如何将仿射变换（4x4矩阵）应用于ndgrid / meshgrid结果？ [英] How to apply an affine transformation (4x4 matrix) to ndgrid/meshgrid results?

问题描述

$ $ p是一个仿射变换矩阵，它可以将坐标从一个坐标系转换到另一个坐标系：

  v_old = [x_old y_old z_old 1]; 
 v_new = M * v_old; 
％v_new包含[x_new y_new z_new 1] 
  

现在， ndgrid / meshgrid的形式：

  [X_old Y_old Z_old] = ndgrid（0:15）; ％例如
  

如何将这些转换为 X_new etc？

我可以用三个for循环循环（ X_old（i，j，k）对应于上面的 x_old ），但是必须有更好的解决方案。

解决方案

您只需重新排列值，以便每个点在一行中具有四个坐标：



  result = [X_old（ :) Y_old（:) Z_old（:) ones（numel（X_old），1）] * M; 
  

每行结果给出新坐标注意这是因为矩阵 A 乘以矩阵 B $ b / code>与每行乘以 A 次 B 。

或者，如果您无法构建上述矩阵（由于内存限制），请使用

 <$ （1，:) + Y_old（：）* M（2，:) + Z_old（：）* M（3，:) + ones（numel（X_old），1 ）* M（4，:); 
  

在任何一种情况下，如果要重新排列结果以使其与 X_old ， Y_old ， Z_old ，将其作为最后一步：

  s = size（X_old）; 
 X_new = reshape（result（：，1），s）; 
 Y_new = reshape（result（：，2），s）; 
 Z_new = reshape（result（：，3），s）; 
  

M is an affine transformation matrix that can transform coordinates from one coordinate system to another as follows:

v_old = [ x_old y_old z_old 1];
v_new = M * v_old;
% v_new contains [ x_new y_new z_new 1 ]

Now, I've got coordinates on the form of ndgrid/meshgrid:

[ X_old Y_old Z_old ] = ndgrid( 0:15 ); % for instance

How can I convert these to X_new etc?

I could do it with three for-loops loop (X_old(i,j,k) corresponds to x_old above) but there must be a better solution.

解决方案

You just have to rearrange the values so that each point has its four coordinates in one row:

result = [X_old(:) Y_old(:) Z_old(:) ones(numel(X_old),1)] * M;

Each row of result gives the new coordinates of each point.

Note that this works because multiplying matrix A times matrix B is the same as multiplying each row of A times B.

Or, if you can't afford to build the above matrix (because of memory limitations), use

result = X_old(:)*M(1,:) + Y_old(:)*M(2,:) + Z_old(:)*M(3,:) + ones(numel(X_old),1)*M(4,:);

In either case, if you want to rearrange the result so that it has the same size as X_old, Y_old, Z_old, use this as a last step:

s = size(X_old);
X_new = reshape(result(:,1), s);
Y_new = reshape(result(:,2), s);
Z_new = reshape(result(:,3), s);

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