| Discrete Laplacian Operators
It is useful to construct a filter to serve as the Laplacian operator when applied to a discrete-space image.
. Recall that the gradient, which is a vector, required a pair of orthogonal filters. The Laplacian is a scalar.Therefore, a single filter,  , is sufficient for realizing a Laplacian operation. The Laplacian estimate for an image,  is then
One of the simplest Laplacian operators can be derived as follows. First needed is an approximation to the derivation in x, so let us use a simple first difference.
The second derivative in x can be built by applying the first difference to Eq. (5.8.1). However, we discussed earlier how the first difference produces locations errors because its zero crossing lies off grid. This second application of a first difference can be shifted to counteract the error introduced by the previous one:
Combining the two derivative-approximation stages from Eqs. (11) and (12) produces
Processing in an identical manner for y yields.
Combining the x and y second partials of Eqs. (5.8.3) and (5.8.4) produces a filter,  which estimates the La
Other Laplacian estimation filters can be constructed by using this method of designing a pair of appropriate 1-D second derivative filters and combining them into a single 2-D filters. The results depend on the choice of derivative approximator, the size of the desired filter kernel, and the characteristics of any noise-reduction filtering applied. Two other 3  3 example are
In general, a discrete-space smoothed Laplacian filter can be easily constructed by sampling an appropriate continuous-space function, such as the Laplacian of Gaussian. When constructing a Laplacian filter, make sure that the kernel's coefficients sum to zero in order to satisfy the discrete form of zero mean property of continuous space Laplacian filter. Truncation effects may upet this property and create bias. If so, the filter co-efficients should be adjusted in a way that restores proper balance.
Locating zero crossings in the discrete-space image,  is fairly straightforward. Each pixel should be compared to its eight immediate neighbors; a four-way neighborhood comparison, while faster, may yield broken contours. If a pixel, p, differs in sign with its neighbor, q, an edge lies between them. The pixel, p, is classified as a zero crossing if
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