Laplace filter | StrataQuest Glossary

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Definition

The Laplacian of a continuous image is the sum of its second partial derivatives: ∇²I = ∂²I/∂x² + ∂²I/∂y². The discrete approximation uses a small convolution kernel derived from finite differences. The 4-neighbor kernel L₄ = [0,−1,0; −1,4,−1; 0,−1,0] approximates ∇² using only horizontal and vertical neighbors. The 8-neighbor kernel L₈ = [−1,−1,−1; −1,8,−1; −1,−1,−1] includes diagonals and is more isotropic. Both have kernel weights summing to zero, so a uniform region produces zero response — only intensity variation survives. Because the Laplacian amplifies high frequencies (and noise is high-frequency), a Gauss filter pre-smoothing pass is standard; the combination is called the Laplacian of Gaussian (LoG).

Isotropic — responds to changes in any direction

Unlike {{sobel-filter|Sobel}}, which is direction-specific

Zero-crossings trace edge locations

Sign changes mark the edge axis to sub-pixel precision

Amplifies noise — pair with smoothing

Second derivatives are noise-sensitive

Bright dots and blobs produce strong response

Useful as a blob detector

The math, in two registers

The continuous 2D Laplacian is ∇²I = ∂²I/∂x² + ∂²I/∂y². Each second partial derivative is approximated by a central finite difference: ∂²I/∂x² ≈ I(x+1, y) − 2I(x, y) + I(x−1, y), and analogously for y. Summing gives the 4-neighbor discrete Laplacian:

∇²I(x, y) ≈ I(x+1,y) + I(x−1,y) + I(x,y+1) + I(x,y−1) − 4·I(x, y)

Written as a convolution kernel: L₄ = [0, −1, 0; −1, 4, −1; 0, −1, 0]. The 8-neighbor variant L₈ = [−1, −1, −1; −1, 8, −1; −1, −1, −1] includes diagonal contributions; it is closer to rotationally symmetric but slightly more noise-sensitive.

The Laplacian of Gaussian (LoG) precomposes smoothing with the Laplacian. Its analytical form is LoG(x, y; σ) = (−1/πσ⁴) · [1 − (x²+y²)/(2σ²)] · exp(−(x²+y²)/(2σ²)) — a center-positive function surrounded by a negative ring, often called the "Mexican hat." Convolving with the LoG is equivalent to (and computationally similar to) applying Gauss with σ followed by Laplace. The scale parameter σ sets the size of features the operation surfaces.

One subtle property: the Laplacian commutes with the Gaussian (since both are linear convolutions). This means LoG = Gauss(Laplace(I)) = Laplace(Gauss(I)) — order doesn't matter, only the scale matters. This commutativity underpins scale-space theory.

Simplified

The Laplace filter is built from a simple idea applied twice. The first derivative of a 1D signal at point x is approximately (signal at x+1) minus (signal at x−1). The second derivative is the first derivative of the first derivative, which works out to (signal at x+1) − 2×(signal at x) + (signal at x−1) — a sum-and-subtract pattern over three neighboring values.

Doing this in both x and y directions and adding the results gives the discrete Laplacian. A pixel surrounded by neighbors of similar brightness produces a small (near-zero) response. A pixel that's much brighter than its neighbors produces a large positive response. A pixel that's much darker than its neighbors produces a large negative response. The filter is fundamentally measuring how the intensity is curving at each pixel.

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The math, in two registers

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