# characterising the manifold representing images

Assuming that the Manifold Hypothesis is valid, or that real-world high-dimensional data lie on low-dimensional manifolds embedded within the high-dimensional space, How can one describe the hypothetical manifold that, for example, contains all $$W\times L$$ pixel images of a dog?

Let $$D$$ be the set of all images of dogs, such that the intrinsic dimension of any image $$dim_{in}(d)=n$$ for $$d\in D$$. Then the manifold $$(D,\mathcal{T})$$ is the representative manifold of all dogs. Let us define the chart $$p: D\mapsto \mathbb{R}^{WL}$$. We can then see that the common practice of representing a $$W\times L$$ image as a vector $$v\in \mathbb{R}^{WL}$$ is calling this function $$p$$ on an image, or that we are mapping the "true" image (whatever that means) to a euclidean vector representation.

For example, a binary classification neural network is then a function $$f(\cdot,\theta):\mathbb{R}^{WL}\to \mathbb{R}^2$$

This characterization then illustrates the point that the pixel vector representation does not necessarily reflect the intrinsic characteristics of an image. Indeed, if one crops the image from $$W\times L$$ to $$W'\times L$$, then you are simply choosing another chart to map from the dog manifold.

However, there are some questions I have. For one, the intrinsic dimension of an image is claimed to be much less than the extrinsic pixel dimension, i.e. $$n<< WL$$. If so, can the chart $$p$$ as defined above possibly even be homeomorphic?

Secondly, how could one argue somewhat rigorously that the A topological space formed by dog images even is a manifold? i.e. that it has the following properties

1. A Hausdorff Space
2. second-countable
3. has local Euclidean of dimension n
• Please ask only one question per post. I'm not sure what kind of answer you're expecting or hoping for to "How can one describe...?" Are you able to identify a concrete technical question? Statements like "images of dogs form a manifold" are probably often intended as informal statements, not necessarily something that can be proven rigorously.
– D.W.
Commented Jun 13, 2022 at 7:56
• The question is under-specified. All pixel images of a dog obviously form a 0-dimensional manifold, as they are just a finite collection of points in some high-dimensional $\mathbb{R}^n$. That is not what you meant, is it? But then, what did you mean? Commented Jun 15, 2022 at 6:16