LaplacianFilter

LaplacianFilter is an image processing operation that applies the discrete Laplacian operator to an image in order to highlight regions of rapid intensity change. It is widely used for edge detection and feature extraction because the Laplacian emphasizes second-order intensity variations rather than absolute brightness.

In two dimensions, the continuous Laplacian is Lf = ∂^2f/∂x^2 + ∂^2f/∂y^2. In digital images, this is approximated by convolving the image with a small kernel. Common 3x3 kernels include:

0 1 0

1 -4 1

0 1 0

and

-1 -1 -1

-1 8 -1

-1 -1 -1

These kernels sum to zero, producing responses at edges and corresponding positive and negative values. Display often uses the magnitude or absolute value of the response, or rescales to visualize both sides of the edge.

The Laplacian is sensitive to noise because it computes second derivatives. To mitigate noise, it is common to smooth the image first, for example with a Gaussian blur, and then apply the Laplacian; this combination is known as the Laplacian of Gaussian (LoG). Variants use different kernel sizes or shapes to capture edges at multiple scales.

Applications include edge detection, contour extraction, and preliminary feature detection for computer vision tasks. Practical use often involves applying the filter to grayscale images or to each color channel, with explicit border handling (padding, replication, or trimming) and optional post-processing such as normalization, thresholding, or combining with the original image for visualization.