Mixing
how to verify that N points are on the same plane (but may not be perfectly)
$begingroup$
Given a collection of points in 3d. The value of each dimension can be visualized
When we visualize these points in 3d. They clearly are actually only live on a plane (but not perfectly), visualized bellow
My question is, what is a general approach to verify that any number of collection of points live on the same plane. Or more generally, any collection of points of dimenion m, actually live on n dimension space where n < m.
One way of doing such a job is to find 3 points to make a plane and iteratively test every point. However, this is not what I am looking for due to the fact that it is not generalizable for more than 3->2 dimension and if there is a slight amount of noise, the method falls apart. One approach I have tried is to find the SVD decomposition and decide by the eigenvalues. I am certain this problem has been solved principally but I am not sure how, I appreciate any opinion.
linear-algebra matrices linear-transformations
asked Jan 3 at 19:09
ScicareScicare
1134
$endgroup$
add a comment |
$begingroup$
Given a collection of points in 3d. The value of each dimension can be visualized
When we visualize these points in 3d. They clearly are actually only live on a plane (but not perfectly), visualized bellow
My question is, what is a general approach to verify that any number of collection of points live on the same plane. Or more generally, any collection of points of dimenion m, actually live on n dimension space where n < m.
One way of doing such a job is to find 3 points to make a plane and iteratively test every point. However, this is not what I am looking for due to the fact that it is not generalizable for more than 3->2 dimension and if there is a slight amount of noise, the method falls apart. One approach I have tried is to find the SVD decomposition and decide by the eigenvalues. I am certain this problem has been solved principally but I am not sure how, I appreciate any opinion.
linear-algebra matrices linear-transformations
asked Jan 3 at 19:09
ScicareScicare
1134
$endgroup$
2
$begingroup$
Pick 3 points, find the plane between them, then check whether the rest satisfy the plane equation.
$endgroup$
– Don Thousand
Jan 3 at 19:12
$begingroup$
If you can make a matrix 3 by 3 with the coords of 3 of these points, with rank 3, then they lay not in the same plane.
$endgroup$
– dmtri
Jan 3 at 19:25
$begingroup$
Hi @DonThousand and dmtri , I have just edited and add additional info. Thanks for your opinion and I'd appreciate any additional insight.
$endgroup$
– Scicare
Jan 3 at 19:33
2
$begingroup$
Are you sure you're not looking to do Principal Component Analysis instead? Applied to your example data, it would yield an orthonormal transform (rotation) around the coordinate mean, with the figure being on the $xy$ plane (with all $z$ coordinates being almost the same).
$endgroup$
– Nominal Animal
Jan 3 at 21:41
add a comment |
$begingroup$
Given a collection of points in 3d. The value of each dimension can be visualized
When we visualize these points in 3d. They clearly are actually only live on a plane (but not perfectly), visualized bellow
My question is, what is a general approach to verify that any number of collection of points live on the same plane. Or more generally, any collection of points of dimenion m, actually live on n dimension space where n < m.
One way of doing such a job is to find 3 points to make a plane and iteratively test every point. However, this is not what I am looking for due to the fact that it is not generalizable for more than 3->2 dimension and if there is a slight amount of noise, the method falls apart. One approach I have tried is to find the SVD decomposition and decide by the eigenvalues. I am certain this problem has been solved principally but I am not sure how, I appreciate any opinion.
linear-algebra matrices linear-transformations
asked Jan 3 at 19:09
ScicareScicare
1134
$endgroup$
Given a collection of points in 3d. The value of each dimension can be visualized
When we visualize these points in 3d. They clearly are actually only live on a plane (but not perfectly), visualized bellow
My question is, what is a general approach to verify that any number of collection of points live on the same plane. Or more generally, any collection of points of dimenion m, actually live on n dimension space where n < m.
One way of doing such a job is to find 3 points to make a plane and iteratively test every point. However, this is not what I am looking for due to the fact that it is not generalizable for more than 3->2 dimension and if there is a slight amount of noise, the method falls apart. One approach I have tried is to find the SVD decomposition and decide by the eigenvalues. I am certain this problem has been solved principally but I am not sure how, I appreciate any opinion.
linear-algebra matrices linear-transformations
linear-algebra matrices linear-transformations
asked Jan 3 at 19:09
ScicareScicare
1134
asked Jan 3 at 19:09
ScicareScicare
1134
edited Jan 3 at 19:31
Scicare
asked Jan 3 at 19:09
ScicareScicare
1134
asked Jan 3 at 19:09
ScicareScicare
1134
asked Jan 3 at 19:09
ScicareScicare
1134
1134
2
$begingroup$
Pick 3 points, find the plane between them, then check whether the rest satisfy the plane equation.
$endgroup$
– Don Thousand
Jan 3 at 19:12
$begingroup$
If you can make a matrix 3 by 3 with the coords of 3 of these points, with rank 3, then they lay not in the same plane.
$endgroup$
– dmtri
Jan 3 at 19:25
$begingroup$
Hi @DonThousand and dmtri , I have just edited and add additional info. Thanks for your opinion and I'd appreciate any additional insight.
$endgroup$
– Scicare
Jan 3 at 19:33
2
$begingroup$
Are you sure you're not looking to do Principal Component Analysis instead? Applied to your example data, it would yield an orthonormal transform (rotation) around the coordinate mean, with the figure being on the $xy$ plane (with all $z$ coordinates being almost the same).
$endgroup$
– Nominal Animal
Jan 3 at 21:41
add a comment |
2
$begingroup$
Pick 3 points, find the plane between them, then check whether the rest satisfy the plane equation.
$endgroup$
– Don Thousand
Jan 3 at 19:12
$begingroup$
If you can make a matrix 3 by 3 with the coords of 3 of these points, with rank 3, then they lay not in the same plane.
$endgroup$
– dmtri
Jan 3 at 19:25
$begingroup$
Hi @DonThousand and dmtri , I have just edited and add additional info. Thanks for your opinion and I'd appreciate any additional insight.
$endgroup$
– Scicare
Jan 3 at 19:33
2
$begingroup$
Are you sure you're not looking to do Principal Component Analysis instead? Applied to your example data, it would yield an orthonormal transform (rotation) around the coordinate mean, with the figure being on the $xy$ plane (with all $z$ coordinates being almost the same).
$endgroup$
– Nominal Animal
Jan 3 at 21:41
2
2
$begingroup$
Pick 3 points, find the plane between them, then check whether the rest satisfy the plane equation.
$endgroup$
– Don Thousand
Jan 3 at 19:12
$begingroup$
Pick 3 points, find the plane between them, then check whether the rest satisfy the plane equation.
$endgroup$
– Don Thousand
Jan 3 at 19:12
$begingroup$
If you can make a matrix 3 by 3 with the coords of 3 of these points, with rank 3, then they lay not in the same plane.
$endgroup$
– dmtri
Jan 3 at 19:25
$begingroup$
If you can make a matrix 3 by 3 with the coords of 3 of these points, with rank 3, then they lay not in the same plane.
$endgroup$
– dmtri
Jan 3 at 19:25
$begingroup$
Hi @DonThousand and dmtri , I have just edited and add additional info. Thanks for your opinion and I'd appreciate any additional insight.
$endgroup$
– Scicare
Jan 3 at 19:33
$begingroup$
Hi @DonThousand and dmtri , I have just edited and add additional info. Thanks for your opinion and I'd appreciate any additional insight.
$endgroup$
– Scicare
Jan 3 at 19:33
2
2
$begingroup$
Are you sure you're not looking to do Principal Component Analysis instead? Applied to your example data, it would yield an orthonormal transform (rotation) around the coordinate mean, with the figure being on the $xy$ plane (with all $z$ coordinates being almost the same).
$endgroup$
– Nominal Animal
Jan 3 at 21:41
$begingroup$
Are you sure you're not looking to do Principal Component Analysis instead? Applied to your example data, it would yield an orthonormal transform (rotation) around the coordinate mean, with the figure being on the $xy$ plane (with all $z$ coordinates being almost the same).
$endgroup$
– Nominal Animal
Jan 3 at 21:41
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Total Least Squares technique is what you need.
If the points are almost layered into one plane (in any dimension), that method will reveal it
and will give you the measure of the noise.
But if the points layer into multiple planes, then you would need more sophisticated methods
ranging through SVD , low-rank approximation or other methods.
In general, the problem falls into the category of Point Cloud analysis.
answered Jan 3 at 19:55
G CabG Cab
18.4k31237
$endgroup$
$begingroup$
So SVD was one of the right track. Thanks :D
$endgroup$
– Scicare
Jan 5 at 3:26
$begingroup$
@Scicare Yes, indeed there is a strict connection between SVD and Least Squares.
$endgroup$
– G Cab
Jan 5 at 14:37
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Total Least Squares technique is what you need.
If the points are almost layered into one plane (in any dimension), that method will reveal it
and will give you the measure of the noise.
But if the points layer into multiple planes, then you would need more sophisticated methods
ranging through SVD , low-rank approximation or other methods.
In general, the problem falls into the category of Point Cloud analysis.
answered Jan 3 at 19:55
G CabG Cab
18.4k31237
$endgroup$
$begingroup$
So SVD was one of the right track. Thanks :D
$endgroup$
– Scicare
Jan 5 at 3:26
$begingroup$
@Scicare Yes, indeed there is a strict connection between SVD and Least Squares.
$endgroup$
– G Cab
Jan 5 at 14:37
add a comment |
$begingroup$
Total Least Squares technique is what you need.
If the points are almost layered into one plane (in any dimension), that method will reveal it
and will give you the measure of the noise.
But if the points layer into multiple planes, then you would need more sophisticated methods
ranging through SVD , low-rank approximation or other methods.
In general, the problem falls into the category of Point Cloud analysis.
answered Jan 3 at 19:55
G CabG Cab
18.4k31237
$endgroup$
$begingroup$
So SVD was one of the right track. Thanks :D
$endgroup$
– Scicare
Jan 5 at 3:26
$begingroup$
@Scicare Yes, indeed there is a strict connection between SVD and Least Squares.
$endgroup$
– G Cab
Jan 5 at 14:37
add a comment |
$begingroup$
Total Least Squares technique is what you need.
If the points are almost layered into one plane (in any dimension), that method will reveal it
and will give you the measure of the noise.
But if the points layer into multiple planes, then you would need more sophisticated methods
ranging through SVD , low-rank approximation or other methods.
In general, the problem falls into the category of Point Cloud analysis.
answered Jan 3 at 19:55
G CabG Cab
18.4k31237
$endgroup$
Total Least Squares technique is what you need.
If the points are almost layered into one plane (in any dimension), that method will reveal it
and will give you the measure of the noise.
But if the points layer into multiple planes, then you would need more sophisticated methods
ranging through SVD , low-rank approximation or other methods.
In general, the problem falls into the category of Point Cloud analysis.
answered Jan 3 at 19:55
G CabG Cab
18.4k31237
edited Jan 3 at 20:06
answered Jan 3 at 19:55
G CabG Cab
18.4k31237
answered Jan 3 at 19:55
G CabG Cab
18.4k31237
answered Jan 3 at 19:55
G CabG Cab
18.4k31237
18.4k31237
$begingroup$
So SVD was one of the right track. Thanks :D
$endgroup$
– Scicare
Jan 5 at 3:26
$begingroup$
@Scicare Yes, indeed there is a strict connection between SVD and Least Squares.
$endgroup$
– G Cab
Jan 5 at 14:37
add a comment |
$begingroup$
So SVD was one of the right track. Thanks :D
$endgroup$
– Scicare
Jan 5 at 3:26
$begingroup$
@Scicare Yes, indeed there is a strict connection between SVD and Least Squares.
$endgroup$
– G Cab
Jan 5 at 14:37
$begingroup$
So SVD was one of the right track. Thanks :D
$endgroup$
– Scicare
Jan 5 at 3:26
$begingroup$
So SVD was one of the right track. Thanks :D
$endgroup$
– Scicare
Jan 5 at 3:26
$begingroup$
@Scicare Yes, indeed there is a strict connection between SVD and Least Squares.
$endgroup$
– G Cab
Jan 5 at 14:37
$begingroup$
@Scicare Yes, indeed there is a strict connection between SVD and Least Squares.
$endgroup$
– G Cab
Jan 5 at 14:37
add a comment |
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2
$begingroup$
Pick 3 points, find the plane between them, then check whether the rest satisfy the plane equation.
$endgroup$
– Don Thousand
Jan 3 at 19:12
$begingroup$
If you can make a matrix 3 by 3 with the coords of 3 of these points, with rank 3, then they lay not in the same plane.
$endgroup$
– dmtri
Jan 3 at 19:25
$begingroup$
Hi @DonThousand and dmtri , I have just edited and add additional info. Thanks for your opinion and I'd appreciate any additional insight.
$endgroup$
– Scicare
Jan 3 at 19:33
2
$begingroup$
Are you sure you're not looking to do Principal Component Analysis instead? Applied to your example data, it would yield an orthonormal transform (rotation) around the coordinate mean, with the figure being on the $xy$ plane (with all $z$ coordinates being almost the same).
$endgroup$
– Nominal Animal
Jan 3 at 21:41