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Frequency analysis is often performed on wave forms (1D DFT), and images (2D DFT), where the function in question often takes the form:

$f(t): \Re \mapsto \Re$

$f(x,y): \Re^2 \mapsto \Re$

$f(x_1, x_2, \ldots, x_n): \Re^n \mapsto \Re$

However, note that in all 3 cases 'f' maps to a single real value. If, however, f takes the form:

$f(t): \Re \mapsto \Re^n$

... it isn't clear to me whether the Fourier transform can be used to perform any kind of frequency analysis that would provide any information across dimensions.

If the dimensions are spatially correlated, eg, the sample $f_1(t_0)$ is physically adjacent to $f_2(t_0)$, and circularly adjacent to $f_n(t_0)$, then intuitively it would make sense to use the 2D DFT to perform the desired analysis. This interpretation, in essence, transforms f into the form $f(t,x): \Re^2 \mapsto \Re$.

However, if no such relationship can be imposed on the set $\lbrace f_i\rbrace$, does there exist an analog of the FFT/DFT for this sort of problem?

To put this another way: if $f(t): \Re \mapsto \Re^n$, and $f(t)$ is transformed in a similar fashion -- eg, $f(t,i): \Re^2 \mapsto \Re$, where $i$ indexes into the dimensions of $f(t)$ -- is there a generalized approach to Fourier analysis that can make use of the index variable without making the assumption that $i=1$ and $i=2$ have any spatial relationship?

A vector-valued function can be used in the expression for computing a Fourier Transform, but unfortunately that results in computing the Fourier transform of each component of the vector-valued function without making use of information available in other dimensions of the range. In other words, if $f(t): \Re \mapsto \Re^3$, then $F\lbrace f(t)\rbrace = (F\lbrace f_1(t)\rbrace, F\lbrace f_2(t)\rbrace, F\lbrace f_3(t)\rbrace)$. The question isn't whether this is possible, but whether more can be done than just this level of analysis.

A somewhat recent paper by Thomas Batard may answer the question, but I don't have the expertise to know whether it does. His paper, A Metric Approach to nD Images Edge Detection with Clifford Algebras, demonstrates a technique for performing analysis on color images, where the mapping might take a form similar to $(c,m,y,k) = f(x,y): \Re^2 \mapsto \Re^4$.

As I have time, I would like to study Clifford algebras, but if this is a good path for me to go down it would give me more incentive to do so earlier rather than later.

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Regarding notation, we would usually say that $f$ maps $\mathbf{R} \to \mathbf{R}^n$ and that $t$ is a coordinate on the source space. –  S. Carnahan Aug 13 '10 at 16:19

2 Answers 2

It is reasonably straightforward to generalize traditional Fourier analysis to vector-valued functions $f$, because the transform process only requires the operations of adding and taking scalar multiples (or suitable limits thereof, like integrals). You end up with a vector-valued Fourier transform function $\check{f}$, and if you have a preferred basis like in your examples, then you can get the components of $\check{f}$ simply by transforming the components of $f$. I do not know what this has to do with Clifford algebras.

I should point out that the dimension of the DFT is defined as the dimension of the source space, not the target.

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You can certainly plug a vector valued function into the standard expression of a Fourier transform, and as you mentioned you end up with something of the form: F{g(x)} = <F{g_1(x)}, F{g_2(x)}, ..., F{g_n(x)}> ... but you're still doing filtering (or other analysis) in one dimension. In Thomas Batard's paper, he came up with an approach for generalizing typical edge detection techniques that employ DFTs, but is able to do so using information in the entire color spectrum, instead of a black & white subset of an image. –  Brian Vandenberg Aug 13 '10 at 17:34

I discovered this discussion used Google because I was wondering about problems similar to the following: Suppose that you want to use Fourier analysis to analyse a photograph of (for example) some trees and you want to create a camouflage pattern. One way to do this would be to Fourier transform a number of photographs of the trees and then use the Fourier spectrum to generate patterns that have the same spatial frequency content. This would work fine in black and white, but, what do you do when you have three channels of colour? Treating each colour channel separately does not seem quite right.

One idea that has occurred to me (but I haven't tried it yet) would be to perform something like a principal component analysis (or proper orthogonal decomposition) on the colour values for the pixels, and then analyse the Fourier spectra of the spatial variation of the amplitudes of these components.

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