Useful Definitions

Anisotropic Being directionally dependent. An anisotropic surface changes appearance as rotated around a geometric normal (ie: brushed metal). Anisotropic scaling uses different amounts of scale along different directions

Covariance How much variables vary together, when X varies strongly in the positive and so does Y, then the covariance is high, when they vary strongly in different directions the covariance is low.

Diffraction Blending or spreading of waves when they meet a barrier.

Fresnel Effect The amount of reflection that you see from a surface depends on the viewing angle (ie: pool of water—easy to see through from the top, very reflective and hard to see through from a glancing angle).

Interference When two waves intersect to create a new wave (constructive interference the two add to each other, destructive they cancel each other out—happens in Thin Films a lot, this is why you see colors on soap bubbles).

Level Sets

Sigmoid A function which is useful in classification, it can give probabilities of a binary classification. y = 1/(1 + e-x)

Tensor Mathematical entity, generalization of scalar, vector, matrix, etc. Arbitrary dimensional structure with n entries in each dimension.

Watershed Method

Wavelet A multi-resolution method for representing information. A course definition of the function to be represented along with the detail coefficients needed to reconstruct the original function.

Matrices

Eigen values of a matrix are the solutions to Ax = λx. It represents a direction in the n-dimensional space along which the matrix A only has an effect of scalar multiplication. These eigen values form a set λ(A) known as the spectrum of A. The spectrum has a spectrual radius equal to max{|λ| : λ ∈ λ(A)} (maximus modulus of the eigen values).

Eigen vectors which correspond to the eigen values of the matrix are the directions along which the matrix A has only scalar multiplication effects.

Hessian is the matrix of second derivatives of a function with respect to n-dimensional components, or the jacobian matrix of the gradient (∇ƒ). If the partial derivates of ƒ are continuous then ∂2ƒ/∂xi∂xj

Invertible matrices have determinant not equal to 0.

Jacobian is the matrix of first derivatives of a function with respect to the n-dimensional components = ∂ƒi(x)/∂xj

Positive Definite: xTAx > 0 for all x ≠o. True if all of the eigen values are positive, this matrix is always nonsingular (det(A) = 0). The definition of positive definiteness is equivalent to the requirement that the determinants associated with all upper-left submatrices are positive. These matrices can be solved efficiently with Cholesky decomposition.

Rank of a matrix the dimension of the image (image = range) or the number of linearly indepent rows or columbus of the matrix.

Singular matrices are square matrices with determinant &ne 0. Therefore they also do not have have inverses.