confusion over averaging rotation matrices as described
I am currently reading the paper on automatic calibration of traffic cameras (https://www.microsoft.com/en-us/research/wp-content/uploads/2017/09/AutoCalib.pdf). At some point, the authors compute multiple calibrations and compute an average of the rotation part of these matrices.
So, the text describes the process as follows:
Finally, we compute the “average” of the remaining calibrations. We
average the Z axis unit vector across all filtered calibrations and
compute two mutually orthogonal X and Y axis unit vectors.
So, let us assume I have two rotation matrices that I want to average. So something like:
r1 = [[-0.64375223, 0.63471324, 0.42746014],
[-0.52107859, -0.77267423, 0.3625626 ],
[ 0.56041072, 0.01066016, 0.82814624]]
r2 = [[-0.31267459, -0.0464914 , 0.94872185],
[-0.88839581, -0.33914166, -0.30941205],
[ 0.3361361 , -0.93958581, 0.06473821]]
So, first I compute the z-axes vector average as:
import numpy as np
c = [r1, r2]
z_avg = np.mean(c, axis=0)[:, 2]
This creates the average z-axes as:
array([0.688091 , 0.02657528, 0.44644223])
Then I wrote the function to compute the basis vectors as:
def basis(v):
v = v / np.linalg.norm(v)
if v[0] > 0.9:
b1 = np.asarray([0.0, 1.0, 0.0])
else:
b1 = np.asarray([1.0, 0.0, 0.0])
b1 -= v * np.dot(b1, v)
b1 /= np.linalg.norm(b1)
b2 = np.cross(v, b1)
return b1, b2, v
I can compute the orthogonal basis vectors as:
x, y, z = basis(z_avg)
Now, the next steps are what has me confused. The text goes on to say:
We then compute the Rotation Matrix for these three unit vectors,
which forms our (averaged) final Rotation Matrix for the calibration
estimate
I am really not sure what is meant by "computing the rotation matrix for these three unit vectors". I am also failing to see how this process is somehow "averaging" these rotation matrices. Any insight you can give me would be really appreciated!
image-processing computer-vision coordinates transform coordinate-transformation
asked Nov 23 '18 at 22:46
LucaLuca
3,22352779
add a comment |
I am currently reading the paper on automatic calibration of traffic cameras (https://www.microsoft.com/en-us/research/wp-content/uploads/2017/09/AutoCalib.pdf). At some point, the authors compute multiple calibrations and compute an average of the rotation part of these matrices.
So, the text describes the process as follows:
Finally, we compute the “average” of the remaining calibrations. We
average the Z axis unit vector across all filtered calibrations and
compute two mutually orthogonal X and Y axis unit vectors.
So, let us assume I have two rotation matrices that I want to average. So something like:
r1 = [[-0.64375223, 0.63471324, 0.42746014],
[-0.52107859, -0.77267423, 0.3625626 ],
[ 0.56041072, 0.01066016, 0.82814624]]
r2 = [[-0.31267459, -0.0464914 , 0.94872185],
[-0.88839581, -0.33914166, -0.30941205],
[ 0.3361361 , -0.93958581, 0.06473821]]
So, first I compute the z-axes vector average as:
import numpy as np
c = [r1, r2]
z_avg = np.mean(c, axis=0)[:, 2]
This creates the average z-axes as:
array([0.688091 , 0.02657528, 0.44644223])
Then I wrote the function to compute the basis vectors as:
def basis(v):
v = v / np.linalg.norm(v)
if v[0] > 0.9:
b1 = np.asarray([0.0, 1.0, 0.0])
else:
b1 = np.asarray([1.0, 0.0, 0.0])
b1 -= v * np.dot(b1, v)
b1 /= np.linalg.norm(b1)
b2 = np.cross(v, b1)
return b1, b2, v
I can compute the orthogonal basis vectors as:
x, y, z = basis(z_avg)
Now, the next steps are what has me confused. The text goes on to say:
We then compute the Rotation Matrix for these three unit vectors,
which forms our (averaged) final Rotation Matrix for the calibration
estimate
I am really not sure what is meant by "computing the rotation matrix for these three unit vectors". I am also failing to see how this process is somehow "averaging" these rotation matrices. Any insight you can give me would be really appreciated!
image-processing computer-vision coordinates transform coordinate-transformation
asked Nov 23 '18 at 22:46
LucaLuca
3,22352779
add a comment |
I am currently reading the paper on automatic calibration of traffic cameras (https://www.microsoft.com/en-us/research/wp-content/uploads/2017/09/AutoCalib.pdf). At some point, the authors compute multiple calibrations and compute an average of the rotation part of these matrices.
So, the text describes the process as follows:
Finally, we compute the “average” of the remaining calibrations. We
average the Z axis unit vector across all filtered calibrations and
compute two mutually orthogonal X and Y axis unit vectors.
So, let us assume I have two rotation matrices that I want to average. So something like:
r1 = [[-0.64375223, 0.63471324, 0.42746014],
[-0.52107859, -0.77267423, 0.3625626 ],
[ 0.56041072, 0.01066016, 0.82814624]]
r2 = [[-0.31267459, -0.0464914 , 0.94872185],
[-0.88839581, -0.33914166, -0.30941205],
[ 0.3361361 , -0.93958581, 0.06473821]]
So, first I compute the z-axes vector average as:
import numpy as np
c = [r1, r2]
z_avg = np.mean(c, axis=0)[:, 2]
This creates the average z-axes as:
array([0.688091 , 0.02657528, 0.44644223])
Then I wrote the function to compute the basis vectors as:
def basis(v):
v = v / np.linalg.norm(v)
if v[0] > 0.9:
b1 = np.asarray([0.0, 1.0, 0.0])
else:
b1 = np.asarray([1.0, 0.0, 0.0])
b1 -= v * np.dot(b1, v)
b1 /= np.linalg.norm(b1)
b2 = np.cross(v, b1)
return b1, b2, v
I can compute the orthogonal basis vectors as:
x, y, z = basis(z_avg)
Now, the next steps are what has me confused. The text goes on to say:
We then compute the Rotation Matrix for these three unit vectors,
which forms our (averaged) final Rotation Matrix for the calibration
estimate
I am really not sure what is meant by "computing the rotation matrix for these three unit vectors". I am also failing to see how this process is somehow "averaging" these rotation matrices. Any insight you can give me would be really appreciated!
image-processing computer-vision coordinates transform coordinate-transformation
asked Nov 23 '18 at 22:46
LucaLuca
3,22352779
I am currently reading the paper on automatic calibration of traffic cameras (https://www.microsoft.com/en-us/research/wp-content/uploads/2017/09/AutoCalib.pdf). At some point, the authors compute multiple calibrations and compute an average of the rotation part of these matrices.
So, the text describes the process as follows:
Finally, we compute the “average” of the remaining calibrations. We
average the Z axis unit vector across all filtered calibrations and
compute two mutually orthogonal X and Y axis unit vectors.
So, let us assume I have two rotation matrices that I want to average. So something like:
r1 = [[-0.64375223, 0.63471324, 0.42746014],
[-0.52107859, -0.77267423, 0.3625626 ],
[ 0.56041072, 0.01066016, 0.82814624]]
r2 = [[-0.31267459, -0.0464914 , 0.94872185],
[-0.88839581, -0.33914166, -0.30941205],
[ 0.3361361 , -0.93958581, 0.06473821]]
So, first I compute the z-axes vector average as:
import numpy as np
c = [r1, r2]
z_avg = np.mean(c, axis=0)[:, 2]
This creates the average z-axes as:
array([0.688091 , 0.02657528, 0.44644223])
Then I wrote the function to compute the basis vectors as:
def basis(v):
v = v / np.linalg.norm(v)
if v[0] > 0.9:
b1 = np.asarray([0.0, 1.0, 0.0])
else:
b1 = np.asarray([1.0, 0.0, 0.0])
b1 -= v * np.dot(b1, v)
b1 /= np.linalg.norm(b1)
b2 = np.cross(v, b1)
return b1, b2, v
I can compute the orthogonal basis vectors as:
x, y, z = basis(z_avg)
Now, the next steps are what has me confused. The text goes on to say:
We then compute the Rotation Matrix for these three unit vectors,
which forms our (averaged) final Rotation Matrix for the calibration
estimate
I am really not sure what is meant by "computing the rotation matrix for these three unit vectors". I am also failing to see how this process is somehow "averaging" these rotation matrices. Any insight you can give me would be really appreciated!
image-processing computer-vision coordinates transform coordinate-transformation
image-processing computer-vision coordinates transform coordinate-transformation
asked Nov 23 '18 at 22:46
LucaLuca
3,22352779
asked Nov 23 '18 at 22:46
LucaLuca
3,22352779
asked Nov 23 '18 at 22:46
LucaLuca
3,22352779
asked Nov 23 '18 at 22:46
LucaLuca
3,22352779
asked Nov 23 '18 at 22:46
LucaLuca
3,22352779
3,22352779
add a comment |
add a comment |
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