Calibrating a Camera

Every lens bends light slightly, and straight lines curve near the edges. If you want to measure in millimetres, you need to know exactly how your lens behaves. There's an elegant way to find out. Photograph a checkerboard.

• Every lens bends light
• Straight lines curve
• Photograph a checkerboard

Calibration

A checkerboard has corners at exact, known positions. Here we simulate twelve photos of a board, using a known camera, and known distortion, with a little noise. In real life, you'd detect the corners in real photos. Then calibrate camera solves for everything at once. Run it. Reprojection error tells you how well the model explains the photos, and anything under about half a pixel is good. It's point two one. The focal length came back as eight hundred and one, and the truth was eight hundred. And the distortion matches too. From a dozen photos, we've worked out the inner workings of the camera.

import cv2
import numpy as np

board = np.zeros((6 * 9, 3), np.float32)
board[:, :2] = np.mgrid[0:9, 0:6].T.reshape(-1, 2) * 25.0
K = np.array([[800, 0, 320], [0, 800, 240], [0, 0, 1]], float)
dist = np.array([-0.25, 0.1, 0, 0, 0], float)
rng = np.random.default_rng(0)

views_3d, views_2d = [], []
for i in range(12):
    rvec = rng.uniform(-0.4, 0.4, 3)
    tvec = np.array([rng.uniform(-80, -40), rng.uniform(-70, -30), rng.uniform(350, 500)])
    pts, _ = cv2.projectPoints(board, rvec, tvec, K, dist)
    views_3d.append(board)
    views_2d.append((pts + rng.normal(0, 0.15, pts.shape)).astype(np.float32))

err, K2, d2, _, _ = cv2.calibrateCamera(views_3d, views_2d, (640, 480), None, None)
print(f"Reprojection error: {err:.2f} pixels")
print(f"Focal length: {K2[0, 0]:.0f} (true 800)")
print(f"Distortion k1: {d2[0, 0]:.2f} (true -0.25)")

Reprojection error: 0.21 pixels
Focal length: 801 (true 800)
Distortion k1: -0.25 (true -0.25)

Doing it for real

Use fifteen to twenty photos, with the board at different angles, and covering the corners of the frame. Keep the board perfectly flat. Lock the focus and zoom, because changing either one means calibrating again. Then undistort every frame before measuring. And recheck it regularly, as a knock can shift a lens.

Summary