3D最小二乘飞机 [英] 3D Least Squares Plane

问题描述

什么算法，用于计算在（X，Y，Z）空间中的最小二乘平面，给定一组的三维数据点？换句话说，如果我有一堆点像（1，2，3），（4，5，6），（7，8，9），等等，如何将一去计算最合适的飞机˚F （X，Y）=斧+由+ C？什么是算法求A，B和C出一套三维点？

What's the algorithm for computing a least squares plane in (x, y, z) space, given a set of 3D data points? In other words, if I had a bunch of points like (1, 2, 3), (4, 5, 6), (7, 8, 9), etc., how would one go about calculating the best fit plane f(x, y) = ax + by + c? What's the algorithm for getting a, b, and c out of a set of 3D points?

推荐答案

如果您有n个数据点（X [I]，Y [I]，Z [I]），计算3×3的对称矩阵A的项目有：

If you have n data points (x[i], y[i], z[i]), compute the 3x3 symmetric matrix A whose entries are:

sum_i x[i]*x[i],    sum_i x[i]*y[i],    sum_i x[i]
sum_i x[i]*y[i],    sum_i y[i]*y[i],    sum_i y[i]
sum_i x[i],         sum_i y[i],         n

另外计算3元向量b：

Also compute the 3 element vector b:

{sum_i x[i]*z[i],   sum_i y[i]*z[i],    sum_i z[i]}

然后解决Ax = b的为给定的a和b。解向量的三个分量是系数的最小二乘拟合平面{A，B，C}

Then solve Ax = b for the given A and b. The three components of the solution vector are the coefficients to the least-square fit plane {a,b,c}.

请注意，这是普通最小二乘适合，这是适当的，只有当z被预期为x和y的线性函数。如果您正在寻找更多的一般在3维空间最佳拟合平面，你可能想了解几何最小二乘。

Note that this is the "ordinary least squares" fit, which is appropriate only when z is expected to be a linear function of x and y. If you are looking more generally for a "best fit plane" in 3-space, you may want to learn about "geometric" least squares.

另请注意，这将失败，如果你的点是在一条线上，因为你的榜样点。

Note also that this will fail if your points are in a line, as your example points are.

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