ABSTRACT
X random variable x realization of X , observation X random (column) vector x realization of X (vector observation) X data design matrix for X G, g,G, g,G as above, designating pixel gray-values x⊤ transposed form of the vector x (row vector) ‖x‖ length or (2-)norm of x x⊤y inner product of two vectors (scalar) x · y Hadamard (component-by-component) product xy⊤ outer product of two vectors (matrix) C matrix x⊤Cy quadratic form (scalar) |C| determinant of C tr(C) trace of C I identity matrix 0 column vector of zeroes 1 column vector of ones Λ Diag(λ1 . . . λN ) (diagonal matrix of eigenvalues) ∂f(x) ∂x partial derivative of f(x) with respect to vector x f(x)|x=x∗ f(x) evaluated at x = x∗ i
√−1 |z| absolute value of real or complex number z z∗ complex conjugate of complex number z Ω sample space in probability theory Pr(A | B) probability of event A conditional on event B P (x) distribution function for random variable X P (x) joint distribution function for random vector X p(x) probability density function for X p(x) joint probability density function for X 〈X〉, µ mean (or expected) value var(X), σ2 variance 〈X〉, µ mean vector Σ variance-covariance matrix S estimator for Σ (random matrix) s estimate of Σ (realization of S)
µˆ, Σˆ maximum likelihood estimates of µ, Σ K kernel matrix with elements k(x,x′) Φ(x) standard normal probability distribution function φ(x) standard normal probability density function pχ2;n(x) chi-square density function with n degrees of freedom pt;n(x) Student-t density function with n degrees of freedom pf ;m,n(x) F -density function with m and n degrees of freedom pW(x,m) Wishart distribution with m degrees of freedom pWC (x,m) complex Wishart distribution with m degrees of freedom (xˆ(0), xˆ(1) . . .) discrete Fourier transform of array (x(0), x(1) . . .) x ∗ y discrete convolution of vectors x and y 〈φ, ψ〉 inner product ∫ φ(x)ψ(x)dx of functions φ and ψ Ni neighborhood of ith pixel {x | c(x)} set of elements x that satisfy condition c(x) K set of class labels {1 . . .K} ℓ vector representation of a class label u ∈ U u is an element of the set U U ⊗ V Cartesian product set V ⊂ U V is a (proper) subset of the set U argmaxx f(x) the set of x which maximizes f(x) f : A 7→ B the function f which maps the set A to the set B IR set of real numbers Z set of integers =: equal by definition
end of a proof
