Editing Discrete Laplace operator (section)

====Implementation via operator discretization====
For one-, two- and three-dimensional signals, the discrete Laplacian can be given as [[convolution]] with the following kernels:
:1D filter: <math>\vec{D}^2_x=\begin{bmatrix}1 & -2 & 1\end{bmatrix}</math>,
:2D filter: <math>\mathbf{D}^2_{xy}=\begin{bmatrix}0 & 1 & 0\\1 & -4 & 1\\0 & 1 & 0\end{bmatrix}</math>.
<math>\mathbf{D}^2_{xy}</math> corresponds to the ([[Five-point stencil]]) finite-difference formula seen previously. It is stable for very smoothly varying fields, but for equations with rapidly varying solutions a more stable and isotropic form of the Laplacian operator is required,<ref name="Provatas Elder p. ">{{cite book 
| last1=Provatas 
| first1=Nikolas 
| last2=Elder 
| first2=Ken 
| title=Phase-Field Methods in Materials Science and Engineering 
| publisher=Wiley-VCH Verlag GmbH & Co. KGaA 
| publication-place=Weinheim, Germany 
| date=2010-10-13 
| isbn=978-3-527-63152-0 
| url=http://www.physics.mcgill.ca/~provatas/papers/Phase_Field_Methods_text.pdf 
| doi=10.1002/9783527631520 
| page=219}}</ref> such as the [[nine-point stencil]], which includes the diagonals:

:2D filter: <math>\mathbf{D}^2_{xy}=\begin{bmatrix}0.25 & 0.5 & 0.25\\0.5 & -3 & 0.5\\0.25 & 0.5 & 0.25\end{bmatrix}</math>,
:3D filter: <math>\mathbf{D}^2_{xyz}</math> using [[seven-point stencil]] is given by: 
::first plane = <math>\begin{bmatrix}0 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{bmatrix}</math>; second plane = <math>\begin{bmatrix}0 & 1 & 0\\1 & -6 & 1\\0 & 1 & 0\end{bmatrix}</math>; third plane = <math>\begin{bmatrix}0 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{bmatrix}</math>.
:and using [[27-point stencil]] by:<ref name="OReilly2006">{{cite journal
|last1=O'Reilly
|first1=H.
|last2=Beck
|first2=Jeffrey M. 
|title=A Family of Large-Stencil Discrete Laplacian Approximations in Three Dimensions
|journal=International Journal for Numerical Methods in Engineering
|year=2006
|pages=1–16 |url=http://psych.colorado.edu/~oreilly/papers/OReillyBeckIP_lapl.pdf

}}</ref>
::first plane = <math>\frac{1}{26}\begin{bmatrix}2 & 3 & 2\\3 & 6 & 3\\2 & 3 & 2\end{bmatrix}</math>; second plane = <math>\frac{1}{26}\begin{bmatrix}3 & 6 & 3\\6 & -88 & 6\\3 & 6 & 3\end{bmatrix}</math>; third plane = <math>\frac{1}{26}\begin{bmatrix}2 & 3 & 2\\3 & 6 & 3\\2 & 3 & 2\end{bmatrix}</math>.
:''{{var|n}}D filter'': For the element <math>a_{x_1, x_2, \dots , x_n}</math> of the kernel <math>\mathbf{D}^2_{x_1, x_2, \dots , x_n},</math>
::<math>a_{x_1, x_2, \dots , x_n} = \left\{\begin{array}{ll}
-2n & \text{if } s = n, \\
1 & \text{if } s = n - 1, \\
0 & \text{otherwise,}
\end{array}\right.</math>
:where {{math|{{var|x}}{{sub|{{var|i}}}}}} is the position (either {{math|−1}}, {{math|0}} or {{math|1}}) of the element in the kernel in the {{var|i}}-th direction, and {{math|{{var|s}}}} is the number of directions {{math|{{var|i}}}} for which {{math|{{var|x}}{{sub|{{var|i}}}} {{=}} 0}}.

Note that the ''n''D version, which is based on the graph generalization of the Laplacian, assumes all neighbors to be at an equal distance, and hence leads to the following 2D filter with diagonals included, rather than the version above:
:2D filter: <math>\mathbf{D}^2_{xy}=\begin{bmatrix}1 & 1 & 1\\1 & -8 & 1\\1 & 1 & 1\end{bmatrix}.</math>

These kernels are deduced by using discrete differential quotients.

It can be shown<ref name=lin90>[http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A472968&dswid=-3163 Lindeberg, T., "Scale-space for discrete signals", PAMI(12), No. 3, March 1990, pp. 234–254.]</ref><ref name=lin94>[http://www.csc.kth.se/~tony/book.html Lindeberg, T., Scale-Space Theory in Computer Vision, Kluwer Academic Publishers, 1994], {{isbn|0-7923-9418-6}}.</ref> that the following discrete approximation of the two-dimensional Laplacian operator as a convex combination of difference operators

:<math>\nabla^2_{\gamma}= (1 - \gamma) \nabla^2_{5} + \gamma \nabla ^2_{\times} 
         = (1 - \gamma) \begin{bmatrix}0 & 1 & 0\\1 & -4 & 1\\0 & 1 & 0\end{bmatrix}
            + \gamma \begin{bmatrix}1/2 & 0 & 1/2\\0 & -2 & 0\\1/2 & 0 & 1/2\end{bmatrix}
</math>

for γ ∈ [0, 1] is compatible with discrete scale-space properties, where specifically the value γ = 1/3 gives the best approximation of rotational symmetry.<ref name=lin90/><ref name=lin94/><ref name="PatraKarttunen2006">{{cite journal|last1=Patra|first1=Michael|last2=Karttunen|first2=Mikko|title=Stencils with isotropic discretization error for differential operators|journal=Numerical Methods for Partial Differential Equations|volume=22|issue=4|year=2006|pages=936–953|issn=0749-159X|doi=10.1002/num.20129|s2cid=123145969 }}</ref> Regarding three-dimensional signals, it is shown<ref name=lin94/> that the Laplacian operator can be approximated by the two-parameter family of difference operators

:<math>
\nabla^2_{\gamma_1,\gamma_2} 
         = (1 - \gamma_1 - \gamma_2) \, \nabla_7^2  + \gamma_1 \, \nabla_{+^3}^2 +  \gamma_2 \, \nabla_{\times^3}^2 ),
</math>

where

:<math>
  (\nabla_7^2 f)_{0, 0, 0}
       =
        f_{-1, 0, 0} + f_{+1, 0, 0} + f_{0, -1, 0} + f_{0, +1, 0} + f_{0, 0, -1} + f_{0, 0, +1} - 6 f_{0, 0, 0},
</math>
:<math>
  (\nabla_{+^3}^2 f)_{0, 0, 0}
      = \frac{1}{4}
          (f_{-1, -1, 0} + f_{-1, +1, 0} + f_{+1, -1, 0} + f_{+1, +1, 0} 
        + f_{-1, 0, -1} + f_{-1, 0, +1} + f_{+1, 0, -1} + f_{+1, 0, +1} 
        + f_{0, -1, -1} + f_{0, -1, +1} + f_{0, +1, -1} + f_{0, +1, +1}
          - 12 f_{0, 0, 0}),
</math>
:<math>
   (\nabla_{\times^3}^2 f)_{0, 0, 0}
  = \frac{1}{4}
          (f_{-1, -1, -1} + f_{-1, -1, +1} + f_{-1, +1, -1} + f_{-1, +1, +1}
         + f_{+1, -1, -1} + f_{+1, -1, +1} + f_{+1, +1, -1} + f_{+1, +1, +1}
           -  8 f_{0, 0, 0}).
</math>
It can be shown by Taylor series analysis that combinations of values of <math>\gamma_1</math> and <math>\gamma_2</math> for which <math>3\gamma_1 + 6\gamma_2 = 2</math> give the best approximations of rotational symmetry.