What is the relation between eigen values and principal axes length for 3D data?
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For a region in an binary image, I have calculated co-variance matrix using co-ordinates of region. Using co-variance matrix, I got two eigen values. Later I have calculated major axis length and minor axis length using the formula $4sqrt{lambda_i}$ where the $lambda_i$ are the eigenvalues. This formula is mentioned in the below given link 3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix. I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching for 2D object.
Now, I want to calculate principal axes lengths (major, middle and minor axis lengths) for 3D object data. For that I want a relation between eigen values obtained from 3D object co-ordinates (using same procedure mentioned above) and principal axes lengths for 3D data. 'regionprops' command in MATLAB do not work for 3D object.
eigenvalues-eigenvectors
edited Apr 13 '17 at 12:21
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asked Jun 3 '15 at 5:42
shrikant mehre
12
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For a region in an binary image, I have calculated co-variance matrix using co-ordinates of region. Using co-variance matrix, I got two eigen values. Later I have calculated major axis length and minor axis length using the formula $4sqrt{lambda_i}$ where the $lambda_i$ are the eigenvalues. This formula is mentioned in the below given link 3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix. I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching for 2D object.
Now, I want to calculate principal axes lengths (major, middle and minor axis lengths) for 3D object data. For that I want a relation between eigen values obtained from 3D object co-ordinates (using same procedure mentioned above) and principal axes lengths for 3D data. 'regionprops' command in MATLAB do not work for 3D object.
eigenvalues-eigenvectors
edited Apr 13 '17 at 12:21
Community♦
1
asked Jun 3 '15 at 5:42
shrikant mehre
12
1
What is "principal axes length" and where the formula 2*2*sqrt(eigenValue) comes from?
– enzotib
Jun 3 '15 at 6:01
@enzotib: Three dimensional object will have major axis, middle axis and minor axis. These axes are called principal axes. Formula mentioned above is taken from [link] (math.stackexchange.com/questions/911792/…). I have verified lengths of axes using mentioned formula and using 'regionprops' from Matlab.
– shrikant mehre
Jun 3 '15 at 11:04
So you're saying: "I have accomplished this in 2D, and now I'm wondering how I can do it for 3D images"?
– GPerez
Jun 3 '15 at 12:50
@GPerez: Yes. Exactly. 'regionprops' from MATLAB will give axes length for 2D. I want to calculate it for 3D data.
– shrikant mehre
Jun 3 '15 at 13:11
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
For a region in an binary image, I have calculated co-variance matrix using co-ordinates of region. Using co-variance matrix, I got two eigen values. Later I have calculated major axis length and minor axis length using the formula $4sqrt{lambda_i}$ where the $lambda_i$ are the eigenvalues. This formula is mentioned in the below given link 3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix. I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching for 2D object.
Now, I want to calculate principal axes lengths (major, middle and minor axis lengths) for 3D object data. For that I want a relation between eigen values obtained from 3D object co-ordinates (using same procedure mentioned above) and principal axes lengths for 3D data. 'regionprops' command in MATLAB do not work for 3D object.
eigenvalues-eigenvectors
edited Apr 13 '17 at 12:21
Community♦
1
asked Jun 3 '15 at 5:42
shrikant mehre
12
For a region in an binary image, I have calculated co-variance matrix using co-ordinates of region. Using co-variance matrix, I got two eigen values. Later I have calculated major axis length and minor axis length using the formula $4sqrt{lambda_i}$ where the $lambda_i$ are the eigenvalues. This formula is mentioned in the below given link 3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix. I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching for 2D object.
Now, I want to calculate principal axes lengths (major, middle and minor axis lengths) for 3D object data. For that I want a relation between eigen values obtained from 3D object co-ordinates (using same procedure mentioned above) and principal axes lengths for 3D data. 'regionprops' command in MATLAB do not work for 3D object.
eigenvalues-eigenvectors
eigenvalues-eigenvectors
edited Apr 13 '17 at 12:21
Community♦
1
asked Jun 3 '15 at 5:42
shrikant mehre
12
edited Apr 13 '17 at 12:21
Community♦
1
asked Jun 3 '15 at 5:42
shrikant mehre
12
edited Apr 13 '17 at 12:21
Community♦
1
edited Apr 13 '17 at 12:21
Community♦
1
edited Apr 13 '17 at 12:21
Community♦
1
1
asked Jun 3 '15 at 5:42
shrikant mehre
12
asked Jun 3 '15 at 5:42
shrikant mehre
12
asked Jun 3 '15 at 5:42
shrikant mehre
12
12
1
What is "principal axes length" and where the formula 2*2*sqrt(eigenValue) comes from?
– enzotib
Jun 3 '15 at 6:01
@enzotib: Three dimensional object will have major axis, middle axis and minor axis. These axes are called principal axes. Formula mentioned above is taken from [link] (math.stackexchange.com/questions/911792/…). I have verified lengths of axes using mentioned formula and using 'regionprops' from Matlab.
– shrikant mehre
Jun 3 '15 at 11:04
So you're saying: "I have accomplished this in 2D, and now I'm wondering how I can do it for 3D images"?
– GPerez
Jun 3 '15 at 12:50
@GPerez: Yes. Exactly. 'regionprops' from MATLAB will give axes length for 2D. I want to calculate it for 3D data.
– shrikant mehre
Jun 3 '15 at 13:11
add a comment |
1
What is "principal axes length" and where the formula 2*2*sqrt(eigenValue) comes from?
– enzotib
Jun 3 '15 at 6:01
@enzotib: Three dimensional object will have major axis, middle axis and minor axis. These axes are called principal axes. Formula mentioned above is taken from [link] (math.stackexchange.com/questions/911792/…). I have verified lengths of axes using mentioned formula and using 'regionprops' from Matlab.
– shrikant mehre
Jun 3 '15 at 11:04
So you're saying: "I have accomplished this in 2D, and now I'm wondering how I can do it for 3D images"?
– GPerez
Jun 3 '15 at 12:50
@GPerez: Yes. Exactly. 'regionprops' from MATLAB will give axes length for 2D. I want to calculate it for 3D data.
– shrikant mehre
Jun 3 '15 at 13:11
1
1
What is "principal axes length" and where the formula 2*2*sqrt(eigenValue) comes from?
– enzotib
Jun 3 '15 at 6:01
What is "principal axes length" and where the formula 2*2*sqrt(eigenValue) comes from?
– enzotib
Jun 3 '15 at 6:01
@enzotib: Three dimensional object will have major axis, middle axis and minor axis. These axes are called principal axes. Formula mentioned above is taken from [link] (math.stackexchange.com/questions/911792/…). I have verified lengths of axes using mentioned formula and using 'regionprops' from Matlab.
– shrikant mehre
Jun 3 '15 at 11:04
@enzotib: Three dimensional object will have major axis, middle axis and minor axis. These axes are called principal axes. Formula mentioned above is taken from [link] (math.stackexchange.com/questions/911792/…). I have verified lengths of axes using mentioned formula and using 'regionprops' from Matlab.
– shrikant mehre
Jun 3 '15 at 11:04
So you're saying: "I have accomplished this in 2D, and now I'm wondering how I can do it for 3D images"?
– GPerez
Jun 3 '15 at 12:50
So you're saying: "I have accomplished this in 2D, and now I'm wondering how I can do it for 3D images"?
– GPerez
Jun 3 '15 at 12:50
@GPerez: Yes. Exactly. 'regionprops' from MATLAB will give axes length for 2D. I want to calculate it for 3D data.
– shrikant mehre
Jun 3 '15 at 13:11
@GPerez: Yes. Exactly. 'regionprops' from MATLAB will give axes length for 2D. I want to calculate it for 3D data.
– shrikant mehre
Jun 3 '15 at 13:11
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
If I understand correctly your question, your formula seems to be wrong.
In three dimensions the distribution of mass could be represented by the inertia ellipsoid, whose equation, in a principal frame is
$$
I_x x^2 + I_y y^2 + I_z z^2 = 1,
$$
so that the length of the axes of the ellipsoid are $a_i = 1/sqrt{I_i} = 1/sqrt{lambda_i}$ (the principal moments of inertia are the eigenvalues of the inertia matrix).
See section 4.5 of http://www.eng.auburn.edu/~marghitu/MECH2110/C_4.pdf .
answered Jun 3 '15 at 12:14
enzotib
5,80821430
I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching with the value obtained using mentioned formula for 2D object.
– shrikant mehre
Jun 7 '15 at 12:42
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
If I understand correctly your question, your formula seems to be wrong.
In three dimensions the distribution of mass could be represented by the inertia ellipsoid, whose equation, in a principal frame is
$$
I_x x^2 + I_y y^2 + I_z z^2 = 1,
$$
so that the length of the axes of the ellipsoid are $a_i = 1/sqrt{I_i} = 1/sqrt{lambda_i}$ (the principal moments of inertia are the eigenvalues of the inertia matrix).
See section 4.5 of http://www.eng.auburn.edu/~marghitu/MECH2110/C_4.pdf .
answered Jun 3 '15 at 12:14
enzotib
5,80821430
I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching with the value obtained using mentioned formula for 2D object.
– shrikant mehre
Jun 7 '15 at 12:42
add a comment |
up vote
0
down vote
If I understand correctly your question, your formula seems to be wrong.
In three dimensions the distribution of mass could be represented by the inertia ellipsoid, whose equation, in a principal frame is
$$
I_x x^2 + I_y y^2 + I_z z^2 = 1,
$$
so that the length of the axes of the ellipsoid are $a_i = 1/sqrt{I_i} = 1/sqrt{lambda_i}$ (the principal moments of inertia are the eigenvalues of the inertia matrix).
See section 4.5 of http://www.eng.auburn.edu/~marghitu/MECH2110/C_4.pdf .
answered Jun 3 '15 at 12:14
enzotib
5,80821430
I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching with the value obtained using mentioned formula for 2D object.
– shrikant mehre
Jun 7 '15 at 12:42
add a comment |
up vote
0
down vote
up vote
0
down vote
If I understand correctly your question, your formula seems to be wrong.
In three dimensions the distribution of mass could be represented by the inertia ellipsoid, whose equation, in a principal frame is
$$
I_x x^2 + I_y y^2 + I_z z^2 = 1,
$$
so that the length of the axes of the ellipsoid are $a_i = 1/sqrt{I_i} = 1/sqrt{lambda_i}$ (the principal moments of inertia are the eigenvalues of the inertia matrix).
See section 4.5 of http://www.eng.auburn.edu/~marghitu/MECH2110/C_4.pdf .
answered Jun 3 '15 at 12:14
enzotib
5,80821430
If I understand correctly your question, your formula seems to be wrong.
In three dimensions the distribution of mass could be represented by the inertia ellipsoid, whose equation, in a principal frame is
$$
I_x x^2 + I_y y^2 + I_z z^2 = 1,
$$
so that the length of the axes of the ellipsoid are $a_i = 1/sqrt{I_i} = 1/sqrt{lambda_i}$ (the principal moments of inertia are the eigenvalues of the inertia matrix).
See section 4.5 of http://www.eng.auburn.edu/~marghitu/MECH2110/C_4.pdf .
answered Jun 3 '15 at 12:14
enzotib
5,80821430
answered Jun 3 '15 at 12:14
enzotib
5,80821430
answered Jun 3 '15 at 12:14
enzotib
5,80821430
answered Jun 3 '15 at 12:14
enzotib
5,80821430
5,80821430
I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching with the value obtained using mentioned formula for 2D object.
– shrikant mehre
Jun 7 '15 at 12:42
add a comment |
I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching with the value obtained using mentioned formula for 2D object.
– shrikant mehre
Jun 7 '15 at 12:42
I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching with the value obtained using mentioned formula for 2D object.
– shrikant mehre
Jun 7 '15 at 12:42
I have verified lengths of major axis and minor axis with the output of 'regionprops' command in MATLAB. It is exactly matching with the value obtained using mentioned formula for 2D object.
– shrikant mehre
Jun 7 '15 at 12:42
add a comment |
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1
What is "principal axes length" and where the formula 2*2*sqrt(eigenValue) comes from?
– enzotib
Jun 3 '15 at 6:01
@enzotib: Three dimensional object will have major axis, middle axis and minor axis. These axes are called principal axes. Formula mentioned above is taken from [link] (math.stackexchange.com/questions/911792/…). I have verified lengths of axes using mentioned formula and using 'regionprops' from Matlab.
– shrikant mehre
Jun 3 '15 at 11:04
So you're saying: "I have accomplished this in 2D, and now I'm wondering how I can do it for 3D images"?
– GPerez
Jun 3 '15 at 12:50
@GPerez: Yes. Exactly. 'regionprops' from MATLAB will give axes length for 2D. I want to calculate it for 3D data.
– shrikant mehre
Jun 3 '15 at 13:11