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Fundamental Matrix (other)
Fundamental matrix may refer to: * Fundamental matrix (computer vision) * Fundamental matrix (linear differential equation) In mathematics, a fundamental matrix of a system of ''n'' homogeneous linear ordinary differential equations \dot(t) = A(t) \mathbf(t) is a matrix-valued function \Psi(t) whose columns are linearly independent solutions of the system. Then ev ... * Fundamental matrix (absorbing Markov chain) {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Matrix (computer Vision)
In computer vision, the fundamental matrix \mathbf is a 3×3 matrix which relates corresponding points in stereo images. In epipolar geometry, with homogeneous image coordinates, x and x′, of corresponding points in a stereo image pair, Fx describes a line (an epipolar line) on which the corresponding point x′ on the other image must lie. That means, for all pairs of corresponding points holds : \mathbf'^ \mathbf = 0. Being of rank two and determined only up to scale, the fundamental matrix can be estimated given at least seven point correspondences. Its seven parameters represent the only geometric information about cameras that can be obtained through point correspondences alone. The term "fundamental matrix" was coined by QT Luong in his influential PhD thesis. It is sometimes also referred to as the "bifocal tensor". As a tensor it is a two-point tensor in that it is a bilinear form relating points in distinct coordinate systems. The above relation which defines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Matrix (linear Differential Equation)
In mathematics, a fundamental matrix of a system of ''n'' homogeneous linear ordinary differential equations \dot(t) = A(t) \mathbf(t) is a matrix-valued function \Psi(t) whose columns are linearly independent solutions of the system. Then every solution to the system can be written as \mathbf(t) = \Psi(t) \mathbf, for some constant vector \mathbf (written as a column vector of height ). One can show that a matrix-valued function \Psi is a fundamental matrix of \dot(t) = A(t) \mathbf(t) if and only if \dot(t) = A(t) \Psi(t) and \Psi is a non-singular matrix for all Control theory The fundamental matrix is used to express the state-transition matrix, an essential component in the solution of a system of linear ordinary differential equations. See also *Linear differential equation *Liouville's formula In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |