python numpy matrix multiplication for coordinate transformation gives a wrong result - Stack Overflow

admin管理员组

文章数量:1392073

Here is my code: I am performing a coordinate transformation on a set of 3D points. T is my 4×4 transformation matrix, which always satisfies the form [[R , t], [ 0, 1]](I am certain of this). points is the set of points I need to transform, with shape (M,3) where M is usually very large(1k~2k).

My problem is this: When I perform the coordinate transformation using Method 1 (homogeneous coordinates), the result is incorrect. Specifically, the last number of some of the transformed points (points_local_homogeneous) is not equal to 1, that's not correct. However, when I use Method 2, all the results are correct.

       # Method 1: wrong result.
       ones = np.ones((points.shape[0], 1))
       points_homogeneous = np.hstack((points, ones))
       points_local_homogeneous = (T @ points_homogeneous.T).T
       points_local = points_local_homogeneous[:, :3]
       # Method 2: Good results.
       points_local= (T[:3, :3] @ points.T).T + T[:3, 3]

There is also a strange phenomenon that reflects the problem: When I first perform the matrix multiplication and then extract a point, the result is wrong. But when I extract the point first and then perform the matrix multiplication, the result is correct.

>> (T @ points_homogeneous.T).T[547]
array([-15.44923687, -0.03295934, -0.07585889, -15.45439878])
>> T @ points_homogeneous[547]
array([6.58938647, 3.43626129, 2.84996901, 1. ])

Can anyone help me understand the reason behind this? Any guidance would be greatly appreciated!

numpy version : 1.23.5

minimal reproducible example

import numpy as np


T =np.array([[-9.99997984e-01,  2.00338436e-03,  1.36517526e-04,2.30385575e+01],
            [-2.00432316e-03, -9.99971609e-01, -7.26382636e-03,3.48879370e+00],
            [ 1.21961414e-04, -7.26408534e-03,  9.99973609e-01,-5.29494007e-02],
            [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00,1.00000000e+00]])
points = np.random.random((1000,3))
print(points)


ones = np.ones((points.shape[0], 1))
points_homogeneous = np.hstack((points, ones))
points_local_homogeneous = (T @ points_homogeneous.T).T
print(points_local_homogeneous)
points_local = points_local_homogeneous[:, :3]
print(points_local)
points_local= (T[:3, :3] @ points.T).T + T[:3, 3]
print(points_local)

Output:

points:
[[0.70431115 0.18240672 0.33961428]
[0.03708955 0.67343087 0.82448419]
[0.56714298 0.73581627 0.6482321 ]
...
[0.57882963 0.04147515 0.05351834]
[0.67357367 0.02644866 0.19563735]
[0.86036017 0.27597764 0.63466743]]

points_local_homogeneous:
[[ 2.23346596e+01 3.30251359e+00 2.85416795e-01 1.00000000e+00]
[ 2.30029297e+01 2.80931870e+00 7.66625690e-01 1.00000000e+00]
[ 2.24729783e+01 2.74715294e+00 5.89989729e-01 1.00000000e+00]
...
[ 4.21157332e-01 -1.46142995e-03 9.03349144e-01 5.65885052e-01]
[ 3.26428767e-01 -1.54217777e-03 5.56317837e-01 4.18719365e-01]
[ 1.39730259e-01 -3.72909348e-03 6.30229075e+00 1.10249332e+00]]

points_local(Method 1):
[[ 2.23346596e+01 3.30251359e+00 2.85416795e-01]
[ 2.30029297e+01 2.80931870e+00 7.66625690e-01]
[ 2.24729783e+01 2.74715294e+00 5.89989729e-01]
...
[ 4.21157332e-01 -1.46142995e-03 9.03349144e-01]
[ 3.26428767e-01 -1.54217777e-03 5.56317837e-01]
[ 1.39730259e-01 -3.72909348e-03 6.30229075e+00]]

points_local(Method 2):
[[2.23346596e+01 3.30251359e+00 2.85416795e-01]
[2.30029297e+01 2.80931870e+00 7.66625690e-01]
[2.24729783e+01 2.74715294e+00 5.89989729e-01]
...
[2.24598194e+01 3.44577082e+00 3.36839801e-04]
[2.23650649e+01 3.45957466e+00 1.42572815e-01]
[2.21788386e+01 3.20648934e+00 5.79801482e-01]]

The last colomn of points_local_homogeneous should be all 1.0000, but it's not, and the result of Method 1 and Method 2 is different. I don't know why?

Here is my code: I am performing a coordinate transformation on a set of 3D points. T is my 4×4 transformation matrix, which always satisfies the form [[R , t], [ 0, 1]](I am certain of this). points is the set of points I need to transform, with shape (M,3) where M is usually very large(1k~2k).

My problem is this: When I perform the coordinate transformation using Method 1 (homogeneous coordinates), the result is incorrect. Specifically, the last number of some of the transformed points (points_local_homogeneous) is not equal to 1, that's not correct. However, when I use Method 2, all the results are correct.

       # Method 1: wrong result.
       ones = np.ones((points.shape[0], 1))
       points_homogeneous = np.hstack((points, ones))
       points_local_homogeneous = (T @ points_homogeneous.T).T
       points_local = points_local_homogeneous[:, :3]
       # Method 2: Good results.
       points_local= (T[:3, :3] @ points.T).T + T[:3, 3]

There is also a strange phenomenon that reflects the problem: When I first perform the matrix multiplication and then extract a point, the result is wrong. But when I extract the point first and then perform the matrix multiplication, the result is correct.

>> (T @ points_homogeneous.T).T[547]
array([-15.44923687, -0.03295934, -0.07585889, -15.45439878])
>> T @ points_homogeneous[547]
array([6.58938647, 3.43626129, 2.84996901, 1. ])

Can anyone help me understand the reason behind this? Any guidance would be greatly appreciated!

numpy version : 1.23.5

minimal reproducible example

import numpy as np


T =np.array([[-9.99997984e-01,  2.00338436e-03,  1.36517526e-04,2.30385575e+01],
            [-2.00432316e-03, -9.99971609e-01, -7.26382636e-03,3.48879370e+00],
            [ 1.21961414e-04, -7.26408534e-03,  9.99973609e-01,-5.29494007e-02],
            [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00,1.00000000e+00]])
points = np.random.random((1000,3))
print(points)


ones = np.ones((points.shape[0], 1))
points_homogeneous = np.hstack((points, ones))
points_local_homogeneous = (T @ points_homogeneous.T).T
print(points_local_homogeneous)
points_local = points_local_homogeneous[:, :3]
print(points_local)
points_local= (T[:3, :3] @ points.T).T + T[:3, 3]
print(points_local)

Output:

points:
[[0.70431115 0.18240672 0.33961428]
[0.03708955 0.67343087 0.82448419]
[0.56714298 0.73581627 0.6482321 ]
...
[0.57882963 0.04147515 0.05351834]
[0.67357367 0.02644866 0.19563735]
[0.86036017 0.27597764 0.63466743]]

points_local_homogeneous:
[[ 2.23346596e+01 3.30251359e+00 2.85416795e-01 1.00000000e+00]
[ 2.30029297e+01 2.80931870e+00 7.66625690e-01 1.00000000e+00]
[ 2.24729783e+01 2.74715294e+00 5.89989729e-01 1.00000000e+00]
...
[ 4.21157332e-01 -1.46142995e-03 9.03349144e-01 5.65885052e-01]
[ 3.26428767e-01 -1.54217777e-03 5.56317837e-01 4.18719365e-01]
[ 1.39730259e-01 -3.72909348e-03 6.30229075e+00 1.10249332e+00]]

points_local(Method 1):
[[ 2.23346596e+01 3.30251359e+00 2.85416795e-01]
[ 2.30029297e+01 2.80931870e+00 7.66625690e-01]
[ 2.24729783e+01 2.74715294e+00 5.89989729e-01]
...
[ 4.21157332e-01 -1.46142995e-03 9.03349144e-01]
[ 3.26428767e-01 -1.54217777e-03 5.56317837e-01]
[ 1.39730259e-01 -3.72909348e-03 6.30229075e+00]]

points_local(Method 2):
[[2.23346596e+01 3.30251359e+00 2.85416795e-01]
[2.30029297e+01 2.80931870e+00 7.66625690e-01]
[2.24729783e+01 2.74715294e+00 5.89989729e-01]
...
[2.24598194e+01 3.44577082e+00 3.36839801e-04]
[2.23650649e+01 3.45957466e+00 1.42572815e-01]
[2.21788386e+01 3.20648934e+00 5.79801482e-01]]

The last colomn of points_local_homogeneous should be all 1.0000, but it's not, and the result of Method 1 and Method 2 is different. I don't know why?

• python
• numpy
• transformation-matrix

Share Improve this question Follow edited Mar 12 at 3:13 pei asked Mar 11 at 17:09 peipei 122 bronze badges 1

• The whole point of homogeneous coordinates is that the last value is not necessarily 1. If it were always 1, we wouldn't bother with keeping 4 values per vector, and 16 values per matrices. That 1 isn't (as we too often read on internet) just a trick to represent affine operation as linear ones (so to have translation matrices, for example). It is also to perform projective geometry operations. That being said, if your are sure that the last row of T is 0,0,0,1, then, both method should have the same result. But without minimal reproducible example, hard to say anything other than "can't reproduce the problem" – chrslg Commented Mar 11 at 22:32

Add a comment  | 

1 Answer 1

Sorted by: Reset to default 0

Running your code in Colab after installing your version of NumPy:

import numpy as np
print(np.__version__)

T =np.array([[-9.99997984e-01,  2.00338436e-03,  1.36517526e-04,2.30385575e+01],
            [-2.00432316e-03, -9.99971609e-01, -7.26382636e-03,3.48879370e+00],
            [ 1.21961414e-04, -7.26408534e-03,  9.99973609e-01,-5.29494007e-02],
            [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00,1.00000000e+00]])
points = np.random.random((1000,3))


ones = np.ones((points.shape[0], 1))
points_homogeneous = np.hstack((points, ones))
points_local_homogeneous = (T @ points_homogeneous.T).T
print(points_local_homogeneous)

Gives

1.23.5
[[22.67514956  3.02269043  0.40258376  1.        ]
 [22.48891076  3.21800509  0.70566574  1.        ]
 [22.83249535  2.76367985  0.40577443  1.        ]
 ...
 [23.0335628   3.05567068  0.0307568   1.        ]
 [22.26727281  2.93475402  0.12775867  1.        ]
 [22.73114851  2.81104189  0.61403673  1.        ]]

So if you're not getting that, I'd suggest reinstalling NumPy.

本文标签： python numpy matrix multiplication for coordinate transformation gives a wrong resultStack Overflow