Inverse-Variance Fusion

Inverse-Variance Fusion

Ada re-derives this chapter’s own numbers step by step, at full precision

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Ada ADA · CALCULATION AUDIT

Inverse-Variance Fusion

The chapter fuses three obstacle-range readings — camera 44.8 m, lidar 45.1 m, radar 45.5 m — each with its own uncertainty, into a single value with a published band of 0.25 m, tighter than any one sensor. That tightness is the whole appeal of inverse-variance fusion, and also its trap. This audit re-runs the weights and the 45.18 m fused range to show when that confident 0.25 m is earned and when it is not.

Companion to the chapter Sensor Fusion Applications — every number here comes from that chapter.

Try

The chapter fuses three obstacle-range readings — camera 44.8 m , lidar 45.1 m , radar 45.5 m — each with its own uncertainty, into a single value with a published band of 0.25 m , tighter than any one sensor. Calculate this case.

Observe

This audit re-runs the weights and the 45.18 m fused range to show when that confident 0.25 m is earned and when it is not. Check shows this.

Explain

That 0.25 m is tighter than any single sensor's sigma, which is exactly why the fused value looks confident. But the same arithmetic assumes the three errors are independent and current, and the freshness check in this section shows the risk: a camera frame 120 ms old on an object moving 1.6 m/s has already slipped 1.6 x 0.120 = 0.192000 m, past a 0.10 m stale budget, whilethe lidar and radar remain within the freshness budget. Check confirms it.

See the relationship before changing it

The figure reads from left to right. The blue input is lidar sigma. The middle card names the page’s rule. The green output is fused sigma. The arrow matters: change the input, apply the rule once, then read the result with its unit.

Lidar Sigma changes fused sigma A three-part teaching diagram connects lidar sigma, the rule sigma = sqrt(1 / (1/1.0^2 + 1/lidar^2 + 1/0.5^2)), and fused sigma. INPUT Lidar sigma APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrow. A smaller sigma gives that sensor more precision weight.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 0.3 m.

  2. 2

    Name the relationship. sigma = sqrt(1 / (1/1.0^2 + 1/lidar^2 + 1/0.5^2))

  3. 3

    Substitute with units. sqrt(1 / (1 + 1/0.3^2 + 4)) = 0.249 m

  4. 4

    Read the result. Keep the unit beside the value, then use the result only inside the technical boundary below.

Predict, then change lidar sigma

Try Predict how fused sigma responds when lidar sigma moves. Calculate lidar sigma; compare fused sigma with that prediction.

0.3 m
Chapter baseline
Fused sigma

Observe Return to 0.3 m. Recheck fused sigma with lidar sigma at its chapter value.

Explain A smaller sigma gives that sensor more precision weight.

Check yourself

What should you do before trusting a moved-slider result?
Answer: Predict its direction, apply the displayed relationship, keep the units, and compare the reset value with the chapter’s worked baseline.
What does this small model leave out?
Answer: Only lidar sigma moves here. The fused sigma calculation excludes field effects listed below.

Honesty note. The fused sigma result holds lidar sigma as the only changing input. It is not a field prediction.

Technical boundaries

Excluded from the “Inverse-Variance Fusion” calculation are correlated sensor errors, bias drift, non-Gaussian noise, outliers, asynchronous sampling, or model mismatch; “Inverse-Variance Fusion” therefore reports only its named fixtures.

Ada: A fused number is trustworthy only when its weights and its published uncertainty come from the same arithmetic. Let me re-run the obstacle-range example above using only its stated values, carry full precision, and round at the end.

The three measurements are camera 44.8 m (sigma 1.0 m), lidar 45.1 m (sigma 0.3 m), and radar 45.5 m (sigma 0.5 m). Precision is the inverse of variance:

  • Camera precision: 1 / 1.0^2 = 1.000000
  • Lidar precision: 1 / 0.3^2 = 1 / 0.09 = 11.111111
  • Radar precision: 1 / 0.5^2 = 1 / 0.25 = 4.000000
  • Total precision: 1.000000 + 11.111111 + 4.000000 = 16.111111, which the chapter rounds to 16.11.

Each normalized weight is a precision divided by that total:

  • Camera: 1.000000 / 16.111111 = 0.062069, rounded to 0.062.
  • Lidar: 11.111111 / 16.111111 = 0.689655, rounded to 0.690.
  • Radar: 4.000000 / 16.111111 = 0.248276, rounded to 0.248.

The fused range is the weighted sum:

  • 44.8 x 0.062069 + 45.1 x 0.689655 + 45.5 x 0.248276
  • = 2.780690 + 31.103448 + 11.296552 = 45.180690 m, rounded to 45.18 m.

The published uncertainty is the standard deviation implied by the total precision:

  • sqrt(1 / 16.111111) = sqrt(0.062069) = 0.249136 m, rounded to 0.25 m.

That 0.25 m is tighter than any single sensor’s sigma, which is exactly why the fused value looks confident. But the same arithmetic assumes the three errors are independent and current, and the freshness check in this section shows the risk: a camera frame 120 ms old on an object moving 1.6 m/s has already slipped 1.6 x 0.120 = 0.192000 m, past a 0.10 m stale budget, while the 40 ms lidar frame has slipped only 1.6 x 0.040 = 0.064000 m. The audit’s lesson is narrow: publish the 0.25 m band only after proving the inputs were independent and fresh, because otherwise the tightest number in the record is the least honest.

Every number above is taken from the chapter’s own material and re-derived step by step.