How are tie points errors defined - Mean, Sigma, RMS?

The following errors for the Ground Control Points (GCPs) and Checkpoints (CPs) in photogrammetry project are calculated:

  • Mean: The average error in each direction (X,Y,Z).
  • Sigma: The standard deviation of the error in each direction (X,Y,Z).
  • RMS: The Root Mean Square error in each direction (X,Y,Z).

In order to calculate the values, tie points (GCPs and CPs) need to have measured coordinates that are compared with their calculated position. To measure the coordinates of points, typically total stations and RTK GNSS receivers are used. 

 
Note: Quality checks can be performed using Pix4Dmapper, Pix4Dmatic, Pix4Dcloud, and Pix4Dengine either directly in the software or in automatically generated quality reports. The error values are given in the project units.

Mean

For a given direction (X,Y or Z) it is defined as: Mean = μ = Σ(ei)/N

Where ei is the error of each point for the given direction.

N: The number of tie points.

The Mean Z error helps to recognize systematic errors due to bad GCP acquisition.

 
Example: If all GCPs have a systematic error of 5 cm in the Z direction, the Mean Z error will be 5cm.

Sigma

For a given direction (X,Y or Z) it is defined as:

Sigma = σ = sqrt(Σ(ei - μ)^2)/N)

Where ei is the error of each point for the given direction.

μ: The mean error for the given direction
N: The number of GCPs

Assuming the error is Gaussian, the Sigma error gives confidence intervals around the Mean error:

  • 68.2% of the points of the project will have an error of +-σ
  • 95.4% of the points of the project will have an error of +-2σ
  • 99.6% of the points of the project will have an error of +-3σ

A graph showing the Gaussian distribution.

 
Example: If all GCPs have a systematic error in Z of 5 cm, the Mean Z error will be 5cm. If the Sigma Z error is 1cm, the probability of a point to have an error in the interval [4,6] cm is 68.2%.

RMS

For a given direction (X,Y or Z) it is defined as:

RMS = sqrt(Σ(ei^2)/N)

Where ei is the error of each point for the given direction.

N: The number of GCPs

The RMS error will take into account the systematic error. If Mean error=0, the RMS error will be equal to the Sigma Z error. The comparison of the RMS error and Sigma error indicates a systematic error.

Of the 3 indicators, the RMS Error is the most representative of the error in the project since it takes into account both the mean error and the variance.

 
Example: If all GCPs have a systematic error in Z of 5 cm, the Mean Z error will be 5cm. The Sigma Z error will show the probability of the points of the project to have an error of +-1cm, +-2 cm, and +-3 cm, if σ = 1 cm (as if there was no systematic error). The RMS error will be bigger than 5 cm, as it does not remove the systematic error.

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