video Mathematical Notation

http://ipa.iwr.uni-heidelberg.de/ipabib/Papers/Lellmann-et-al-09b.pdf

http://lmb.informatik.uni-freiburg.de/Publications/2012/Keu12/Dissertation_Keuper.pdf


.6. Mathematical Notation N the space of natural numbers R the space of real numbers C the space of complex numbers Ω ⊆ R d image domain in d-dimensional space Ω1 t Ω2 union over open regions Ω1 and Ω2 Ω1 ∪ Ω2 union over closed regions Ω1 and Ω2 x d-tuple of real coordinates Γ : [0, 1] → Ω ⊆ R 2 a contour in R 2 |Γ| the length of contour Γ κ(s) the curvature of a contour Γ at position s ∈ [0, 1] Γ : [0, 1] × [0, 1] → Ω ⊆ R 3 a contour in R 3 v T transposed vector v ∇I = ∂I ∂x1 , . . . , ∂I ∂xd T the gradient operator on I : R d → R div(I) = ∂I ∂x1 + · · · + ∂I ∂xd the divergence operator on I : R d → R ∆I = ∂ 2 I ∂x2 1 + · · · + ∂ 2 I ∂x2 d the Laplace operator on I : R d → R G = (V, E) undirected graphical model V = {v1, . . . , v|V|} set of nodes |V| cardinality of the set of nodes N neighborhood system X probability space ` ∈ X label configuration `i label of vertex vi C set of all cliques in graph G Ψ(`) potential functions Φ : R d → R d-dimensional embedding function · , · inner product Gσ a Gaussian normal distribution with standard deviation σ ∗ the convolution operator p(·) probability density function H(·) Heavyside function δ the Dirac impulse TV(·) the TV norm k · k the L2 norm of a vector | · | the absolute value ΠS(·) projection onto the set S o : Ω → R the objective function s : Ω → R the specimen function h : Ω → R the point spread function n : R → R a voxel-wise noise function nG zero mean Gaussian noise function 9 1. Introduction ˆf estimate of function f f m(x) = f(−x) the mirrored function f F(·) the Fourier transform F −1 (·) the inverse Fourier transform f 0 derivative of a differentiable function f x ∗ the complex conjugate of x ∈ C J0 zero order Bessel function of the first kind ı the imaginary unit P m l associated Legendre polynomial of order m and degree l Y m l spherical harmonic basis function of order m and degree l