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Now naturally after completing a physics degree I am very familiar with the form and function of the Fourier Transform (FT) but never have grasped it quite conceptually. I understand that given a function f(x) I can express every functional value as a linear combination of complex sinusoidal functions as in the typical illustration of a square wave and you keep adding sine waves of higher and higher frequencies to get an approximate square wave (that would approach the exact with an infinite number of frequencies). Here is what I struggle with in terms of the continuous FT. When I compute a FT of function I get another function F(k), where k is frequency. I can plot this like any other function but what am I plotting? I think, but I am not positive, that the value of F(k) is the summation of all the linear coefficients C(k) for a particular frequency over all values f(x)? Is this correct?

Second, the real reason I am trying to get everything straight in my head is because I am doing image processing for my research and when I look at the FT, as an image, I am not sure what I am looking at. I know that it represents spatial frequencies in the image, i.e if you make a interference band you get a dot in the center and two dots spaced equidistant from the center. Now, image processing basically comes down to matrix algebra because an image in nothing more than a matrix of intensity values, I(x,y). So when I do the FT I am doing matrix algebra of the image with a matrix composed of several different sinusoidal complex functions. Here is where I am confused, what constitutes a frequency in an image? On a pixel basis, say I have a white dot followed by a black dot pattern that would constitute a certain frequency, I think. However, how does it look from a linear algebra standpoint?

If I have a i by j image A, how do I interpret the image of the FT? So if I follow my logic from the 1D continuous case, assuming it is the correct interpretation, looking at pixels (1,1:j) in A and taking the FT, B=TA, where T is the FT transformation matrix the first matrix element is going to be the summation of all the pixels intensities in that row with a corresponding T element. But when all is said and done and I pull up the image of the Fourier Transform on MATLAB I don't think I am looking at matrix B but instead each pixel (x,y) corresponds to frequency in the x and y directions, i.e if I look at pixel (1,1) in the FT, the intensity corresponds to "how much" a white pixel is followed by a black pixel in the image?

If you have read all this, thanks, I am hoping my interpretation is correct but just want to make sure.