Error Estimation in Volume Calculation

 
Important: In order to understand the concepts below in this article, read first the following article about how Pix4Dmapper calculates the volume: 202559239.

Pix4Dmapper estimates the error of a volume calculation. To calculate the error of a volume, the error for each cell of the volume has to be computed first.

 

volume_pixel_3d.jpg
Figure 1. Volume of one cell

 

The altitude (Z) of a 3D point is computed with an accuracy of 1 to 3 times the GSD. The average error for the altitude of each 3D point is 1.5 times the GSD. As each cell of the volume base has a width and a height equal to the GSD, the error on the X and Y direction is estimated to be 0.

The error for one cell of the volume is given by:

error in X direction (length) = 0 cm.
error in Y direction (width) = 0 cm.
error in Z direction (height) = 1.5*GSD.

The volume error for each cell is given by:

Ei =  L* Wi  * Zei = GSD * GSD * 1.5*GSD = 1.5 * GSD3

Where: 

Ei = the volume error of one cell.
L= the length of the cell.
Wi = the width of the cell.
Zei = the error in Z direction of the cell.

 

error_pixel.jpg
Figure 2. Volume error of a cell.

 

Once the error for each cell is known, the error of the Cut, Fill and Total volume are computed by summing the error of each cell. 

Cut Volume error = EC1 + EC2 +...ECN

Where:

EC1...N = Cut volume error for cell i..N. If the cell has Fill volume, then the Cut Volume error  Eis 0.

 

Fill Volume error = EF+ EF2 +...EFN

Where:

EF1...N = Fill volume error for cell i...N. If the cell has Cut volume, then the Fill Volume Eerror is 0.

 

Total Volume error = Cut Volume error + Fill Volume error

 

The error of the volume calculations depends on different factors:

  • The GSD: The spatial resolution highly contributes to the error:
    • It determines the accuracy with which each point's coordinates have been computed and therefore determines the volume error.
    • It determines the kind of volume measurements that can be done. If the GSD is high (large value), volume calculations should not be applied for this type of object.
 
Example: In a project with GSD = 50 cm, the volume calculation of an object with 50cm height will have a very large error. 

Volume of one cell:

V= L* Wi * H= 50= 125000 cm3

and the volume error for this cell:

Ei =  L* Wi  * Zei = 187500 cm3

 
Important: Choose the appropriate GSD depending on the height of the volume to be calculated. For short objects, a small GSD value (high spatial resolution) is recommended. This can be achieved by flying lower or by using a narrow focal length.
  • The accuracy of the specific volume/object to be measured. For objects for which few 3D points have been computed, lower volume accuracy should be expected. The error estimation of Pix4Dmapper does not take this into account. Increasing the accuracy of the object s volume can be achieved by increasing the overlap.
  • If it is considered that the volume error is higher or lower than 1.5*GSD, then the formula to calculate volume error of each cell has to be adjusted.  
 
Example: For a project with GSD = 5 cm Pix4D estimates volume error of Ei = 1.5 * GSD3 = 187.5 cm3. However, if it is considered that the Zei (the error in Z direction of the cell) is 10 cm, then the volume error of each cell is equal to:
 
 
Ei = Li * Wi * Zei = GSD * GSD * 10 cm = 250 cm3.

 

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