Edge detection

Gradients on a grid

For a 2D image I(x, y), the gradient ∇I = (∂I/∂x, ∂I/∂y) points in the direction of steepest brightness increase. On a discrete grid we approximate partial derivatives with finite differences—exactly what convolution kernels like Sobel implement. The gradient magnitude |∇I| ≈ √(Gx² + Gy²) highlights edges regardless of orientation; the angle atan2(Gy, Gx) describes edge direction.

Sobel and Scharr

cv2.Sobel computes separable 3×3 (or larger, odd) derivative filters. Use CV_64F (float) to avoid clipping negative slopes, then rescale for display. Scharr is a 3×3 variant with slightly better rotation invariance for equal cost.

import cv2
import numpy as np

gray = cv2.imread("photo.jpg", cv2.IMREAD_GRAYSCALE)
blur = cv2.GaussianBlur(gray, (5, 5), 0)

gx = cv2.Sobel(blur, cv2.CV_64F, 1, 0, ksize=3)
gy = cv2.Sobel(blur, cv2.CV_64F, 0, 1, ksize=3)
mag = np.sqrt(gx * gx + gy * gy)
mag_u8 = np.uint8(255 * mag / (mag.max() + 1e-6))

gx_s = cv2.Scharr(blur, cv2.CV_64F, 1, 0)
gy_s = cv2.Scharr(blur, cv2.CV_64F, 0, 1)

Convert to uint8 only for saving or imshow; keep floats for further math.

Gradient angle (optional)

angle = np.arctan2(gy, gx)  # radians, in [-pi, pi]

Laplacian

The Laplacian is the second derivative (sum of ∂²/∂x² and ∂²/∂y²). It is sensitive to noise but highlights fine detail and zero-crossings near edges. Often used in sharpening (unsharp mask) and as a building block in blob detection.

import cv2

gray = cv2.imread("photo.jpg", cv2.IMREAD_GRAYSCALE)
lap = cv2.Laplacian(gray, cv2.CV_64F, ksize=3)
lap_vis = cv2.convertScaleAbs(lap)

convertScaleAbs maps floats to 8-bit for display after Laplacian/Sobel.

Canny edge detector

Canny is a multi-stage algorithm: Gaussian smoothing, gradient magnitude and angle, non-maximum suppression (thin edges along the gradient normal), and hysteresis thresholding with a low and high threshold. Pixels above threshold2 are strong edges; weak edges are kept only if connected to strong ones—this reduces broken fragments.

import cv2

gray = cv2.imread("photo.jpg", cv2.IMREAD_GRAYSCALE)
blur = cv2.GaussianBlur(gray, (5, 5), 0)

# Classic ratio: t_high : t_low often ~ 2:1 or 3:1; tune per image scale
t1, t2 = 50, 150
edges = cv2.Canny(blur, t1, t2, apertureSize=3, L2gradient=False)

More Canny examples

# Finer edges (noisier): lower thresholds
fine = cv2.Canny(blur, 30, 90)

# Fewer, stronger edges: raise thresholds
coarse = cv2.Canny(blur, 100, 200)

# L2 gradient uses sqrt(Gx^2+Gy^2) internally for magnitude
edges_l2 = cv2.Canny(blur, 50, 150, apertureSize=3, L2gradient=True)

# Larger Sobel aperture inside Canny (5 or 7): slightly smoother gradients
edges_ap5 = cv2.Canny(blur, 50, 150, apertureSize=5)

Practical pipeline

For document or industrial images, thresholds are easier to set if intensities are normalized. One pattern: blur → optional CLAHE (see histogram chapter) → Canny. For color images, either convert to grayscale or run Canny per channel and combine with bitwise OR.

import cv2
import numpy as np

bgr = cv2.imread("scene.jpg")
gray = cv2.cvtColor(bgr, cv2.COLOR_BGR2GRAY)
gray = cv2.GaussianBlur(gray, (5, 5), 0)

med = np.median(gray)
t1 = int(max(0, 0.66 * med))
t2 = int(min(255, 1.33 * med))
auto_edges = cv2.Canny(gray, t1, t2)

Median-based heuristic for thresholds—starting point only; validate on your data.