The infinitesimal generator (and hence characteristic operator) of a Brownian motion on R is easily calculated to be ⁠ 1 / 2 ⁠ Δ, where Δ denotes the Laplace operator. In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob and edge detection. This observation is useful in defining Brownian motion on an m -dimensional Riemannian manifold (M, g) : a Brownian motion on M is defined to be a diffusion on M whose characteristic operator A in local coordinates x i, 1 ≤ i ≤ m, is given by ⁠ 1 / 2 ⁠ Δ LB, where Δ LB is the Laplace–Beltrami operator given in local coordinates by Δ Δ L B = 1 det ( g ) ∑ ∑ i = 1 m ∂ ∂ ∂ ∂ x i ( det ( g ) ∑ ∑ j = 1 m g i j ∂ ∂ ∂ ∂ x j ) , where [ g ] = [ g ij ] in the sense of the inverse of a square matrix.