= Scale ambiguity of perspective projection
<Scale ambiguity of perspective projection> means that image coordinates alone cannot determine absolute scene size and distance. In an ideal camera with focal length $f$ and centered image coordinates,
$$
u=f\frac XZ,\qquad v=f\frac YZ.
$$
Replacing $(X,Y,Z)$ by $(sX,sY,sZ)$ for any positive $s$ leaves $(u,v)$ unchanged. A geometrically similar scene twice as large and twice as far away therefore produces the same projection.<Camera Calibration and 3D Reconstruction by OpenCV>[Detailed Description, pinhole projection and camera-coordinate equations]{ref}
For example, a $0.10\,\mathrm m$ object at depth $1\,\mathrm m$ and a $1\,\mathrm m$ object at depth $10\,\mathrm m$ subtend the same image height under this model. Extra metric information, such as a known object length or a measured <camera baseline>, supplies a scale. In <stereo vision>, scaling the reconstructed scene and the assumed baseline together preserves the image observations; measuring the baseline breaks that freedom. <Camera calibration> of focal length alone does not provide an absolute length for an otherwise unknown scene.
Back to article page
