我有一个3D数组(矢量的2D数组),我想用旋转矩阵变换每个矢量。旋转位于两个独立的2D弧度角值数组中,分别称为colsrows

我已经能够让NumPy为我计算角度,而无需Python循环。现在,我正在寻找一种方法,使NumPy也生成旋转矩阵,希望可以大大提高性能。

size = img.shape[:2]

# Create an array that assigns each pixel the percentage of
# the correction (value between -1 and 1, distributed linearly).
cols = np.array([np.arange(size[1]) for __ in range(size[0])])   / (size[1] - 1) * 2 - 1
rows = np.array([np.arange(size[0]) for __ in range(size[1])]).T / (size[0] - 1) * 2 - 1

# Atan distribution based on F-number and Sensor size.
cols = np.arctan(sh * cols / (2 * f))
rows = np.arctan(sv * rows / (2 * f))

### This is the loop that I would like to remove and find a
### clever way to make NumPy do the same operation natively.
for i in range(size[0]):
  for j in range(size[1]):
    ah = cols[i,j]
    av = rows[i,j]

    # Y-rotation.
    mat = np.matrix([
      [ np.cos(ah), 0, np.sin(ah)],
      [0, 1, 0],
      [-np.sin(ah), 0, np.cos(ah)]
    ])

    # X-rotation.
    mat *= np.matrix([
      [1, 0, 0],
      [0, np.cos(av), -np.sin(av)],
      [0, np.sin(av),  np.cos(av)]
    ])

    img[i,j] = img[i,j] * mat

return img


有什么聪明的方法可以重写NumPy操作中的循环吗?

最佳答案

(假设img的形状为(a, b, 3)。)

首先,colsrows不需要完全扩展为(a, b)(您可以写cols[j]代替cols[i,j])。使用np.linspace可以轻松生成它们:

cols = np.linspace(-1, 1, size[1])   # shape: (b,)
rows = np.linspace(-1, 1, size[0])   # shape: (a,)

cols = np.arctan(sh * cols / (2*f))
rows = np.arctan(sv * rows / (2*f))


然后,我们可以预先计算矩阵的成分。

# shape: (b,)
cos_ah = np.cos(cols)
sin_ah = np.sin(cols)
zeros_ah = np.zeros_like(cols)
ones_ah = np.ones_like(cols)

# shape: (a,)
cos_av = np.cos(rows)
sin_av = np.sin(rows)
zeros_av = np.zeros_like(rows)
ones_av = np.ones_like(rows)


然后构造旋转矩阵:

# shape: (3, 3, b)
y_mat = np.array([
    [cos_ah, zeros_ah, sin_ah],
    [zeros_ah, ones_ah, zeros_ah],
    [-sin_ah, zeros_ah, cos_ah],
])

# shape: (3, 3, a)
x_mat = np.array([
    [ones_av, zeros_av, zeros_av],
    [zeros_av, cos_av, -sin_av],
    [zeros_av, sin_av, cos_av],
])


现在让我们看看。如果有一个循环,我们将编写:

for i in range(size[0]):
    for j in range(size[1]):
        img[i, j, :] = img[i, j, :] @ y_mat[:, :, j] @ x_mat[:, :, i]


或者,如果我们展开矩阵乘法:

python - 为Numpy中的值对生成矩阵-LMLPHP

使用np.einsum可以很好地处理此问题(请注意,i,j,k,m,n完全类似于上面的公式):

img = np.einsum('ijk,kmj,mni->ijn', img, y_mat, x_mat)




总结一下:

size = img.shape[:2]

cols = np.linspace(-1, 1, size[1])   # shape: (b,)
rows = np.linspace(-1, 1, size[0])   # shape: (a,)

cols = np.arctan(sh * cols / (2*f))
rows = np.arctan(sv * rows / (2*f))

cos_ah = np.cos(cols)
sin_ah = np.sin(cols)
zeros_ah = np.zeros_like(cols)
ones_ah = np.ones_like(cols)

cos_av = np.cos(rows)
sin_av = np.sin(rows)
zeros_av = np.zeros_like(rows)
ones_av = np.ones_like(rows)

y_mat = np.array([
    [cos_ah, zeros_ah, sin_ah],
    [zeros_ah, ones_ah, zeros_ah],
    [-sin_ah, zeros_ah, cos_ah],
])

x_mat = np.array([
    [ones_av, zeros_av, zeros_av],
    [zeros_av, cos_av, -sin_av],
    [zeros_av, sin_av, cos_av],
])

return np.einsum('ijk,kmj,mni->ijn', img, y_mat, x_mat)

10-07 14:05
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