Camera Calibration Inputs
Formula Used
Xc = R Xw + t
P = K [ R | t ]
[ u' v' w' ]T = P [ X Y Z 1 ]T
u = u' / w', v = v' / w'
The calculator first builds the intrinsic matrix K. It then creates the rotation matrix R from roll, pitch, and yaw. Translation t is appended to create [R | t]. The final 3 by 4 projection matrix is P.
r² = x² + y²
radial = 1 + k1r² + k2r⁴ + k3r⁶
xd = x radial + 2p1xy + p2(r² + 2x²)
yd = y radial + p1(r² + 2y²) + 2p2xy
How to Use This Calculator
- Enter fx, fy, skew, cx, and cy from your camera calibration.
- Select the angle unit and rotation order used by your pose data.
- Enter roll, pitch, yaw, and translation values.
- Add optional distortion coefficients when you want adjusted image points.
- Paste world points as X,Y,Z or Label,X,Y,Z.
- Press the calculate button to show the matrix, graph, and exports.
Example Data Table
| Input | Example Value | Meaning |
|---|---|---|
| fx, fy | 1200, 1180 | Focal lengths in pixel units. |
| cx, cy | 640, 360 | Principal point near image center. |
| Roll, Pitch, Yaw | 0, 0, 12 degrees | Camera rotation values. |
| Translation | 0, 0, 5 | World origin position in camera coordinates. |
| World Point | A,0,0,0 | Named 3D point for projection. |
Camera Projection Matrix Guide
Why Projection Matters
Camera projection links 3D world geometry to 2D image pixels. It is a core step in camera calibration, augmented reality, robotics, photogrammetry, and visual measurement. This calculator builds the 3 x 4 projection matrix from intrinsic values, rotation angles, and translation values. It also projects sample world points, so you can inspect image coordinates quickly.
Intrinsic Matrix
The intrinsic matrix stores focal lengths, image center, and skew. Focal lengths control scale. The principal point sets the pixel location of the optical axis. Skew is usually zero, but it is useful for older sensors or special scans.
Extrinsic Block
The extrinsic block describes camera pose. Rotation aligns the world axes with the camera axes. Translation moves the world origin into camera coordinates. Together, these values form [R | t]. Multiplying K by [R | t] gives P, the final projection matrix.
Calibration Checks
A good calibration has stable focal lengths, sensible principal points, and small distortion. The determinant of R should stay close to one. The camera center should also match the expected scene position. These checks help detect wrong angle units, swapped axes, or inverted translation.
Point Projection
Use several 3D points across the scene. Points near one plane are convenient, but deeper points improve checks. If a projected point has negative depth, it is behind the camera. If it falls far outside the image, review focal length, pose, or point units.
Reports and Graphs
The graph shows projected image locations. The table shows depth, normalized coordinates, pixel output, and distortion offset. CSV export is useful for spreadsheets. PDF export is useful for notes and reports.
Practical Scope
This tool does not replace bundle adjustment or a full checkerboard solver. It is a practical calculator for testing a known calibration, explaining formulas, and reviewing matrix behavior before using vision libraries.
Better Input Practice
For better results, keep all distance units consistent. Use millimeters or meters, but do not mix them. Use degrees only when the angle selector is set to degrees. Save the output after each trial, then compare rows. Small changes in yaw, pitch, or translation can move image points strongly. That behavior is normal for close objects, wide lenses, and large focal lengths. Review the plot after every edit to catch entry mistakes early and clearly.
FAQs
1. What is a camera projection matrix?
It is a 3 by 4 matrix that maps 3D world points into 2D image coordinates using camera intrinsics, rotation, and translation.
2. What does K mean in calibration?
K is the intrinsic matrix. It stores focal lengths, skew, and the principal point of the camera sensor.
3. What does [R | t] represent?
It represents the extrinsic camera pose. R rotates world points, and t translates them into the camera coordinate system.
4. Why is det(R) important?
A valid rotation matrix should have a determinant near one. Large differences may indicate wrong angles or an invalid matrix.
5. What happens when depth is negative?
Negative camera depth means the point is behind the camera. Its image projection is usually not useful for normal viewing.
6. Does distortion change the projection matrix?
Lens distortion is nonlinear. The projection matrix gives the linear pinhole projection, while distortion adjusts normalized image coordinates separately.
7. Which rotation order should I choose?
Use the same order as your pose source. Many vision setups use yaw, pitch, roll as ZYX, but conventions vary.
8. Can I export the calculated values?
Yes. Use the CSV button for spreadsheet data, or use the PDF button for a simple report with matrix and point results.