Math Bridge: Why does camera range lose weight with distance?

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Why does camera range lose weight with distance?

Trace each sensor sigma to its physical measurement before inverse-variance fusion combines them.

Data Dora, the guideData Dora guides
The one targetExplain why camera uncertainty grows with range while timing-based uncertainty need not.
The chapter caseCamera 1.00 m, lidar 0.300 m, radar 0.500 m near 44.8 m.
What it buys youFusion weights with a physical audit trail.

A field team faces an unresolved physical question: Why does camera range lose weight with distance? They must answer it before changing target range on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is target range. The middle card applies this page's relationship. The green card is lidar timing sigma. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Target range changes lidar timing sigma An input card leads through the page relationship to the lidar timing sigma result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Stereo sensitivity contains r², so distance amplifies pixel uncertainty. Lidar timing and the fixed radar bandwidth examples remain linear teaching limits.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for target range is 44.8.

  2. 2

    Name the relationship. Lidar σ=0.300 m → timing σ=2.00 ns; radar B=300 MHz → ΔR=0.500 m Stereo f=1,000 px, B=0.300 m, σd=0.150 px: σcamera(44.8 m)=1.00 m; σcamera(89.6 m)=4.02 m

  3. 3

    Substitute the chapter fixture. Set target range to 44.8. The page ledger gives lidar timing sigma as 2.0 ns.

  4. 4

    Read the result. Keep ns beside the value. Use it only inside the technical boundary on this page.

Predict, then change target range

Try Predict the direction of lidar timing sigma. Move one control, calculate, then check your prediction.

44.8
Chapter baseline
Lidar timing sigma

Observe Stereo sensitivity contains r², so distance amplifies pixel uncertainty. Lidar timing and the fixed radar bandwidth examples remain linear teaching limits. Reset the control to 44.8 and compare lidar timing sigma.

Explain Only target range moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only target range moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only target range. It replays the chapter fixture; it is not a field prediction.

1. Start with what sigma means

Sigma is an uncertainty scale, not a confidence vote. Lidar and radar infer range from travel time or bandwidth; stereo infers range from pixel disparity. Their errors follow different laws.

Data Dora: Before weighting a sensor, ask which measurement its sigma came from.

2. Name every algebra move

1

Undo round tripr=ct/2, so σt=2σr/c.

2

Invert radar bandwidthΔR=c/(2Bsweep).

3

Invert stereo disparityr=fB/d.

4

Propagate pixel errorσcamera=σd r²/(fB).

5

Build precision weightsw=1/σ² and σfused=√[1/Σw].

3. Reproduce the chapter case

Lidar σ=0.300 m → timing σ=2.00 ns; radar B=300 MHz → ΔR=0.500 m
Stereo f=1,000 px, B=0.300 m, σd=0.150 px: σcamera(44.8 m)=1.00 m; σcamera(89.6 m)=4.02 m

At 44.8 m, inverse-variance weights are about 11.11 for lidar, 4.00 for radar, and 0.99 for camera, with an ideal fused sigma of 0.249 m.

4. Try the target range

TryMove the target farther away and watch the same 0.150-pixel mismatch grow into a larger camera-range sigma.

Target range
Lidar timing sigma
Lidar weight
Radar resolution
Radar weight
Camera sigma
Camera weight
Doubled range
Doubled camera sigma
Ideal fused sigma

ObserveAt 44.8 m, camera sigma is 1.004 m and its weight is about 0.99. At 89.6 m the same pixel precision gives 4.014 m—four times the sigma from twice the range.

ExplainStereo sensitivity contains r², so distance amplifies pixel uncertainty. Lidar timing and the fixed radar bandwidth examples remain linear teaching limits.

Technical boundaries.

This compact engine shows ideal independent precision; it is not a fusion safety case.

Camera
Fixed focal length, baseline, and disparity sigma omit texture, occlusion, calibration, and lighting
Radar/lidar
Resolution and timing sigma omit target reflectivity, clutter, multipath, and estimator bias
Fusion
Inverse variance assumes aligned, unbiased, independent measurements of the same state

Validate timing, covariance, bias, operating conditions, and failure correlation in the application.

5. Decide whether weights are allowed

Only combine measurements after confirming common state, units, timestamp, coordinate frame, calibration, and an uncertainty estimate valid at the current range and conditions.

6. Keep the fusion record

Record sensor model, range, bandwidth or timing precision, focal length, baseline, disparity error, calibration, timestamps, coordinate transform, covariance, biases, operating condition, owner, and retest trigger.

7. Check yourself

Why does 0.300 m lidar sigma imply 2.00 ns?
Answer: Round-trip timing gives σt=2σr/c=0.600/(3.00×10⁸)=2.00 ns.
Why does doubling stereo range quadruple sigma?
Answer: With fixed optics and pixel uncertainty, σcamera is proportional to r², so (2r)²=4r².
Can the smallest sigma always receive the largest weight?
Answer: Only when sigmas are valid, aligned, unbiased, independent uncertainty estimates for the same state.
Honesty boundary.

The sigmas and 44.8 m range come from the chapter; sensor design values are labelled catalog-typical.

0.300/0.500/1.00 m
The chapter's lidar, radar, and camera uncertainty example
44.8 m
The chapter's fused range case
300 MHz, 1,000 px, 0.300 m, 0.150 px
Catalog-typical teaching radar and stereo parameters

Correct, not complete: application evidence determines whether these uncertainty models hold.

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