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Structure Tensor
In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant respect the observing coordinates. The structure tensor is often used in image processing and computer vision. J. Bigun and G. Granlund (1986), ''Optimal Orientation Detection of Linear Symmetry''. Tech. Report LiTH-ISY-I-0828, Computer Vision Laboratory, Linkoping University, Sweden 1986; Thesis Report, Linkoping studies in science and technology No. 85, 1986. The 2D structure tensor Continuous version For a function I of two variables , the structure tensor is the 2×2 matrix : S_w(p) = \begin \int w(r) (I_x(p-r))^2\,d r & \int w(r) I_x(p-r)I_y(p-r)\,d r \\0pt\int w(r) I_x(p-r)I_y(p-r)\,d r & \int w(r) (I_y(p-r))^2\,d r \end where I_x and I_y are the partial derivatives of I with respect to ''x'' and ''y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity ( stress–energy tensor, cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
