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High Dimensional Spaces for Images

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Problem Detail: 

Can anyone explain why pictures are not considered 2D, but rather high dimensional? Especially with regards to CV and AI.

Asked By : Jonathan
Answered By : D.W.

From one perspective, a picture is a 2D image, because it has height and width.

But from a machine learning perspective, we can think of a picture as a point in a high-dimensional space. In particular, suppose we have a greyscale picture that is $m\times n$ pixels, i.e., $m$ pixels wide and $n$ pixels high. Then there are a total of $mn$ pixels in the image. Each pixel has a greyscale intensity, which we can think of as a real number in the interval $[0,1]$. Therefore, we can think of the picture as being a collection of $mn$ real numbers. In other words, the picture can be treated as a $mn$-dimensional vector -- as an element of $\mathbb{R}^{mn}$. Thus, any particular picture can be thought of as an element of a high-dimensional space.

The latter perspective arises natural for some machine learning approaches to computer vision, e.g., where we feed the pixels of the image into the machine learning algorithm, where each pixel value is treated as a separate pixel.

(A color image can be thought of as an element of $\mathbb{R}^{3mn}$: for each pixel, we have three numbers, corresponding to the intensity in the red, green, and blue channels.)

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