Second-order derivatives is better suited for most applications for sharpening.

By constructing a filter based on discrete formulation of second- order derivatives.

Simplest isotropic derivative operator: Laplacian

Laplacian Filter is created or designed as:

Following figure.11 shows various Laplacian Masks taking 4-neighbours and 8-neighbours.

Figure 11

Problem: While applying Laplacian highlights fine detail, it de-emphasizes smooth regions (e.g., background features). It results in featureless background with grayish fine details.

Solution: Add original image to recover background features.

$g(x,y) = f(x,y)- \triangledown^{2}f(x,y) $, negative filter center

$\hspace{12mm}$ $= f(x,y)- \triangledown^{2}f(x,y) $, positive filter center

Sharpening using Unsharp Masking:

Subtract blurred version of image from the image itself to produce sharp image.

$g(x,y) = f(x,y) - \overline f(x,y)$