that frequency and average. Edges are interesting; they are integrated impulses which introduce impulses in the frequency domain with 1/f rolloff. Thus they introduce higher harmonics but not as fast as real impulses (which introduce constant terms).

Read this critically, it's early in the morning and I have been know to make egregious errors then (well worse than other times).

Ray