Nyquist Criterion in higher dimensions

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 do not have a reference for you, but you are sampling among every one of the dimensions and the Nyquist Criterion must be satisfied for each one.

so fix $v$ to an arbitrary value $v_0$, then check if $\tilde{f}(u,v_0)$ is bandlimited some bandlimit, we'll call $B_u$. find the max value of $B_u$ for all possible $v_0$, double that and that is the minimum sample rate along the $u$-axis.

then swap the roles of $u$ and $v$, fix $u$ to $u_0$, find the bandlimit of $\tilde{f}(u_0,v)$ for every possible value of $u_0$. find the maximum (over all $u_0$) bandlimit $B_v$, double that and that is the minimum-sample rate along the $v$-axis.

if your curve can move at an arbitrary angle w.r.t. the $u$-axis and $v$-axis, then pick the maximum of $B_u$ and $B_v$ and sample both axes at double of that rate (or slightly higher). then, along the path length, you must sample at a minimum of that rate, per unit of arc length, to satisfy Nyquist in two dimensions.

• Actually, looking at your answer made me realize I have made a mistake in my original question, as the reparameterization of $t$ and $f$ makes $(t,\tilde f)$ look like a surface of points when in fact I intended for $(t,\tilde f)$ to be a 1 dimensional line in a 2D plane. I'll update the question. – user27886 Apr 1 '15 at 22:22
• actually, i think the way you wanna word it is that you have this function of two variables (appears as a surface) $\tilde{f}(x,y)$ and the two independent variables are the two coordinates of, what we called in the Calculus of Several Variables course, a "parametric equation": $x(t)$ and $y(t)$. and you're interested in $$f(t) = \tilde{f}\left( x(t),y(t) \right)$$. is that right? – robert bristow-johnson Apr 1 '15 at 23:18
• I am interested in a parametric form of some function involving f, but not it will not be equal to the original $f(t)$, since $f(t) \in \mathcal{R}^1$ and the line I am talking has one parameter, but exists in 2 dimensions, or in a plane. For a 3D example, imagine if this spring were infinitely thin, that it were a 1-D manifold ( a line $l(u)$) in in 3D space, where $u$ is a scalar parameter that takes me from the beginning of the spring to the end of it. How can i sample this line to allow perfect reconstruction from the samples? – user27886 Apr 1 '15 at 23:29
• sounds like you're defining an 3-D parametric equation, $x(t)$, $y(t)$, and $z(t)$. is that right? and you want to know how often you need to sample that curve? it would be the largest of the three minimum sample rates derived from each of $x(t)$, $y(t)$, and $z(t)$. – robert bristow-johnson Apr 1 '15 at 23:44
• Okay, I see. That is definitely a starting point. I feel like in some scenarios, though, the true rate of sampling that needs to be found would be lower than the max nyquist rate on any of the three dimensions. Lines in higher spaces with a very large radius of curvature (lower freq) could appear appear much higher in frequency when considering its projection onto any one of its three coordinates. – user27886 Apr 2 '15 at 0:44