问题
Contour lines (aka isolines) are curves that trace constant values across a 2D scalar field. For example, in a geographical map you might have contour lines to illustrate the elevation of the terrain by showing where the elevation is constant. In this case, let's store contour lines as lists of points on the map.
Suppose you have map that has several contour lines at known elevations, and otherwise you know nothing about the elevations of the map. What algorithm would you use to fill in additional contour lines to approximate the unknown elevations of the map, assuming the landscape is continuous and doesn't do anything surprising?
It is easy to find advise about interpolating the elevation of an individual point using contour lines. There are also algorithms like Marching Squares for turning point elevations into contour lines, but none of these exactly capture this use case. We don't need the elevation of any particular point; we just want the contour lines. Certainly we could solve this problem by filling an array with estimated elevations and then using Marching Squares to estimate the contour lines based on the array, but the two steps of that process seem unnecessarily expensive and likely to introduce artifacts. Surely there is a better way.
回答1:
IMO, about all methods will amount to somehow reconstructing the 3D surface by interpolation, even if implicitly.
You may try by flattening the curves (turning them to polylines) and triangulating the resulting polygons thay they will define. (There will be a step of closing the curves that end on the border of the domain.)
By intersection of the triangles with a new level (unsing linear interpolation along the sides), you will obtain new polylines corresponding to new isocurves. Notice that the intersections with the old levels recreates the old polylines, which is sound.
You may apply a post-smoothing to the curves, but you will have no guarantee to retrieve the original old curves and cannot prevent close surves to cross each other.
Beware that increasing the density of points along the curves will give you a false feeling of accuracy, as the error due to the spacing of the isolines will remain (indeed the reconstructed surface will be cone-like, with one of the curvatures being null; the surface inside the bottommost and topmost lines will be flat).
Alternatively to using flat triangles, one may think of a scheme where you compute a gradient vector at every vertex (f.i. from a least square fit of a plane on the vertex and its neighbors), and use this information to generate a bivariate polynomial surface in the triangle. You must do this in such a way that the values along a side will coincide for the two triangles that share it. (Unfortunately, I have no formula to give you.)
The isolines are then obtained by a further subdivision of the triangle in smaller triangles, with a flat approximation.
Actually, this is not very different from getting sample points, (Delaunay) triangulating them and fitting picewise continuous patches to the triangles.
Whatever method you will use, be it 2D or 3D, it is useful to reason on what happens if you sweep the range of z values in a continous way. This thought experiment does reconstruct a 3D surface, which will possess continuity and smoothness properties.
A possible improvement over the crude "flat triangulation" model could be to extend every triangle side between to iso-polylines with sides leading to the next iso-polylines. This way, higher order interpolation (cubic) can be achieved, giving a smoother reconstruction.
Anyway, you can be sure that this will introduce discontinuities or other types of artifacts.
回答2:
A mixed method:
flatten the isolines to polylines;
triangulate the poygons formed by the polylines and the borders;
on every node, estimate the surface gradient (least-square fit of a plane to the node and its neighborrs);
in every triangle, consider the two sides along which you need to interpolate and compute the derivative at endpoints (from the known gradients and the side directions);
use Hermite interpolation along these sides and solve for the desired iso-levels;
join the points obtained on both sides.
This method should be a good tradeoff between complexity and smoothness. It does reconstruct a continuous surface (except maybe for the remark below).
Note that is some cases, yo will obtain three solutions of the cubic. If there are three on each side, join them in order. Otherwise, make a decision on which to join and use the remaining two to close the curve.
来源：https://stackoverflow.com/questions/44994214/interpolating-missing-contour-lines-between-existing-contour-lines