Laplacian Filter Tool: Edge Detection, Laplacian of Gaussian and Image Sharpening

Key facts

• Laplacian: ∇²f = ∂²f/∂x² + ∂²f/∂y². It is a second derivative, rotation invariant, and zero on flat areas and linear ramps.
• Discrete kernels: 4-neighbour [0 1 0; 1 −4 1; 0 1 0] ≈ ∇²; 8-neighbour [1 1 1; 1 −8 1; 1 1 1] ≈ 3∇². Every Laplacian kernel sums to 0.
• Edges are zero crossings: at a blurred step the Laplacian swings from positive to negative; the edge lies where it crosses zero.
• Noise: the response grows like frequency squared, so noise is amplified (×√20 ≈ 4.5 in standard deviation for the 4-neighbour kernel). Smooth first: LoG = ∇²(G_σ * f) = (∇²G_σ) * f.
• LoG kernel: ∇²G = (x² + y² − 2σ²)/(2πσ⁶)·e^{−(x²+y²)/2σ²}, zero at r = √2σ; DoG with k = 1.6 approximates it.
• Sharpening: g = f − c∇²f (negative-centre kernel). With c = 1 this is [0 −1 0; −1 5 −1; 0 −1 0].

1. Choose an operator. Start with the 4-neighbour kernel on the shapes image. Grey means zero, white is positive and black is negative.
2. Add noise. Move the noise slider to 10 and watch the plain Laplacian drown in noise. Then switch to LoG with σ = 2 (or give the 3 × 3 kernel a pre-smoothing σ).
3. Find edges. Choose Edges to mark zero crossings in red. Raise the threshold to drop weak crossings caused by noise and texture.
4. Sharpen. Choose Sharpen and set the amount. Look at the row profile to see the overshoot that makes edges look crisp.
5. Inspect. The kernel tab shows the effective kernel, the frequency tab shows which detail sizes the operator responds to, and clicking a pixel on the input shows its calculation in the step-by-step tab.

Σk² is the sum of squared kernel weights: the factor by which the variance of white noise is multiplied. The LoG values are for SciPy's sampled kernel. Compare operators at the same signal response, not just by Σk²: the LoG also attenuates the edge, but much less than it attenuates noise.

1. Smooth with a Gaussian of σ chosen for the scale of the edges you want (larger σ: fewer, smoother, more displaced edges).
2. Take the Laplacian; in practice steps 1 and 2 are a single LoG (or DoG) convolution.
3. Find zero crossings: pixels whose response changes sign relative to a neighbour.
4. Threshold the slope: keep only crossings where the jump between the positive and negative sides is large, which removes the many weak crossings caused by noise and texture.

Zero crossings form closed contours, which is both a strength (no gaps) and a weakness (spurious closed loops in flat noisy areas). They also shift at corners and at curved edges as σ grows. The Canny detector, which uses first derivatives and hysteresis, generally performs better for straight edges, but the LoG remains the standard tool for blob detection: a bright disc of radius R gives the strongest scale-normalized response σ²∇²G at σ = R/√2.

Example 1: A step edge in 1-D

For the signal f = [10, 10, 10, 10, 50, 50, 50, 50], the second difference f(x − 1) − 2f(x) + f(x + 1) is [0, 0, 0, 40, −40, 0, 0, 0]. It is zero on both flat parts, +40 on the dark side and −40 on the bright side of the step. The sign changes between positions 3 and 4, exactly where the edge is.

Example 2: A bright spot in 2-D

A pixel of value 120 has neighbours 53 (above), 50 (right), 51 (below) and 52 (left). The 4-neighbour response is 53 + 50 + 51 + 52 − 4 × 120 = −274. A bright spot gives a strongly negative Laplacian and a dark spot a positive one; a flat area or a linear ramp gives 0.

Example 3: Sharpening creates overshoot

Sharpening the step of Example 1 with c = 1: g = f − ∇²f = [10, 10, 10, −30, 90, 50, 50, 50]. The edge contrast rises from 40 to 120, which the eye reads as sharper. On 8-bit images −30 is clipped to 0, and the overshoot appears as a dark and a bright halo along each edge.

Example 4: Why smooth first?

Add white noise with σₙ = 10 to the step of height 40. The 4-neighbour Laplacian responds to the edge with ±40 but to the noise with a standard deviation of 10√20 ≈ 45: the edge is invisible. A LoG with σ = 2 gives an edge peak of about ±2.3 (grey levels per pixel²) and a noise standard deviation of only 10 × √0.00249 ≈ 0.50, a signal-to-noise ratio of about 4.6 instead of 0.9.

Example 5: Choosing σ for blobs

For a disc of radius R, the scale-normalized LoG at its centre is σ²∇²G * disc = −(R²/σ²) e^{−R²/(2σ²)}. Setting the derivative with respect to σ to zero gives σ = R/√2 and a peak value of −2/e ≈ −0.736 per unit of contrast. Trees about 8 pixels in radius in a 0.5 m aerial image are therefore found best with σ ≈ 5.7 pixels.

import numpy as np, cv2
from scipy import ndimage
f = img.astype(float)

lap4 = ndimage.laplace(f)                       # 4-neighbour kernel, mode='reflect'
k8 = np.array([[1, 1, 1], [1, -8, 1], [1, 1, 1]], float)
lap8 = ndimage.correlate(f, k8)
log = ndimage.gaussian_laplace(f, sigma=2.0)    # LoG, separable, truncate=4
dog = ndimage.gaussian_filter(f, 1.6 * 2) - ndimage.gaussian_filter(f, 2)

lap_cv = cv2.Laplacian(f, cv2.CV_64F, ksize=1)   # ksize=1: [0 1 0; 1 -4 1; 0 1 0]
# careful: ksize=3 is NOT the 8-neighbour kernel; it is [2 0 2; 0 -8 0; 2 0 2]

sharp = np.clip(f - 1.0 * lap4, 0, 255)         # sharpening, c = 1
focus = cv2.Laplacian(gray, cv2.CV_64F).var()   # variance of the Laplacian (blur measure)

% MATLAB
L  = imfilter(double(I), [0 1 0; 1 -4 1; 0 1 0], 'replicate');
h  = fspecial('log', 13, 2);          % LoG kernel, size 13, sigma 2 (adjusted to sum to 0)
L2 = imfilter(double(I), h, 'replicate');
E  = edge(I, 'log', [], 2);           % Marr-Hildreth zero-crossing edges
% note: del2(I) returns the discrete Laplacian divided by 4

• No direction or strength by itself. The Laplacian does not give the edge orientation, and its value at the zero crossing is zero by definition; edge strength must come from the slope across the crossing or from the gradient magnitude.
• Double edges and spurious loops. Lines (thin features) produce two zero crossings, one on each side; noise produces closed loops unless thresholded.
• Scale matters. σ that is too small leaves noise; too large merges nearby edges and shifts corners. Real scenes, such as a satellite image with field boundaries and rivers of different widths, usually need several scales.
• Sharpening is not restoration. It boosts existing high frequencies, including noise and compression artefacts; it cannot recover detail that the optics or the sensor removed (that needs deconvolution with a known PSF).
• In this tool the Laplacian is computed on luma (0.299R + 0.587G + 0.114B); sharpening is applied to each RGB channel. Uploaded images are resized to at most 512 pixels and never leave your browser.