通过解决 procrustes 问题的线性投影矩阵

Question

我有两个矩阵：

target = np.array([[1, 1, 1, 1, 1],
               [2, 2, 2, 2, 2],
               [3, 3, 3, 3, 3]])

source = np.array([[11, 11, 11, 11, 11],
               [22, 22, 22, 22, 22],
               [33, 33, 33, 33, 33]])

我想创建一个转换矩阵，将源矩阵投影到目标矩阵。

我发现 Scipy 库提供了一个功能来做到这一点：

from scipy.spatial import procrustes
mtx1, mtx2, disparity = procrustes(target, source)

基于documentation，它说：

因此，mtx2 是投影矩阵。

如果我有另一个数据，我想使用 Scipy 用于将 源 矩阵投影到目标一个？

我如何使用 Scipy 做到这一点？

Answer 1

您需要修改函数才能返回转换矩阵（R）。

移除 cmets 后的源代码如下所示：

def procrustes(data1, data2):
    mtx1 = np.array(data1, dtype=np.double, copy=True)
    mtx2 = np.array(data2, dtype=np.double, copy=True)

    if mtx1.ndim != 2 or mtx2.ndim != 2:
        raise ValueError("Input matrices must be two-dimensional")
    if mtx1.shape != mtx2.shape:
        raise ValueError("Input matrices must be of same shape")
    if mtx1.size == 0:
        raise ValueError("Input matrices must be >0 rows and >0 cols")

    # translate all the data to the origin
    mtx1 -= np.mean(mtx1, 0)
    mtx2 -= np.mean(mtx2, 0)

    norm1 = np.linalg.norm(mtx1)
    norm2 = np.linalg.norm(mtx2)

    if norm1 == 0 or norm2 == 0:
        raise ValueError("Input matrices must contain >1 unique points")

    # change scaling of data (in rows) such that trace(mtx*mtx') = 1
    mtx1 /= norm1
    mtx2 /= norm2

    # transform mtx2 to minimize disparity
    R, s = orthogonal_procrustes(mtx1, mtx2)
    mtx2 = np.dot(mtx2, R.T) * s    # HERE, the projected mtx2 is estimated.

    # measure the dissimilarity between the two datasets
    disparity = np.sum(np.square(mtx1 - mtx2))

    return mtx1, mtx2, disparity, R

来源：https://github.com/scipy/scipy/blob/v1.3.0/scipy/spatial/_procrustes.py#L17-L132