A Kalman filter sounds like graduate-school machinery, but the idea behind it is almost embarrassingly practical. You maintain a guess about where you are and how wrong that guess might be. When you take a step, you predict a new position and deliberately widen your uncertainty because steps are never perfect. When a sensor offers a measurement — compass heading, perhaps, or a hint that you stopped moving — you blend that measurement with your guess, weighted by how much you trust each source. The math is linear algebra; the philosophy is humility.

Wheria uses a reduced extended Kalman filter on a four-element state vector: horizontal position x and y in local meters, heading theta in radians, and a slowly drifting gyroscope bias term. We call it extended because heading updates involve trigonometry, but the implementation on phone is still a handful of matrix multiplies per step, not a supercomputer job. The filter never runs in isolation; it sits between step detection and the confidence badge you see in Find mode.

Prediction when the user walks

Each detected step triggers a prediction. Position advances by stride length times the cosine and sine of current heading. The covariance matrix P — which encodes how uncertain we are about each state element — grows by a process noise matrix Q. Q is not pulled from a textbook appendix; we tuned it from field logs in real garages. Stride uncertainty sits around five centimeters standard deviation per step. Heading process noise is about two degrees per step when compass trust is high, and closer to eight degrees in steel-heavy zones where magnetometer corrections are unreliable.

That asymmetry matters. In an open-air slab garage, compass measurements arrive frequently and the filter tightens heading after almost every few steps. In a spiral ramp wrapped in rebar, we inflate measurement noise for magnetometer updates by an order of magnitude, which effectively tells the filter to believe the gyroscope integration between steps and treat compass spikes as suspicious guests rather than honored advisors.

Measurements and when to believe them

Magnetometer heading enters as a measurement z equal to the arctangent of the horizontal field components. The update adjusts theta and pulls P down when the field magnitude and inclination look like Earth's expected range. When they do not — when steel columns rotate effective north by forty degrees without the user turning — the filter increases measurement noise R for magnetometer observations and the heading correction becomes gentle instead of violent.

Barometer floor hints are handled more cautiously. A pressure drop since parking can suggest you descended one slab, but elevators, HVAC gusts, and weather fronts also move pressure. We often surface barometer inference as parallel UI guidance — "likely B3" with a plus-or-minus one floor band — rather than hard-fusing altitude into the Kalman state where a false lock would corrupt horizontal confidence users rely on for distance-to-car.

Zero-velocity updates are a classic pedestrian trick: if no step has registered for about two seconds, assume the user stood still and damp any legacy velocity drift in integration paths that still carry speed states from earlier prototypes. It is a small stabilizer, not a cure for escalators.

Why not a particle filter or full SLAM

Particle filters represent uncertainty with clouds of hypothetical positions and weights. They shine when ambiguity is multimodal — you genuinely do not know whether the user turned left or right at a fork. They also burn CPU and complicate parity between Swift and Kotlin implementations that must behave identically for ktuyen's regression matrix. For walks under roughly one hundred twenty meters in a single garage, an extended Kalman filter with adaptive measurement noise has been sufficient. We would revisit particles if we added map matching to painted parking lines in open lots where fork ambiguity is common.

Simultaneous localization and mapping with camera or lidar is even farther from our constraints. Parking garages change signage, lighting, and parked car layouts daily. Phone-only SLAM without infrastructure beacons fails the offline, no-account, privacy-first requirements described in our on-device post.

What users actually see

The trace of the position covariance block maps to the green, yellow, and red confidence chip on the Find screen. When uncertainty grows beyond a threshold, the UI switches from a precise arrow to a coarse distance ring: walk about sixty meters, then reassess. Outdoors, if a GPS segment is still valid, the two-by-two position block of P can draw an uncertainty ellipse on the map instead of a lying pinpoint dot.

If you want to play with the matrices interactively, Phyzix includes a Kalman demonstration that steps through predict and update with sliders for Q and R. The full indoor pipeline — step detection, compass fusion, barometer hints — is documented in indoor navigation when GPS dies. Step odometry details live in step detection and IMU odometry; magnetometer failure modes in magnetometers and compass nightmares.