Since = . . n . t e In this notebook, we use Kronecker products to construct a 2d finite-difference approximation of the Laplacian operator \(-\nabla^2\) with Dirichlet (zero) boundary conditions, via the standard 5-point stencil (centered differences in \(x\) and \(y\)).. Notice the ordinary Laplacian is a generalized Laplacian. ) ) : is in general not hermitian, it has real eigenvalues. ≤ In other words, at steady state, the value of In other words, the discrete Laplacian filter of any size can be generated conveniently as the sampled Laplacian of Gaussian with spatial size befitting the needs of a particular application as controlled by its variance. ( be the (sparse) cotangent matrix with entries. (which is hermitian). ) V . 下面是维基百科上的描述： Jump to search In the mathematical field of graph theory, the Laplacian matrix, sometimes called admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph.The Laplacian matrix can be used to find many useful properties of a graph. For one-, two- and three-dimensional signals, the discrete Laplacian can be given as convolution with the following kernels: D ¯ and with ) ( − M On regular lattices, the operator typically has both traveling-wave as well as Anderson localization solutions, depending on whether the potential is periodic or random. ϕ the Laplacian matrix is defined as \[L := D - A\] This definition is super simple, but it describes something quite deep: it’s the discrete analog to the Laplacian operator on multivariate continuous functions. whose are given by. {\displaystyle C} lim A {\displaystyle {\bar {r}}\in R^{n}} c M (19.12). + ( oriented incidence matrix M with element Mev for edge e (connecting vertex i and j, with i > j) and vertex v given by. V g Together with Kirchhoff's theorem, it can be used to calculate the number of spanning trees for a given graph. N δ ) sym w 234–254. L is a positive semi-definite matrix and so all its eigenvalues are no negative. ( f 2 denotes the neighborhood of λ For the convention Thus, for example, on a one-dimensional grid we have. n {\displaystyle K} k , where the coordinate vector i i {\displaystyle P\colon V\rightarrow R} L .). v i ( L So the projection onto the eigenvectors of λ Where Δ Use delsq to generate the discrete Laplacian. Second Eigenvalue: The second smallest eigenvalue. 2 j {\textstyle g} Then, the discrete Laplacian acting on is defined by. . i j (See Discrete Poisson equation)[8] In this interpretation, every graph vertex is treated as a grid point; the local connectivity of the vertex determines the finite difference approximation stencil at this grid point, the grid size is always one for every edge, and there are no constraints on any grid points, which corresponds to the case of the homogeneous Neumann boundary condition, i.e., free boundary. {\textstyle i} {\displaystyle f_{k}\in R} is represented via j It shows the process of specifying initial conditions, projecting these initial conditions onto the eigenvalues of the Laplacian Matrix, and simulating the exponential decay of these projected initial conditions. The matrix elements of i for a graph with In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. {\textstyle L^{\text{sym}}} | ϕ of For the discrete equivalent of the Laplace transform, see Z-transform. of L are non-negative, showing that the solution to the diffusion equation approaches an equilibrium, because it only exponentially decays or remains constant. rw vertices, For each element For functions of three or more variables, U(x,y,z,...), the discrete Laplacian del2(U) calculates second-derivatives in each dimension, L = Δ U 2 N = 1 2 N ( ∂ 2 U ∂ x … 1 ¯ ϕ ∈ {\displaystyle n} is a probability distribution of the location of a random walker on the vertices of the graph, then An advantage of using Gaussians as interpolation functions is that they yield linear operators, including Laplacians, that are free from rotational artifacts of the coordinate frame in which = 0 ; that is, one third of the summed areas of triangles incident to w ⟨ Discrete Laplacian: A matrix L such that L i;j= 1 if i6= jand there is an edge between vertices iand jand = 0 if there is no edge. . Here are some examples: The heat equation @u @t = udescribes the distribution of heat in a given region over time. . V ⁡ v = A 0 ⟩ ; i.e., is the probability distribution of the walker after {\displaystyle n} In the mathematical field of graph theory, the Laplacian matrix, sometimes called admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph.The Laplacian matrix can be used to find many useful properties of a graph. [3], Approximations of the Laplacian, obtained by the finite-difference method or by the finite-element method, can also be called discrete Laplacians. to node {\displaystyle \gamma \colon E\to R} T {\displaystyle Lu=(\Delta u)_{i}} − c on -dimensions, and are frequency aware by definition. {\textstyle \nabla ^{2}} {\textstyle M^{\textsf {T}}} Such an interpretation allows one, e.g., to generalise the Laplacian matrix to the case of graphs with an infinite number of vertices and edges, leading to a Laplacian matrix of an infinite size. ( {\textstyle \lambda _{i}} {\displaystyle f_{k}} corresponds to the (Five-point stencil) finite-difference formula seen previously. -dimensions i.e. i deg | 2 ⟩ Lindeberg, T., "Scale-space for discrete signals", PAMI(12), No. {\textstyle \phi (t)=\sum _{i}c_{i}(t)\mathbf {v} _{i}.}. K {\displaystyle \phi } V The discrete Laplacian is an approximation to the continuous Laplacian that is appropriate when data is known or sampled only at finitely many points. {\textstyle L^{\text{sym}}} Both matrices have been extremely well studied from an algebraic point of view. 1 j , e diffusing over time through a graph. are all non-negative. ϕ ( = {\textstyle \lambda _{0}\leq \lambda _{1}\leq \cdots \leq \lambda _{n-1}} cot λ We expect that neighboring elements in the graph will exchange energy until that energy is spread out evenly throughout all of the elements that are connected to each other. V S There are no constraints here on the values of the function f(x, y) on the boundary of the lattice grid, thus this is the case of no source at the boundary, that is, a no-flux boundary condition (aka, insulation, or homogeneous Neumann boundary condition). {\displaystyle i} r {\textstyle A_{\text{reduced}}\equiv D^{-{\frac {1}{2}}}AD^{-{\frac {1}{2}}}} . be a function of the vertices taking values in a ring. ( are orthogonal): As shown before, the eigenvalues ( ϕ are given by, The generalized Laplacian ¯ P