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Perception · Theory · P1

P1-T5 — Noise, Uncertainty, Occlusion and Ambiguity

Noise averages out, bias does not, occlusion hides things, and ambiguity lets several answers fit.

  • Lesson 5 of 6
  • Module 1: Perception Foundations
  • Theory
  • About 9 minutes to read

Every measurement is wrong in some way, and the way matters. Random noise scatters the readings and averages out. A bias shifts all readings the same way and does not. Outliers and dropouts are rare large errors. Occlusion hides part of the scene entirely. Ambiguity means that more than one scene fits the data. A perception system that treats them all as “noise” will apply the wrong remedy. This lesson separates the kinds, shows how a small angular error becomes a large position error at range, and states what each one means for the stages that follow.

Learning objectives

After this lesson, you should be able to:

  • Classify a measurement error as noise, bias, outlier, dropout, occlusion or ambiguity.
  • Explain why averaging reduces noise but not bias, and compute the effect of averaging n readings.
  • Propagate angular noise into lateral position error as a function of range.
  • State what a missing measurement does and does not imply about the scene.

The idea: different errors, different remedies

Weigh yourself five times on a bathroom scale. The readings differ by a few hundred grams: that is noise. Averaging the five readings gives a better estimate. Now suppose the scale is a kilogram off. Every reading is wrong by the same amount; averaging a hundred readings gives you a very precise wrong answer. That is bias. If the scale occasionally shows 300 kg, that is an outlier; if it sometimes shows nothing, a dropout. If a bag you forgot about is hanging on the scale, that is a true measurement of something you did not intend to measure, which is closer to ambiguity: the scale cannot tell you your weight from your weight plus the bag.

KindWhat it looks likeAveraging helps?Typical remedy
Random noiseSmall scatter around the true value, different each timeYes: error falls as 1/√nFilter, average, model the variance
Systematic biasEvery reading shifted the same wayNoCalibrate against a reference
OutlierRare, large, wrong valuesMakes it worse (mean) unless a robust statistic is usedReject with a robust rule, use the median, robust loss
DropoutNo measurement for a sample or a directionNot applicableTreat as unknown; handle explicitly
OcclusionPart of the scene cannot be seen at allNoMove, use another sensor, remember what was seen
AmbiguityThe data fit more than one explanationNoMore data, more views, prior knowledge
Two failures of a measurementLeft: a plot of lateral position error against range, three straight lines for angular noise of 0.2, 0.5 and 1.0 degrees, steeper for larger noise. Right: a top view where a sensor looks past a box; a second smaller object stands in the shadow behind it and is not measured.01020304000.10.20.30.4range r (m)lateral error (m) 0.2°0.5°1.0° noise: error grows with rangeocclusion: not measured is not absent sensor box hidden the box casts a shadow: nothing behind it is measured
Left: for a fixed angular noise, the lateral error grows linearly with range. Right: a box hides what is behind it; the lack of a measurement behind the box is not evidence of emptiness.

Name the error before choosing the remedy. A filter for noise does nothing for bias, and an outlier rule applied to occluded data can delete the only evidence you have.

The mathematics: variance, bias and propagation

Model a reading as the true value plus a bias plus random noise:

a simple measurement error model
zk=x+b+nkz_{k}=x+b+n_{k}
nk∼N(0, σ2)n_{k}\sim N(0,\,\sigma ^{2})
zk
the k-th reading.
x
the true value being measured.
b
the bias: a constant error present in every reading.
nk
random noise on reading k, here Gaussian with zero mean and standard deviation σ, independent between readings.

The mean of n readings is then

averaging n readings
mean=x+b+(1n) Σ nk\text{mean}=x+b+(\frac{1}{n})\,\Sigma \,n_{k}
error=b±σn\text{error}=b\pm \frac{\sigma }{\sqrt{n}}

The random part shrinks as 1/√n: to halve it you need four times as many readings. The bias is untouched. After 10 000 readings with σ = 3 cm, the random part is 0.03 cm, and a 10 cm bias is still a 10 cm error.

How noise in the inputs becomes noise in the output

Most perception outputs are computed from measurements by a function, so measurement noise is carried through it. For a function y = f(x) of one noisy input with standard deviation σx, to first order,

first-order error propagation
σy≈∣f′(x)∣σx\sigma _{y}\approx \lvert f^{\prime}(x)\rvert \sigma _{x}

For several independent inputs the contributions add as squares: σy2 ≈ Σ (∂f/∂xi)2 σi2. This is the basis of covariance propagation in estimation.

A range sensor reports range r and bearing φ; the lateral position (perpendicular to the beam) is y = r sin φ. For small angular noise σφ and a point near the beam axis,

position error of a range-bearing measurement
σlateral≈r σϕ\sigma _{\text{lateral}}\approx r\,\sigma _{\phi }
σalong≈σr\sigma _{\text{along}}\approx \sigma _{r}
σφ
bearing noise in radians (convert from degrees by multiplying by π/180).
σr
range noise in metres, often nearly constant with range for a LiDAR.
σlateral
position error perpendicular to the beam: it grows in proportion to range.
σalong
position error along the beam.

The two components differ, so the error region is not a circle but an ellipse that is long across the beam and short along it at large range. A planner that assumes a circular uncertainty will be wrong about which direction matters.

Occlusion and ambiguity are not noise

Noise has a distribution that you can write down. Occlusion and ambiguity do not fit that picture. When a surface is occluded, the sensor reports nothing about it, and the information is absent, not noisy. When the data are ambiguous, the likelihood p(z | x) is high for several states x; a second measurement (another view, another time) or a prior is needed to choose between them.

A worked example: how far off is the object at range?

A 2D LiDAR has angular noise σφ = 0.25° = 0.00436 rad and range noise σr = 3 cm. It sees a pole at two ranges.

Range rLateral error r σφAlong-beam errorWhich dominates?
2 m2 × 0.00436 = 0.9 cm3 cmRange noise
10 m10 × 0.00436 = 4.4 cm3 cmLateral error
40 m40 × 0.00436 = 17.5 cm3 cmLateral error (6 times larger)

At 40 m, the pole’s position is uncertain by 17.5 cm sideways and only 3 cm along the beam. If a robot wants to pass 30 cm from a pole at that range, the uncertainty is more than half the margin, and the planner needs to know it.

Averaging and its limits

Suppose the robot averages 25 scans of a stationary wall at 6 m with σr = 3 cm. The random error of the mean is 3/√25 = 0.6 cm. If the sensor has a 4 cm bias (a slightly wrong scale factor in the electronics), the result is 6.04 m ± 0.006 m: very precise, and 4 cm wrong. Precision (small scatter) and accuracy (closeness to the truth) are different. A calibration against a known reference, such as a target at a surveyed 6.000 m, finds the bias; no amount of averaging does.

What a missing measurement tells you

Consider the right-hand diagram: the sensor sees a box, and behind it a smaller object. There is no measurement at all of the second object. If the robot builds a point cloud and then asks “is anything behind the box?”, the answer from the data is “unknown”, not “no”. Treating the shadow as free space is a classic cause of collisions in cluttered scenes, and the same logic applies to a dark object that returns no light (Lab P1-L1).

Ambiguity has a visual version. A white wall seen by a camera gives no way to estimate its distance; a window and a photograph of a window look alike; and a repeating pattern, such as a fence or a row of identical shelves, matches itself at several offsets. In each case the correct answer is not “pick one” but “more than one fits, so collect more information”, and Module 4 builds methods that notice this.

Engineering implications and limitations

Characterise each sensor’s errors separately. Measure noise by recording a static scene (the scatter gives σ), bias by comparing with a reference (the mean error), and dropout rate by counting missing values. Datasheets give nominal values under ideal conditions; the field gives different ones.

Make the uncertainty directional. Report a covariance, not a single σ, when errors differ by direction, as they do for range-bearing sensors and for cameras (large along the viewing ray). Downstream filters and planners rely on it.

Use robust statistics by default. Median instead of mean, robust loss functions (Huber, Cauchy) instead of squared error, and rejection rules for gating. Real data contain outliers; assume them.

Represent unknown explicitly. Distinguish free, occupied and unknown. A planner that treats unknown as free will drive into what it could not see. A planner that treats unknown as occupied will refuse to move. The correct treatment depends on the task and on how long the region has been unobserved.

Plan for occlusion. Move to see behind, combine sensors with different viewpoints, and remember recently seen objects for a short time so that an object that is momentarily hidden does not vanish (Module 7).

Report ambiguity. When several explanations fit, return them with their scores instead of the top one only. A second-best hypothesis with 40% is information a safety layer can use.

Common misconceptions

“Noise and error are the same thing.”

Noise is the random part. Bias, outliers and occlusion are errors with different causes and need different treatment.

“More samples always make the result more accurate.”

They make it more precise. Accuracy also needs the absence of bias, which only calibration against a reference can establish.

“The mean is the safest summary.”

One outlier of 3 m among 80 readings moves the mean by about 4 cm. The median ignores it.

“If the sensor shows nothing there, there is nothing there.”

Absence of measurement can mean free space, a surface that does not reflect, an occluded region, or a dropout. The sensor cannot tell you which.

Engineering takeaways

  1. Classify errors: random noise, bias, outliers, dropouts, occlusion and ambiguity each have different causes and different remedies.
  2. Averaging n readings shrinks random error as σ/√n and leaves bias unchanged; accuracy needs calibration against a reference.
  3. First-order propagation σy ≈ |f′| σx carries input noise to outputs; a range-bearing measurement has lateral error r·σφ that grows with range.
  4. Uncertainty is directional: report covariance, not one number, when error differs by direction.
  5. Treat missing data as unknown, plan for occlusion, and report ambiguity instead of hiding it.

References and further reading

Standard references on measurement uncertainty; the sensor numbers in the example are illustrative.

  • Thrun, S., Burgard, W. and Fox, D. (2005). Probabilistic Robotics. MIT Press: sensor noise models, range-bearing measurement models and uncertainty.
  • Barfoot, T. D. (2017). State Estimation for Robotics. Cambridge University Press: propagation of uncertainty and robust estimation.
  • Joint Committee for Guides in Metrology (2008). Evaluation of measurement data: Guide to the expression of uncertainty in measurement (GUM): the standard treatment of random and systematic error and propagation.

Knowledge check

Three conceptual questions. Write an answer, then reveal the explanation. Your answers stay in your browser.

Question 1

A sensor has σ = 2 cm. How many independent readings are needed to bring the random error of the mean down to 0.5 cm, and does this fix a 5 cm bias?

Explanation

σ/√n = 0.5 cm gives √n = 4, so n = 16. It does not touch the 5 cm bias: the mean would be 5 cm off, very precisely. The bias needs a calibration reference.

Question 2

A LiDAR has bearing noise 0.5° (0.0087 rad). What is the lateral error of a point at 20 m, and why might the along-beam error be less important there?

Explanation

Lateral error ≈ 20 × 0.0087 = 0.17 m. Range noise is usually a few centimetres and nearly independent of range, so at 20 m the lateral error is several times larger than the along-beam error and dominates the position uncertainty.

Question 3

A robot’s grid has a block of empty cells behind a parked van. The planner routes through them and meets a pedestrian. What was wrong with the representation?

Explanation

The cells were marked as free when they were only unobserved: the van occluded them. The representation lacked an “unknown” state (or the planner treated it as free). Occluded regions must be unknown, and the plan should slow down or reveal them before committing.

What’s next

Next: how to evaluate a perception system, which means choosing metrics, data and tests that reflect what the robot needs.

Practice this lesson

Find the Misleading Measurements

Identify noisy, missing and misleading measurements in a log and decide which to trust.

Next lesson

P1-T6 — How to Evaluate a Perception System

Choose metrics and test data that match what the robot needs, and recognise misleading evaluations.

Nothing below is required to finish this lesson.