Talk:Discrete Laplace operator

Correspondence for the divergence?

Is any correspondence of the discrete Laplacian for the divergence? This should necessarily be the case, since both the divergence and the Laplacian are used heavily in PDE:s. --Kri (talk) 16:47, 6 April 2011 (UTC)

Yes, of course, e.g., P. McDonald and R. Meyers. Diffusions on graphs, Poisson problems and spectral geometry. Trans. Amer. Math. Soc., 354(12):5111–5136 (electronic), 2002. 2andrewknyazev (talk) 00:28, 8 April 2011 (UTC)

More on stencils

Approximations of the continuous Laplacian under Definitions should contain more on larger-stencil approximations, as in [paper]; See in particular illustrations starting on page 3. A proper explanation should facilitate comprehension of equation 2 under section 3.2. Reference FDTD Scheme in [article]. ([Screenshot of equation.]) 80.232.11.13 (talk) 14:24, 3 January 2012 (UTC)

Implementation in Image Processing

In the section Implementation in Image Processing I always struggle with which of the 3 recommended one now to use. While the MatLab help actually gives both a rule how to construct them and a recommendation which one to use. $\mathbf {D} _{xy}^{2}={\frac {1}{\alpha +1}}{\begin{bmatrix}\alpha &1-\alpha &\alpha \\1-\alpha &-4&1-\alpha \\\alpha &1-\alpha &\alpha \end{bmatrix}}$.

While the 3 filters mentioned in this article just have an $\alpha =0$ leading to $\mathbf {D} _{xy}^{2}={\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}$.

$\alpha =1/3$ leading to $\mathbf {D} _{xy}^{2}=1/2{\begin{bmatrix}0.5&1&0.5\\1&-6&1\\0.5&1&0.5\end{bmatrix}}$.

And $\alpha =0.5$ leading to $\mathbf {D} _{xy}^{2}=1/3{\begin{bmatrix}1&1&1\\1&-8&1\\1&1&1\end{bmatrix}}$.

MatLab itself (for probably some very good reason otherwise MatLab wouldn't do that, however, they don't cite this reason unfortunately) recommends $\alpha =0.2$ leading to $\mathbf {D} _{xy}^{2}=1/6{\begin{bmatrix}1&4&1\\4&-20&4\\1&4&1\end{bmatrix}}$.

Everybody having a MatLab to their abuse can try this with fspecial('laplacian',alpha). (Wow, this was the most complicated maths I ever edited here.) ;-) Peterthewall (talk) 17:16, 20 September 2012 (UTC)

The stencil that you point out is recommended by MATLAB is also recommended in the book Horn, B.K.P., Robot Vision. There is an exercise in that book that asks the reader to show by Taylor series analysis that that stencil gives the best approximation of rotational symmetry. A couple of the references in this Wikipedia article also show that. 72.74.36.69 (talk) 17:52, 8 October 2024 (UTC)

Laplace Filter

So, by far the most common use of this is the laplace filtering used for image processing. This article shows lots of nice math about this, but however completely fails at describing anything about how this is used for image processing. To contrast, the German page seems to be vastly superior for this (I am not very good at german though). Would say this needs a separate page at Laplace filter instead of the current redirect to this page. Not sure I will find much time for it, so any help is very welcome.

I would rather argue that there is a bunch of other applications from applied engineering to theoretical physics ;) But never mind, yes two different pages would make sense. In English and in German, though. W.pseudon (talk) 20:17, 22 August 2019 (UTC)

The embedding of Laplace filter in Discrete Laplace operator even could look like the embedding of "Diskreter Laplace-Operator" in Laplace-Operator W.pseudon (talk) 20:21, 22 August 2019 (UTC)

Discrete heat equation

In the section 'discrete heat equation' there is a mismatch between the development of the solution and the example computer code given at the end. Specifically, the solution development explicitly names the left eigenvectors defined by ${\textstyle L\mathbf {v} _{i}=\lambda _{i}\mathbf {v} _{i}}$, but the computer code uses right eigenvectors produced by Matlab command 'eig', which are defined by ${\textstyle L\mathbf {v} _{i}=\mathbf {v} _{i}\lambda _{i}}$ . For clarity and kindness to the reader, the development and the example should match.

One way to improve this section would be to revise the development with matrix notation and right eigenvectors, supported by appropriate literature, to better fit the example written in Matlab (which facilitates matrix operations). Presently the literature cited for the development is: Mark Newman (2010). Networks: An Introduction. Oxford University Press. ISBN 978-0199206650, however sources for a matrix development are available. Are there other suggestions for improving this section? Uncole (talk) 13:45, 8 May 2022 (UTC)