This post reviews some most commonly used matrices in computer vision, mainly refers to the Appendix B of Peter Corke’s book[1]. In addition, I would like to recommend No bullshit guide to linear algebra [2] as a quick linear algebra refresher.
Important concepts for matrices
Singular
The determinant of a square matrix A is denoted \(det(A)\). The matrix determinant is undefined for a non-square matrix [3].
A square matrix A is singular if \(det(A) = 0\).
A singular matrix is non-invertible. A non-singular matrix is invertible.
Symmetric / skew-symmetric
If matrix A is symmetric, then \(A^T = A\).
If matrix A is skew-symmetric, then \(A^T = -A\).
A symmetric matrix and skew-symmetric matrix will always be square.
In Peter Corke’s book it is said that “…the matrix is skew-symmetric or anti-symmetric. Such a matrix has a zero diagonal, is always singular…”, but this statement is imprecisely. The correct statement should be: if A is an \(n \times n\) skew-symmetric matrix, and n is odd, then A is singular.
Orthogonal
If matrix A is orthogonal, then \(A^{-1} = A^T\).
It can be seen that an orthogonal matrix must be non-singular (\(det(A) \neq 0\)) and invertible. Actually, The determinant of an orthogonal matrix is either +1 or −1.
The column vectors (and row vectors) of an orthogonal matrix must be of unit length and orthogonal to each other. This property is useful in focal length estimation [4].
Normal
A matrix A is normal if it satisfies the equation \(A^TA = AA^T\).
All symmetric, skew-symmetric and orthogonal matrices are normal matrices.
All normal matrices are diagonalizable.
Positive definite / semi-definite
Real symmetric matrices can be classified according to the sign of their eigenvalues \(\lambda_{i}\):
\[\lambda_{i} > 0, \forall i \quad positive \ definite\] \[\lambda_{i} \geq 0, \forall i \quad positive \ semi-definite\] \[\lambda_{i} < 0, \forall i \quad negative definite\] \[otherwise \quad indefinite\]The determinant is equal to the product of the eigenvalues, thus a positive definite matrix is non-singular.
Lemma (Least Squares and Postive Semidefiniteness) : Let \(A \in \mathbb{R}^{n \times m}\) be an \(n \times m\) matrix with \(n > m\). The matrix \(B = A^TA\) is positive semidefinite. If further the columns of \(A\) are linearly independent, then the matrix \(B\) is positive definite [5].
Theorem (Cholesky Decomposition) : Any positive definite matrix \(B \in \mathbb{R}^{n \times n}\) can be factorized uniquely as where \(L \in \mathbb{R}^{n \times n}\) is a lower triangular matrix with positive entries on the diagonal [5].
Applications in computer vision
Rotation matrix
- Orthogonal
Essential matrix
- The essential matrix is singular, has a rank of 2, and has two equal nonzero singular values and one of zero. The essential matrix has only 5 DoF and is completely defined by 3 rotational and 2 translational parameters [1]. The magnitude of translation cannot be recovered due to the scale ambiguity.
Fundamental matrix
- Singular, rank of 2, 7 DoF
Homography matrix
- Non-singular, rank of 3, 8 DoF
Covariance matrix / fisher information matrix
- Symmetric, positive semi-definite
- Under certain standard assumptions, the Fisher information matrix is the inverse of the covariance matrix [6].
References
[1] Corke, P. (2017). Robotics, Vision and Control : Fundamental Algorithms In MATLAB® Second, Completely Revised, Extended And Updated Edition. Cham: Springer International Publishing.
[2] Savov, I. (2017). No bullshit guide to linear algebra. Montréal, Québec: Minireference Co.
[3] Ii, R. (n.d.). A Brief Review of Matrices and Linear Algebra. [online] Available at: https://www.ohio.edu/mechanical-faculty/williams/html/PDF/MatricesLinearAlgebra.pdf [Accessed 30 Aug. 2022].
[4] Benosman, R. and Kang, S.B. (2001). Panoramic vision : sensors, theory, and applications. New York: Springer.
[5] Roch, S. (n.d.). TOPIC 1 Least squares: Cholesky, QR and Householder 3 Overdetermined systems, positive semidefinite matrices and Cholesky decomposition 3.1 Solving an overdetermined linear system. [online] Available at: https://people.math.wisc.edu/~roch/mmids/qr-3-overdetermined.pdf [Accessed 30 Aug. 2022].
[6] Wittman, D. (n.d.). Fisher Matrix for Beginners. [online] Available at: https://wittman.physics.ucdavis.edu/Fisher-matrix-guide.pdf [Accessed 30 Aug. 2022].