Search for projection on a special matrix space with regard to Frobenius norm(computer vision background)
$begingroup$
Background
Define essential space as
$$varepsilon={E in mathbb R^{3times3}|E=hat{T}R}$$
$$hat{T}in{Sin mathbb R^{3times3}|S^T=-S}$$
$$Rin{Ainmathbb R^{3times3}|A^TA=I,det(A)=1}$$
that is any matrix $E in varepsilon$ is skew-symmetric matrix $hat T$ post-multiplied by a rotation matrix R.
According to this paper, $E$ resides in $varepsilon$ if and only if
$$E=USigma V^T,Sigma = diag(sigma,sigma,0).$$
Optimization Problem
For a matrix $F in mathbb R^{3times 3}$, searching it's projection on the essential space can be considered as solving following optimization problem:
$$argmin_{E} |F-E|_F^2 $$
$$s.t. qquad E in varepsilon.$$
Suppose $F=Udiag(lambda_1,lambda_2,lambda_3) V^T$, then solution to above problem is
$$E = Udiag(sigma,sigma,0) V^T, sigma = frac{lambda_1+lambda_2}{2}$$
Question
Let $E=U_1diag(sigma,sigma,0)V_1^T$ be any matrix in $varepsilon$. I have derived above solution if $U_1=U,V_1=V$, but stuck on cases in which $U_1ne U$ or $V_1 ne V$.
matrices rotations numerical-optimization svd
asked Dec 26 '18 at 8:19
FinleyFinley
400213
$endgroup$
|
show 1 more comment
$begingroup$
Background
Define essential space as
$$varepsilon={E in mathbb R^{3times3}|E=hat{T}R}$$
$$hat{T}in{Sin mathbb R^{3times3}|S^T=-S}$$
$$Rin{Ainmathbb R^{3times3}|A^TA=I,det(A)=1}$$
that is any matrix $E in varepsilon$ is skew-symmetric matrix $hat T$ post-multiplied by a rotation matrix R.
According to this paper, $E$ resides in $varepsilon$ if and only if
$$E=USigma V^T,Sigma = diag(sigma,sigma,0).$$
Optimization Problem
For a matrix $F in mathbb R^{3times 3}$, searching it's projection on the essential space can be considered as solving following optimization problem:
$$argmin_{E} |F-E|_F^2 $$
$$s.t. qquad E in varepsilon.$$
Suppose $F=Udiag(lambda_1,lambda_2,lambda_3) V^T$, then solution to above problem is
$$E = Udiag(sigma,sigma,0) V^T, sigma = frac{lambda_1+lambda_2}{2}$$
Question
Let $E=U_1diag(sigma,sigma,0)V_1^T$ be any matrix in $varepsilon$. I have derived above solution if $U_1=U,V_1=V$, but stuck on cases in which $U_1ne U$ or $V_1 ne V$.
matrices rotations numerical-optimization svd
asked Dec 26 '18 at 8:19
FinleyFinley
400213
$endgroup$
$begingroup$
In what sense can $U_1neq U$? haven't you just defined them as equal?
$endgroup$
– user617446
Dec 26 '18 at 13:41
$begingroup$
@user617446 Not equal in definition. To my understanding, the only constraint is $E in varepsilon $ (i.e. $E=Udiag(sigma,sigma,0)V^T$) which does not necessarily make $F=Udiag(lambda_1,lambda_2,lambda_3)V^T$
$endgroup$
– Finley
Dec 26 '18 at 13:57
$begingroup$
Are you asking why $E$ is the optimal solution given $F$? Your definition of $E$ is based on $F$ and hence on $U,V$, so $E$ must be $E = Udiag(sigma,sigma,0) V^T, sigma = frac{lambda_1+lambda_2}{2}$
$endgroup$
– user617446
Dec 26 '18 at 14:35
$begingroup$
@user617446 I didn't mention definition of $E$ is based on $F$ in this post, the only constraint is $E in varepsilon$. I derived the solution when I enforcing additional constraints that $U=U_1,V=V_1$.
$endgroup$
– Finley
Dec 27 '18 at 3:20
$begingroup$
@user617446 I wonder how to arrive without above additional constraints.
$endgroup$
– Finley
Dec 27 '18 at 3:26
|
show 1 more comment
$begingroup$
Background
Define essential space as
$$varepsilon={E in mathbb R^{3times3}|E=hat{T}R}$$
$$hat{T}in{Sin mathbb R^{3times3}|S^T=-S}$$
$$Rin{Ainmathbb R^{3times3}|A^TA=I,det(A)=1}$$
that is any matrix $E in varepsilon$ is skew-symmetric matrix $hat T$ post-multiplied by a rotation matrix R.
According to this paper, $E$ resides in $varepsilon$ if and only if
$$E=USigma V^T,Sigma = diag(sigma,sigma,0).$$
Optimization Problem
For a matrix $F in mathbb R^{3times 3}$, searching it's projection on the essential space can be considered as solving following optimization problem:
$$argmin_{E} |F-E|_F^2 $$
$$s.t. qquad E in varepsilon.$$
Suppose $F=Udiag(lambda_1,lambda_2,lambda_3) V^T$, then solution to above problem is
$$E = Udiag(sigma,sigma,0) V^T, sigma = frac{lambda_1+lambda_2}{2}$$
Question
Let $E=U_1diag(sigma,sigma,0)V_1^T$ be any matrix in $varepsilon$. I have derived above solution if $U_1=U,V_1=V$, but stuck on cases in which $U_1ne U$ or $V_1 ne V$.
matrices rotations numerical-optimization svd
asked Dec 26 '18 at 8:19
FinleyFinley
400213
$endgroup$
Background
Define essential space as
$$varepsilon={E in mathbb R^{3times3}|E=hat{T}R}$$
$$hat{T}in{Sin mathbb R^{3times3}|S^T=-S}$$
$$Rin{Ainmathbb R^{3times3}|A^TA=I,det(A)=1}$$
that is any matrix $E in varepsilon$ is skew-symmetric matrix $hat T$ post-multiplied by a rotation matrix R.
According to this paper, $E$ resides in $varepsilon$ if and only if
$$E=USigma V^T,Sigma = diag(sigma,sigma,0).$$
Optimization Problem
For a matrix $F in mathbb R^{3times 3}$, searching it's projection on the essential space can be considered as solving following optimization problem:
$$argmin_{E} |F-E|_F^2 $$
$$s.t. qquad E in varepsilon.$$
Suppose $F=Udiag(lambda_1,lambda_2,lambda_3) V^T$, then solution to above problem is
$$E = Udiag(sigma,sigma,0) V^T, sigma = frac{lambda_1+lambda_2}{2}$$
Question
Let $E=U_1diag(sigma,sigma,0)V_1^T$ be any matrix in $varepsilon$. I have derived above solution if $U_1=U,V_1=V$, but stuck on cases in which $U_1ne U$ or $V_1 ne V$.
matrices rotations numerical-optimization svd
matrices rotations numerical-optimization svd
asked Dec 26 '18 at 8:19
FinleyFinley
400213
asked Dec 26 '18 at 8:19
FinleyFinley
400213
edited Dec 27 '18 at 3:28
Finley
asked Dec 26 '18 at 8:19
FinleyFinley
400213
asked Dec 26 '18 at 8:19
FinleyFinley
400213
asked Dec 26 '18 at 8:19
FinleyFinley
400213
400213
$begingroup$
In what sense can $U_1neq U$? haven't you just defined them as equal?
$endgroup$
– user617446
Dec 26 '18 at 13:41
$begingroup$
@user617446 Not equal in definition. To my understanding, the only constraint is $E in varepsilon $ (i.e. $E=Udiag(sigma,sigma,0)V^T$) which does not necessarily make $F=Udiag(lambda_1,lambda_2,lambda_3)V^T$
$endgroup$
– Finley
Dec 26 '18 at 13:57
$begingroup$
Are you asking why $E$ is the optimal solution given $F$? Your definition of $E$ is based on $F$ and hence on $U,V$, so $E$ must be $E = Udiag(sigma,sigma,0) V^T, sigma = frac{lambda_1+lambda_2}{2}$
$endgroup$
– user617446
Dec 26 '18 at 14:35
$begingroup$
@user617446 I didn't mention definition of $E$ is based on $F$ in this post, the only constraint is $E in varepsilon$. I derived the solution when I enforcing additional constraints that $U=U_1,V=V_1$.
$endgroup$
– Finley
Dec 27 '18 at 3:20
$begingroup$
@user617446 I wonder how to arrive without above additional constraints.
$endgroup$
– Finley
Dec 27 '18 at 3:26
|
show 1 more comment
$begingroup$
In what sense can $U_1neq U$? haven't you just defined them as equal?
$endgroup$
– user617446
Dec 26 '18 at 13:41
$begingroup$
@user617446 Not equal in definition. To my understanding, the only constraint is $E in varepsilon $ (i.e. $E=Udiag(sigma,sigma,0)V^T$) which does not necessarily make $F=Udiag(lambda_1,lambda_2,lambda_3)V^T$
$endgroup$
– Finley
Dec 26 '18 at 13:57
$begingroup$
Are you asking why $E$ is the optimal solution given $F$? Your definition of $E$ is based on $F$ and hence on $U,V$, so $E$ must be $E = Udiag(sigma,sigma,0) V^T, sigma = frac{lambda_1+lambda_2}{2}$
$endgroup$
– user617446
Dec 26 '18 at 14:35
$begingroup$
@user617446 I didn't mention definition of $E$ is based on $F$ in this post, the only constraint is $E in varepsilon$. I derived the solution when I enforcing additional constraints that $U=U_1,V=V_1$.
$endgroup$
– Finley
Dec 27 '18 at 3:20
$begingroup$
@user617446 I wonder how to arrive without above additional constraints.
$endgroup$
– Finley
Dec 27 '18 at 3:26
$begingroup$
In what sense can $U_1neq U$? haven't you just defined them as equal?
$endgroup$
– user617446
Dec 26 '18 at 13:41
$begingroup$
In what sense can $U_1neq U$? haven't you just defined them as equal?
$endgroup$
– user617446
Dec 26 '18 at 13:41
$begingroup$
@user617446 Not equal in definition. To my understanding, the only constraint is $E in varepsilon $ (i.e. $E=Udiag(sigma,sigma,0)V^T$) which does not necessarily make $F=Udiag(lambda_1,lambda_2,lambda_3)V^T$
$endgroup$
– Finley
Dec 26 '18 at 13:57
$begingroup$
@user617446 Not equal in definition. To my understanding, the only constraint is $E in varepsilon $ (i.e. $E=Udiag(sigma,sigma,0)V^T$) which does not necessarily make $F=Udiag(lambda_1,lambda_2,lambda_3)V^T$
$endgroup$
– Finley
Dec 26 '18 at 13:57
$begingroup$
Are you asking why $E$ is the optimal solution given $F$? Your definition of $E$ is based on $F$ and hence on $U,V$, so $E$ must be $E = Udiag(sigma,sigma,0) V^T, sigma = frac{lambda_1+lambda_2}{2}$
$endgroup$
– user617446
Dec 26 '18 at 14:35
$begingroup$
Are you asking why $E$ is the optimal solution given $F$? Your definition of $E$ is based on $F$ and hence on $U,V$, so $E$ must be $E = Udiag(sigma,sigma,0) V^T, sigma = frac{lambda_1+lambda_2}{2}$
$endgroup$
– user617446
Dec 26 '18 at 14:35
$begingroup$
@user617446 I didn't mention definition of $E$ is based on $F$ in this post, the only constraint is $E in varepsilon$. I derived the solution when I enforcing additional constraints that $U=U_1,V=V_1$.
$endgroup$
– Finley
Dec 27 '18 at 3:20
$begingroup$
@user617446 I didn't mention definition of $E$ is based on $F$ in this post, the only constraint is $E in varepsilon$. I derived the solution when I enforcing additional constraints that $U=U_1,V=V_1$.
$endgroup$
– Finley
Dec 27 '18 at 3:20
$begingroup$
@user617446 I wonder how to arrive without above additional constraints.
$endgroup$
– Finley
Dec 27 '18 at 3:26
$begingroup$
@user617446 I wonder how to arrive without above additional constraints.
$endgroup$
– Finley
Dec 27 '18 at 3:26
|
show 1 more comment
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$begingroup$
In what sense can $U_1neq U$? haven't you just defined them as equal?
$endgroup$
– user617446
Dec 26 '18 at 13:41
$begingroup$
@user617446 Not equal in definition. To my understanding, the only constraint is $E in varepsilon $ (i.e. $E=Udiag(sigma,sigma,0)V^T$) which does not necessarily make $F=Udiag(lambda_1,lambda_2,lambda_3)V^T$
$endgroup$
– Finley
Dec 26 '18 at 13:57
$begingroup$
Are you asking why $E$ is the optimal solution given $F$? Your definition of $E$ is based on $F$ and hence on $U,V$, so $E$ must be $E = Udiag(sigma,sigma,0) V^T, sigma = frac{lambda_1+lambda_2}{2}$
$endgroup$
– user617446
Dec 26 '18 at 14:35
$begingroup$
@user617446 I didn't mention definition of $E$ is based on $F$ in this post, the only constraint is $E in varepsilon$. I derived the solution when I enforcing additional constraints that $U=U_1,V=V_1$.
$endgroup$
– Finley
Dec 27 '18 at 3:20
$begingroup$
@user617446 I wonder how to arrive without above additional constraints.
$endgroup$
– Finley
Dec 27 '18 at 3:26