How a Laplacian matrix different from Laplacian operator.

For to impose smoothing laplace operator is used, but how a laplace matrix is different from laplace operator.

[Image Analyst]

Why do you think a laplacian smooths and image? It does not. It's like an edge detector and it will sharpen/steepen edges and that's it's main purpose. Sure it will smooth areas that have no edges, setting them to zero, but that's not really smoothing, it's more like erasing.

Answer 1

[John D'Errico]

One is an operator, that would operate on a function. The other a matrix. As such a matrix is an APPROXIMATION, to a discretized, sampled function. Since the matrix will use a finite difference APPROXIMATION for the second partial derivatives, it is of course an approximation. That does not make it any less useful as a tool, but it is not the same thing as a Laplacian operator, any more than a picture or video of a dog is the same thing as the dog itself.

[Roshan Jameer Patan]

Here is a laplacian operator

then how a laplacian matrix look like,suppose a size 32*32(M*M)

[John D'Errico]

No. That is NOT a laplacian operator. That is a matrix that happens to contain a template for a finite difference approximation TO a laplacian operator. It is also not properly scaled to compute a true laplacian, since there would potentially be a divisor to yield the actual derivatives.

In context, you might choose to call the above a Laplacian matrix. You might often use the above matrix with conv2, for example, to do computations on a single channel image.

A matrix contains numbers. They have meaning only in context.

If you wish an example of a Laplacian matrix, look here . (Given that Google found this as the first hit I tried, surely you could have done the same and saved a lot of time.)