Editing Discrete Laplace operator (section)

====Implementation via continuous reconstruction====
A discrete signal, comprising images,  can be viewed as a discrete representation of a continuous function <math>f(\bar r)</math>, where the coordinate vector <math>\bar r \in R^n </math> and the value domain is real <math>f\in R</math>.
Derivation operation is therefore directly applicable to the continuous function, <math>f</math>.
In particular any discrete image, with reasonable presumptions on the discretization process, e.g. assuming band limited functions, or wavelets expandable functions, etc. can be reconstructed by means of well-behaving interpolation functions underlying the reconstruction formulation,<ref name="bigun06vd">{{cite book
 |author1=Bigun, J.
 | year = 2006
 | title =  Vision with Direction
 | publisher = Springer
 | doi=10.1007/b138918
| isbn = 978-3-540-27322-6
 }}</ref>
:<math>
f(\bar r)=\sum_{k\in K}f_k \mu_k(\bar r) 
</math>
where <math>f_k\in R</math> are discrete representations of <math>f</math> on grid <math>K</math> and <math>\mu_k </math> are interpolation functions specific to the grid <math>K</math>. On a uniform grid, such as images,  and for bandlimited functions,   interpolation functions are shift invariant amounting to  <math>\mu_k(\bar r)= \mu(\bar r-\bar r_k) </math> with <math>\mu </math> being an appropriately dilated [[sinc function]] defined in <math>n</math>-dimensions i.e. <math>\bar r=(x_1,x_2...x_n)^T</math>. Other approximations of <math>\mu</math> on uniform grids, are appropriately dilated [[Gaussian function]]s in <math>n</math>-dimensions.  Accordingly, the discrete Laplacian becomes a discrete version of the Laplacian  of the continuous <math>f(\bar r)</math>  
:<math>
\nabla^2 f(\bar r_k)= \sum_{k'\in K}f_{k'} (\nabla^2 \mu(\bar r-\bar r_{k'}))|_{\bar r= \bar r_k}
</math>
which in turn is a convolution with the Laplacian of the interpolation function on the uniform (image) grid <math>K</math>.  
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 <math>f</math> is represented via <math>f_k</math>, in <math>n</math>-dimensions, and are frequency aware by definition.  A linear operator has not only a limited range in the <math>\bar r</math> domain but also an effective range in the frequency domain (alternatively Gaussian scale space) which can be controlled explicitly via the variance of the Gaussian in a principled manner. The resulting filtering can be implemented by separable filters and [[decimation (signal processing)]]/[[pyramid (image processing)]] representations for further computational efficiency in <math>n</math>-dimensions. 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. Monomials which are non-linear operators can also be implemented using a similar reconstruction and approximation approach provided that the signal is sufficiently over-sampled.  Thereby, such non-linear operators e.g.  [[Structure Tensor]], and [[Generalized Structure Tensor]] which are used in pattern recognition for their total least-square optimality in orientation estimation, can be realized.