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I was just wondering if there were any "obvious" resources, papers, or textbooks I should look into regarding the nyquist-criterion in higher dimensions. For example, this would be relevant if there is some curve that is i.e. not 1-to-1 w.r.t. time: $f: t \in \mathbb{R}^1 \rightarrow f(t) \in \mathbb{R}^1 $, but instead for some other parameterization, we have a line $l: u \in \mathbb{R}^1 \rightarrow l(u) = \big(\tilde t(u),\tilde f(u)\big) \in \mathbb{R}^2$, it is 1-to-1, but not onto, as $(\tilde t,\tilde f)$ is on a 1-dimensional manifold (a line) in a 2-D plane.

Then I'd like to know what sampling scheme I need to be able to recreate the curve through reconstruction. Using this approach, you could use nyquist's theorem for general spatial surfaces and manifolds as well.

This might be a math question, so I can migrate this. But I figured some experts on this sub-stack might know.

EDIT: several small edits for clarity/correctness.

EDIT 2: Fixed parameterization!!!

I was just wondering if there were any "obvious" resources, papers, or textbooks I should look into regarding the nyquist-criterion in higher dimensions. For example, this would be relevant if there is some curve that is i.e. not 1-to-1 w.r.t. time: $f: t \in \mathbb{R}^1 \rightarrow f(t) \in \mathbb{R}^1 $, but instead for some other parameterization, we have a line $l: u \in \mathbb{R}^1 \rightarrow l(u) = \big(\tilde t(u),\tilde f(u)\big) \in \mathbb{R}^2$, it is 1-to-1, but not onto, as $(\tilde t,\tilde f)$ is on a 1-dimensional manifold (a line) in a 2-D plane.

Then I'd like to know what sampling scheme I need to be able to recreate the curve through reconstruction. Using this approach, you could use nyquist's theorem for general spatial surfaces and manifolds as well.

This might be a math question, so I can migrate this. But I figured some experts on this sub-stack might know.

EDIT: several small edits for clarity/correctness.

EDIT 2: Fixed parameterization!!!

EDIT 3: I want to emphasize that although I am parameterizing by $t$, it may be more useful to think of $t$ as a spatial dimension, since time can only move forward.

I was just wondering if there were any "obvious" resources, papers, or textbooks I should look into regarding the nyquist-criterion in higher dimensions. For example, this would be relevant if there is some curve that is i.e. not 1-to-1 w.r.t. time: $f: t \in \mathbb{R}^1 \rightarrow f(t) \in \mathbb{R}^1 $, but instead for some other parameterization $(u,v) \in \mathbb{R}^2 \rightarrow \big(t(u,v),\tilde f(u,v)\big) \in \mathbb{R}^2$, it is 1-to-1, but your independent variables are no longer single dimensional.

Then I'd like to know what sampling scheme I need to be able to recreate the curve through reconstruction. Using this approach, you could use nyquist's theorem for general spatial surfaces and manifolds as well.

This might be a math question, so I can migrate this. But I figured some experts on this sub-stack might know.

EDIT: several small edits for clarity/correctness.

I was just wondering if there were any "obvious" resources, papers, or textbooks I should look into regarding the nyquist-criterion in higher dimensions. For example, this would be relevant if there is some curve that is i.e. not 1-to-1 w.r.t. time: $f: t \in \mathbb{R}^1 \rightarrow f(t) \in \mathbb{R}^1 $, but instead for some other parameterization, we have a line $l: u \in \mathbb{R}^1 \rightarrow l(u) = \big(\tilde t(u),\tilde f(u)\big) \in \mathbb{R}^2$, it is 1-to-1, but not onto, as $(\tilde t,\tilde f)$ is on a 1-dimensional manifold (a line) in a 2-D plane.

Then I'd like to know what sampling scheme I need to be able to recreate the curve through reconstruction. Using this approach, you could use nyquist's theorem for general spatial surfaces and manifolds as well.

This might be a math question, so I can migrate this. But I figured some experts on this sub-stack might know.

EDIT: several small edits for clarity/correctness.

EDIT 2: Fixed parameterization!!!

I was just wondering if there were any "obvious" resources, papers, or textbooks I should look into regarding the nyquist-criterion in higher dimensions. For example, this would be relevant if there is some curve that is i.e. not 1-to-1 w.r.t. time: $f: t \in \mathcal{R}^1 \rightarrow f(t) \in \mathcal{R}^1 $, but instead for some other parameterization $(u,v) \in \mathcal{R}^2 \rightarrow \big(t(u,v),\tilde f(u,v)\big) \in \mathcal{R}^2$, it is 1-to-1, but your independent variables are no longer single dimensional.

Then I'd like to know what sampling scheme I need to be able to recreate the curve through reconstruction. Using this approach, you could use nyquist's theorem for general spatial surfaces and manifolds as well.

This might be a math question, so I can migrate this. But I figured some experts on this sub-stack might know.

EDIT: several small edits for clarity/correctness.

I was just wondering if there were any "obvious" resources, papers, or textbooks I should look into regarding the nyquist-criterion in higher dimensions. For example, this would be relevant if there is some curve that is i.e. not 1-to-1 w.r.t. time: $f: t \in \mathbb{R}^1 \rightarrow f(t) \in \mathbb{R}^1 $, but instead for some other parameterization $(u,v) \in \mathbb{R}^2 \rightarrow \big(t(u,v),\tilde f(u,v)\big) \in \mathbb{R}^2$, it is 1-to-1, but your independent variables are no longer single dimensional.

Then I'd like to know what sampling scheme I need to be able to recreate the curve through reconstruction. Using this approach, you could use nyquist's theorem for general spatial surfaces and manifolds as well.

This might be a math question, so I can migrate this. But I figured some experts on this sub-stack might know.

EDIT: several small edits for clarity/correctness.