How many dimensions is a RGB image?

Question

I first thought it would be 3 dimensions (one for R, one for G, one for B). However what about the width and height?

More a physics/image processing question I think.

So, I have a few possible answers to this question myself: 3, 5, 9.

3 is from where you just count the R, G and B as a separate dimension. But then you are ignoring the width and height of the image.

5 is from considering width and height on top of each RGB channel.

9 is from considering width, height and intensity for each channel, so it becomes 3+3+3 hence 9 dimensions.

Any clarity in what is considered a 'dimension' in regards to images will help my understanding, thanks!

Answer 1

There are many discussions about this topic in this site; just search for "signal dimension".

In my view, the number of "coordinates" of the "output" is what determines the number of dimensions. An image is a mapping $$\mathbb{R}^2 \mapsto \mathbb{R}^3.$$ Then, an image has three dimensions.

Another definition of "dimension" is related to basis functions. Assume the signal $f$ can be represented as $$f(t) = \sum_{n=1}^N c_n\phi_n(t)$$ where $c_n$ are scalars. This means that the signal $f$ is actually a linear combination of $N$ "basis functions" $\phi_n(t)$. Here, $N$ would be the dimension of $f$.

For example, signals that can be represented with exactly three Fourier coefficients would have dimension 3. In this sense, the number of dimensions is unrelated to the number of coordinates. As another example, one could say a black image has dimension 1, because it can be represented with the single basis function $\phi(x,y) = \text{black}$.