如何从等值线生成 3-D 表面? [英] How Do I Generate a 3-D Surface From Isolines?

问题描述

我有一组等值线点(或轮廓点)，例如:

等值线上的每个点都有其各自的 X、Y 和 Z 坐标.由于它们是等值线，这意味着每个点将具有唯一的 X-Y 对，但同一条线上的点将具有相同的 Z 坐标.

现在，是否有任何算法或任何软件包(在 C# 或 C++ 或 MATLAB 中)可用于将等值线点插入到完整的 3-D 表面中?

P/S:我不仅对最终输出感兴趣，我还对获取插值表面数据感兴趣，以便我可以自己绘制表面.

也欢迎 C++ 解决方案.

解决方案

在 MATLAB 中，您可以使用函数

请注意，两个结果之间几乎没有区别(至少在这个范围内).还要注意，由于这些点的轮廓数据稀疏，插值表面在拐角附近有空白区域.

I have a set of isoline points (or contour points) such as this:

Each point on an isoline has its own respective X, Y, and Z coordinate. Since they are isolines, that means that each point will have a unique X-Y pair, but points on the same line will have the same Z coordinate.

Now, is there any algorithm or any software packages (either in C# or C++ or MATLAB) that I can use to interpolate the isoline points into a full 3-D surface?

P/S: I am not just interested in the final output, I am interested in getting the interpolated surface data so that I can plot the surface myself.

Edit: C++ solutions are welcomed as well.

解决方案

In MATLAB you can use either the function griddata or the TriScatteredInterp class (Note: as of R2013a scatteredInterpolant is the recommended alternative). Both of these allow you to fit a surface of regularly-spaced data to a set of nonuniformly-spaced points (although it appears griddata is no longer recommended in newer MATLAB versions). Here's how you can use each:

• griddata:

  [XI,YI,ZI] = griddata(x,y,z,XI,YI)
  

  where x,y,z each represent vectors of the cartesian coordinates for each point (in this case the points on the contour lines). The row vector XI and column vector YI are the cartesian coordinates at which griddata interpolates the values ZI of the fitted surface. The new values returned for the matrices XI,YI are the same as the result of passing XI,YI to meshgrid to create a uniform grid of points.

• TriScatteredInterp class:

  [XI,YI] = meshgrid(...);
  F = TriScatteredInterp(x(:),y(:),z(:));
  ZI = F(XI,YI);
  

  where x,y,z again represent vectors of the cartesian coordinates for each point, only this time I've used a colon reshaping operation (:) to ensure that each is a column vector (the required format for TriScatteredInterp). The interpolant F is then evaluated using the matrices XI,YI that you must create using meshgrid.

Example & Comparison

Here's some sample code and the resulting figure it generates for reconstructing a surface from contour data using both methods above. The contour data was generated with the contour function:

% First plot:

subplot(2,2,1);
[X,Y,Z] = peaks;  % Create a surface
surf(X,Y,Z);
axis([-3 3 -3 3 -8 9]);
title('Original');

% Second plot:

subplot(2,2,2);
[C,h] = contour(X,Y,Z);  % Create the contours
title('Contour map');

% Format the coordinate data for the contours:

Xc = [];
Yc = [];
Zc = [];
index = 1;
while index < size(C,2)
  Xc = [Xc C(1,(index+1):(index+C(2,index)))];
  Yc = [Yc C(2,(index+1):(index+C(2,index)))];
  Zc = [Zc C(1,index).*ones(1,C(2,index))];
  index = index+1+C(2,index);
end

% Third plot:

subplot(2,2,3);
[XI,YI] = meshgrid(linspace(-3,3,21));  % Generate a uniform grid
ZI = griddata(Xc,Yc,Zc,XI,YI);          % Interpolate surface
surf(XI,YI,ZI);
axis([-3 3 -3 3 -8 9]);
title('GRIDDATA reconstruction');

% Fourth plot:

subplot(2,2,4);
F = TriScatteredInterp(Xc(:),Yc(:),Zc(:));  % Generate interpolant
ZIF = F(XI,YI);                             % Evaluate interpolant
surf(XI,YI,ZIF);
axis([-3 3 -3 3 -8 9]);
title('TriScatteredInterp reconstruction');

Notice that there is little difference between the two results (at least at this scale). Also notice that the interpolated surfaces have empty regions near the corners due to the sparsity of contour data at those points.

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