Scale observability

A monocular camera measures only bearings (directions to points), so the absolute size of the reconstructed world is unobservable: scaling the whole scene and all translations by any factor produces exactly the same images. Stereo removes this ambiguity structurally. The known, calibrated baseline $B$ between the two cameras is a physical length placed inside every stereo measurement: through $Z = fB/d$ , each matched keypoint yields a depth in meters. Metric scale is observable from a single stereo frame, with no motion, no IMU, and no scene assumptions.

Why monocular scale is unobservable

Take any monocular reconstruction $\{\mathbf{X}_j, (\mathbf{R}_i, \mathbf{t}_i)\}$ and scale it by $s > 0$ : points become $s\mathbf{X}_j$ , translations become $s\mathbf{t}_i$ (rotations unchanged). Every projection is untouched, because the pinhole projection is invariant to scaling of homogeneous coordinates:

$\pi\!\left(\mathbf{K}(\mathbf{R}_i\, s\mathbf{X}_j + s\mathbf{t}_i)\right) = \pi\!\left(s\,\mathbf{K}(\mathbf{R}_i \mathbf{X}_j + \mathbf{t}_i)\right) = \pi\!\left(\mathbf{K}(\mathbf{R}_i \mathbf{X}_j + \mathbf{t}_i)\right).$

The measurement likelihood is identical for every $s$ — no amount of monocular data can distinguish a small world traversed slowly from a large world traversed quickly. Scale is a gauge freedom, exactly like the choice of global position and orientation.

What makes scale observable

Scale becomes observable the moment any measurement carries a metric unit:

Stereo baseline — a calibrated physical length inside every frame’s measurement (this page’s topic).
IMU accelerometer — measures metric specific force; integrating it predicts metric displacement, so comparing that to the visual (up-to-scale) displacement reveals scale. But this needs acceleration excitation: under constant velocity or hover the accelerometer only sees gravity and scale becomes weakly observable again.
Depth cameras / LiDAR — direct metric range measurements.
Wheel odometry — metric translation between poses.
Scene priors — known object sizes, known camera height above a ground plane (common in autonomous driving), fiducial markers of known dimensions.

Consequences for the SLAM estimator

Instant initialization. A stereo SLAM system can build a metric local map from its very first frame, whereas monocular SLAM needs a translating motion and still only recovers structure up to scale (and monocular VIO needs sufficient IMU excitation before scale converges — VI-DSO even keeps scale as an explicit optimization variable so it can start with an arbitrary value and converge later).
No scale drift. Monocular odometry accumulates scale error over time (scale drift), which pose-graph methods must correct with $\mathrm{Sim}(3)$ (7-DoF) loop closures. Stereo constrains scale at every keyframe, so loop closure reduces to ordinary $\mathrm{SE}(3)$ (6-DoF) alignment.
Fewer unobservable directions in VIO. In monocular VIO, scale becomes observable only through IMU accelerations; four directions (global position + yaw) remain unobservable. Stereo VIO gets scale directly from the baseline, keeping the estimator well-conditioned during low-excitation motion (hover, constant velocity) where monocular VIO scale is weakly observable.
Caveat — scale is only as good as your geometry. The metric information decays with range ( $\Delta Z \propto Z^2$ ), so at distances far beyond the baseline the system degrades gracefully toward monocular behavior. Baseline calibration errors also translate directly into a global scale bias.

Common pitfalls

Assuming IMU = scale, always: a drone hovering or a car cruising at constant speed provides almost no acceleration excitation; monocular VIO scale can drift or converge slowly in exactly these common regimes.
Far-field stereo is monocular: on a highway, most of the scene may be beyond the reliable range of a roof-mounted rig; the estimator should not count those points as metric anchors.
Uncalibrated baseline changes: thermal expansion or mechanical flex changes $B$ ; since $B$ multiplies every depth, this appears as a slow, global scale change that loop closure alone cannot explain.

A useful way to think about it: monocular SLAM estimates the world in “units of the first translation”; stereo SLAM estimates it in meters because the ruler (the baseline) is bolted to the robot.

Why it matters for SLAM

Robots plan, control, and avoid obstacles in meters, so metric scale is non-negotiable for navigation. Understanding which sensor configurations make scale observable — stereo baseline, IMU accelerations, depth cameras, wheel odometry — explains many system design choices, from why autonomous cars use wide-baseline stereo to why AR headsets pair cameras with IMUs.