Download Solving a Homogeneous Linear Equation System

Solving a Homogeneous Linear Equation System
A standard problem in computer vision (and in engineering in general) is
to solve a set of homogeneous linear equations. A homogeneous linear equation
system is given by the expression
Ax = 0 ,
(1)
where x is the vector of N unknowns, and A is the matrix of (M ×N ) coefficients.
We will assume here that there are enough equations to solve for the unknowns,
i.e. M ≥ N .
The equations are linear in all unknowns, and the right side is 0. Hence,
every linear equation system allows the trivial solution x = 0. This is normally
not the desired solution. The standard way to avoid it is to constrain the vector
x to a fixed length, for example
|x|2 = 1 .
(2)
The constraint is not P
linear, but quadratic in the elements of x, because the
2
vector norm is |x| = x2i .
The numerically best way to solve the equations (1) subject to the constraint
(2) is to perform singular value decomposition on the matrix A. Singular Value
Decomposition (SVD) factors the matrix into a diagonal matrix D and two
orthogonal matrices U, V, such that
A = UDVT
(3)
The diagonal entries of D are related to eigenvalues of AT A. If the number of
linearly independent equations in (1) is the same as the number of unknowns,
M = N , then D will have exactly one diagonal entry di = 0. The matrices
are often reordered such that the diagonal entries of D are in descending order
(“ordered SVD”) – then this is the last (rightmost) entry. The exact solution
of the equation system is given by the column of V, which corresponds to the
diagonal entry di = 0 (in ordered SVD, this is the last column of V).
If M > N , i.e. the equation system is over-constrained, then there is no exact
solution, and no diagonal entry di = 0. Usually one is interested in the solution,
which minimizes the sum of squared errors over all variables (under Gaussian
white noise, this is the most likely solution). The least-squares solution is given
by the column of V, which corresponds to the smallest diagonal entry of D – in
ordered SVD again the last column.
In MATLAB, ordered singular value decomposition comes as a pre-canned
routine. The equation system is solved using the commands
[U,D,V] = svd(A,0);
x = V(:,end);