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The precision of photogrammetric models

PoG 16, G.H. Ligterink, The precision of photogrammetric models

G.H. Ligterink

Publications on Geodesy 16 (Vol. 4 Nr. 3). Delft, 1972. 86 pagina's.
ISBN-13: 978 90 6132 216 0. ISBN-10: 90 6132 216 2.


The influence of random observation errors on the machine coordinates of model points can be represented by a covariance matrix. In this investigation the covariance matrix of a limited number of model points has been determined in two different ways:

1. by executing repeated measurements of points in photogrammetric models; in some cases these repeated measurements are combined with repeated relative orientations or with repeated relative and inner orientations; from the series of these measurements the estimated covariance matrices (σ2) have been determined.

2. by writing the machine coordinates as functions of the initial observations, e.g. X- and y-parallaxes; by means of the standard deviations of these initial observations, and applying the law of propagation of errors, the covariance matrices (σ2) have been computed.

1. Eight experiments have been executed to determine the (σ2). The following observations were repeated 20 times:

  • for 2 models: the measuring of the coordinates of 8 model points (after the inner and relative orientation was done once).
  • for 3 models: the relative orientation and the measuring of the coordinates of 8 model points (after inner orientation was done once).
  • for 3 models: the inner orientation, the relative orientation and the measuring of the coordinates of 8 model points.

The estimated covariance matrix (σ2) is computed from the 20 repeated observations per experiment. This makes 8 full-matrices of 24 X 24 elements for the 8 X 3 coordinates of the 8 model points.

2. The 8 covariance matrices (σ2) were computed from the standard deviations by applying the law of propagation of errors. The observations are divided into three groups:

  • the measuring of the coordinates of a model point
  • the relative orientation
  • the inner orientation

Sub-matrices of (σ2) and (σ2) represent point standard ellipsoids and relative standard ellipsoids. The shape and position of these ellipsoids are represented and compared in a large number of diagrams showing the projections of the ellipsoids on three perpendicular planes. These projections are standard ellipses and relative standard ellipses.
Interesting correlations are demonstrated, both between coordinates of a single point and between coordinates of different points.

In order to be able to extrapolate these results further investigations will be necessary for better information about the factors which influence the standard deviations of the individual observations.

The structure of the covariance matrix of the coordinates of model points is essential for studies of precision and accuracy in all procedures which use the photogrammetric model as basic unit.


  • Summary  4
  1. Introduction  5
  2. Errors due to measuring of a model point  7
  3. Errors due to relative orientation  22
  4. Errors due to inner orientation  44
  5. Recapitulation and conclusions  64
  • Appendix 1  67
  • Appendix 2  70
  • Appendix 3  78
  • References  86
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