Finding if path exists on 2D surface

Here's a fun problem I ran into. Say you have a given 2D function, f(x,y) (as an example, you can use peaks(30)). Now, I separately have a measured "path", i.e. f_{meas}, but I don't know the x_meas and y_meas that created that path. I can assume that the values of x_meas and y_meas are smooth, no discontinuities. I want, given an input starting x and y, to determine if there is a "path" along the function surface that matches the measured "path", and determine if there are multiple solutions (as there likely will be!). Seems like I can use a minimization or fitting routine here, but it's not clear yet. Interested if anyone has any tricks for doing such a thing.