Laplace operator and Laplacian Matrix

1 LaplacePhysical Meaning of Operators

The Laplace operator is defined as the gradient divergence.

In the Cartesian coordinate system, it can also be expressed:

Or, it is the trace of the Hessian matrix:

Take the heat conduction equation as an example. Because the heat flow is proportional to the temperature gradient, the divergence of the temperature gradient is the loss rate of heat.

It can be seen that the Laplace operator can be used to express the material transport caused by uneven material distribution.

2 LaplaceMathematical significance of Operators

Now, in the one-dimensional space, simply analyze the above formula:

You can also write:

Expand the first and second items of the numerator by Taylor:

It can be seen that the Laplace operator is actually an operator that makes the function take the mean. Multi-dimensional space is similar.

3 LaplaceEquation

If the right side of the Laplace operator is zero, it is called the Laplace equation. The solution of the Laplace equation is called a harmonic function. If the right side is a function, it is called a Poisson equation.

 

4 LaplaceApplication of operators in image processing

Image processing is based on discretization of pixels as follows:

5 LaplacianMatrix

Is a matrix used to represent graphs. Its Dimension is | v |-by-| v | (| v | the number of nodes ). James demmel provides a method to convert incidence matrix to Laplacian matrix.

In (g) is a | v |-by-| E | matrix (| E | Number of edges), where E = (I, j ), this column is zero except row I (+ 1) and row J (-1. According to this definition, E = (I, j) and E = (J, I) are equivalent, it seems that many in graphs are generated (based on different edges ). However, it can be proved that the L graph generated by the in graph is unique no matter how the edge is obtained. That is to say, E = (I, j) and E = (J, I) are irrelevant. How to Use the in graph to generate an L graph:

 

We can know two important properties of the Laplacian Matrix: symmetric arrays. Second, there is a zero feature value (Rank: | v |-1 ). The third is a semi-Definite Matrix. Note that the Laplace operator is negative.

When solving a Equations containing the Laplacian matrix, it is often required to be a positive definite matrix. It is observed that this is because the sum of each column in the Laplacian matrix is equal to zero. In this case, you only need to manually change the first row and the first column (for example, if the first element is set to 1 and the rest is set to zero), destroy its structure, and make the rank equal to | v |.

For non-definite matrices, the derivation is as follows:

Ax-B = 0

Minimization | Ax-B | ^ 2, returns the derivative of X after expansion:

Can be converted to a definite equations.

6 LaplaceRelationship between operators and Laplacian Matrices

Laplace operators can be extended to multidimensional computing. Laplacian matrix is mainly used for graphics computing in less than three dimensions and can represent complex geometric structures. The lapace equation uses the Laplace operator to represent the Laplacian matrix.

Laplace operator and Laplacian Matrix