Module 4/5 · Weeks 10–12 · 27 h

Cameras and calibration

UAT 205 Sensors and Instrumentation Systems

About 90 minDraft, awaiting reviewLast updated 27 September 2026

By the end of this module you will be able to

Explain the pinhole camera model and intrinsics such as focal length in pixels and principal point
Compute the effect of radial lens distortion on pixel position and ground distance
Explain checkerboard camera calibration with Zhang's method
Assess calibration with reprojection error and state the definition used

Prerequisites: UAT 205 Module 1 · UAT 315 Module 1 (images as arrays of numbers)

Why this matters

The camera is the richest sensor on a drone, used for mapping, counting plants and visual navigation. But an image projects a 3D world onto a 2D plane through an imperfect lens. Without knowing the camera’s parameters and lens distortion, distances measured from images are wrong, maps curve and visual navigation drifts. The drone knowledge base’s camera and IMU calibration unit stresses that calibration is not a single number: it belongs to that specific camera, lens and resolution.

The pinhole camera model

The pinhole model assumes every ray passes through a single point, the centre of projection. A world point at $(X, Z)$ lands on the image plane at $x = X / Z$ (normalised coordinates), then converts to pixels using the focal length in pixels $f_{x} = f / p$ , where $p$ is the pixel size, and the principal point $c_{x}$ . These values are the intrinsics, written as the matrix $K$ (Hartley and Zisserman, 2004; Szeliski, 2022).

Side view. The centre of projection on the left, the image plane as a vertical blue line at distance f. A pink world point at X, Z at top right. A pink ray from the world point crosses the image plane at x on its way to the centre of projection. A dashed horizontal line is the optical axis at distance Z — Figure 1 Pinhole camera model

Lens distortion

Real lenses bend light unevenly across the image. The model of Brown (1966), used by OpenCV, has radial distortion $k_{1}, k_{2}, k_{3}$ and tangential distortion $p_{1}, p_{2}$ . Wide-angle drone lenses often have negative $k_{1}$ , called barrel distortion: the farther from the centre, the more the image is pulled inward, so straight lines near the edges curve.

A grey dashed grid is undistorted. A solid blue grid overlays it; in the centre they almost coincide, but toward the edges and corners the blue lines bow toward the centre in a barrel shape. k1 is minus 0.12 and k2 is 0.03 — Figure 2 Barrel distortion of a lens

Example 1 How far distortion moves points on the ground

A camera with 8.8 mm focal length, 3.3 µm pixels and a 4,000-pixel-wide image looks straight down from 100 m. The lens has $k_{1} = - 0.12$ , $k_{2} = 0.03$ (hypothetical values).

FX = 8.8e-3 / 3.3e-6          # focal length in pixels
CX = 2000.0
K1, K2 = -0.12, 0.03
Z = 100.0                     # m altitude
GSD = Z / FX                  # m per pixel at the image centre

print(f"fx {FX:.1f} px, GSD {GSD * 100:.2f} cm/px")
for X in (0, 10, 30, 50, 70):
    x = X / Z
    r2 = x * x
    xd = x * (1 + K1 * r2 + K2 * r2 * r2)
    u_ideal, u_real = FX * x + CX, FX * xd + CX
    shift = u_ideal - u_real
    print(f"X {X:>2} m: ideal u {u_ideal:7.1f}, distorted u {u_real:7.1f}, "
          f"shift {shift:5.1f} px = {shift * GSD:.2f} m on the ground")

fx 2666.7 px, GSD 3.75 cm/px
X  0 m: ideal u  2000.0, distorted u  2000.0, shift   0.0 px = 0.00 m on the ground
X 10 m: ideal u  2266.7, distorted u  2266.3, shift   0.3 px = 0.01 m on the ground
X 30 m: ideal u  2800.0, distorted u  2791.6, shift   8.4 px = 0.32 m on the ground
X 50 m: ideal u  3333.3, distorted u  3295.8, shift  37.5 px = 1.41 m on the ground
X 70 m: ideal u  3866.7, distorted u  3770.4, shift  96.3 px = 3.61 m on the ground

At the centre of the image distortion has almost no effect, but near the edge points shift by nearly a hundred pixels, several metres on the ground. Without calibration, a map stitched from images warps into a bowl or dome.

Calibration with Zhang’s method

Zhang (2000) proposed a method using a planar pattern, such as a checkerboard, photographed in many poses. The software finds the square corners in every image, then solves for $K$ , the distortion coefficients and the board pose in each image so that the reprojected corners best match the detected ones. OpenCV’s calibration tutorial uses this method. Key points:

Enter the number of inner corners and the square size in the correct units.
Photograph the board all over the image, especially at edges and corners, where distortion is largest.
Lock focus and resolution to match how the camera will be used.

The quality measure is the reprojection error, the difference between detected corners and corners reprojected with the estimated parameters. The drone knowledge base’s unit F16 warns that there are two definitions, RMS per point (divided by $n$ ) and RMS per component (divided by $2 n$ ), which differ by a factor of √2, so a report must always state the definition.

Example 2 Per-image reprojection error

Residuals (dx, dy) in pixels for four corners in five images (hypothetical, reduced from a real set with dozens of corners per image).

import math
import statistics as st

residuals = {
    "img01": [(0.2, -0.1), (-0.3, 0.2), (0.1, 0.1), (0.0, -0.2)],
    "img02": [(0.1, 0.3), (-0.2, -0.1), (0.2, 0.0), (-0.1, 0.2)],
    "img03": [(1.8, -1.2), (2.1, 0.9), (-1.6, 1.5), (1.9, -1.4)],
    "img04": [(-0.2, 0.1), (0.3, -0.2), (0.1, 0.2), (-0.1, -0.1)],
    "img05": [(0.2, 0.2), (-0.1, 0.1), (0.0, -0.3), (0.2, -0.1)],
}

def rmse(pts, per_axis=False):
    total = sum(dx * dx + dy * dy for dx, dy in pts)
    return math.sqrt(total / (len(pts) * (2 if per_axis else 1)))

every = [p for pts in residuals.values() for p in pts]
print(f"all images: {rmse(every):.3f} px per point, {rmse(every, True):.3f} px per axis")
per_image = {k: rmse(v) for k, v in residuals.items()}
limit = 2 * st.median(per_image.values())
for k, v in per_image.items():
    print(f"{k}: {v:.3f} px {'<- check this image' if v > limit else ''}")
kept = [p for k, pts in residuals.items() if per_image[k] <= limit for p in pts]
print(f"without flagged images: {rmse(kept):.3f} px per point")

all images: 1.031 px per point, 0.729 px per axis
img01: 0.245 px
img02: 0.245 px
img03: 2.252 px <- check this image
img04: 0.250 px
img05: 0.245 px
without flagged images: 0.246 px per point

A single abnormal image, for example a blurred one or one with misdetected corners, pulls the overall figure up several-fold. Look at that image before deciding to drop it, and check the result with a separate image set not used for calibration. A low reprojection error alone does not prove the calibration is right.

Module lab

Lab: camera calibration with OpenCV

Print a checkerboard, measure the real square size with calipers and mount it on a flat, rigid board.
Take at least 20 images in many poses, covering the whole image including the edges, with focus locked.
Follow OpenCV’s calibration tutorial (or lab L11 in the drone knowledge base), recording $K$ , the distortion coefficients and the reprojection error with its definition.
Use the code from Example 2 to find abnormal images, inspect the cause and recalibrate.
Undistort one aerial image and compare straight lines near the edges before and after.

Common mistakes

Watch out

Entering the square size in the wrong unit.
Photographing the board only in the centre of the image.
Changing resolution or zoom after calibration.
Reporting reprojection error without its definition.
Dropping high-error images without looking at the cause.

Summary

The pinhole model projects points with $f_{x}$ and $c_{x}$ , the camera’s intrinsics.
Real lenses have radial and tangential distortion, strongest at the image edges.
Zhang’s method uses many images of a planar board to find intrinsics and distortion.
Reprojection error must state its definition, be checked per image and be verified with an independent image set.

Check your understanding

A 4.5 mm focal length with 1.5 µm pixels gives what $f_{x}$ ?
A point at $X = 20$ m, $Z = 100$ m with $f_{x} = 3000$ and $c_{x} = 2000$ lands at what pixel $u$ (ignoring distortion)?
What sign does $k_{1}$ have for barrel distortion?
An RMS of 0.5 px per point equals what RMS per component?
Why must the checkerboard also appear near the image edges?

Answers

$4.5 \times 1 0^{- 3} /1.5 \times 1 0^{- 6} = 3000$ px
$3000 \times 0.2 + 2000 = 2600$
Negative.
$0.5/ 2 \approx 0.354$ px
Distortion is largest at the edges; without data there, the estimate is inaccurate in that region.

Key formulas

Pinhole projection	$u = f_{x} \frac{X}{Z} + c_{x}, f_{x} = \frac{f}{p}$
Radial distortion	$x_{d} = x (1 + k_{1} r^{2} + k_{2} r^{4} + k_{3} r^{6})$
Reprojection RMSE per point	$RMSE = \frac{1}{n} \sum_{i} (Δ x_{i}^{2} + Δ y_{i}^{2})$

Key references

Zhang, Z. (2000). A flexible new technique for camera calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11), 1330–1334. link
Brown, D. C. (1966). Decentering distortion of lenses. Photogrammetric Engineering, 32(3), 444–462.
OpenCV. Camera calibration (Python tutorial). OpenCV 4.x documentation. link
Hartley, R., & Zisserman, A. (2004). Multiple view geometry in computer vision (2nd ed.). Cambridge University Press. link
Szeliski, R. (2022). Computer vision: Algorithms and applications (2nd ed.). Springer. link

In class / field

Lab or field practice from worksheets with a safety checklist

Learning evidence: Checked worksheets and quiz results

Module quiz

This is a formative self-check, not a graded exam

Knowledge domain: Control, autopilot and navigation · Sensors and embedded systems · Artificial intelligence and computer vision

Cameras and calibration

By the end of this module you will be able to

Why this matters

The pinhole camera model

Lens distortion

Calibration with Zhang’s method

Module lab

Common mistakes

Summary

Check your understanding

Key formulas

Key references

Further reading

Calibrating cameras and IMUs

Camera calibration with sample images

In class / field

Module quiz

Cameras and calibration

Lesson

By the end of this module you will be able to

Why this matters

The pinhole camera model

Lens distortion

Calibration with Zhang’s method

Module lab

Common mistakes

Summary

Check your understanding

Key formulas

Key references

Further reading

Calibrating cameras and IMUs

Camera calibration with sample images

In class / field

Module quiz