Abstract
The following article presents a new isometric transformation algorithm based on the transformation in the newly normed Hilbert type space. The presented method is based on so-called virtual translations, already known in advance, of two relative oblique orthogonal coordinate systems—interior and exterior orientation of sensors—to a common, known in both systems, point. Each of the systems is translated along its axis (the systems have common origins) and at the same time the angular relative orientation of both coordinate systems is constant. The translation of both coordinate systems is defined by the spatial norm determining the length of vectors in the new Hilbert type space. As such, the displacement of two relative oblique orthogonal systems is reduced to zero. This makes it possible to directly calculate the rotation matrix of the sensor. The next and final step is the return translation of the system along an already known track. The method can be used for big rotation angles. The method was verified in laboratory conditions for the test data set and measurement data (field data). The accuracy of the results in the laboratory test is on the level of 10−6 of the input data. This confirmed the correctness of the assumed calculation method. The method is a further development of the author’s 2017 Total Free Station (TFS) transformation to several centroids in Hilbert type space. This is the reason why the method is called Multi-Centroid Isometric Transformation—MCIT. MCIT is very fast and enables, by reducing to zero the translation of two relative oblique orthogonal coordinate systems, direct calculation of the exterior orientation of the sensors.
Highlights
To describe the position of material points in three-dimensional Euclidean space, orthogonal coordinate systems are usually used
The remainder of this paper propose an algorithm based on the use of three centroids and the generation of the new space of points in which the following elements will be calculated: change scale coefficient, exterior orientation of sensor: rotation matrix and coordinates of sensor expressed in external coordinates system
The method is a further development of the Total Free Station (TFS) transformation [15] to several centroids in Hilbert type space
Summary
To describe the position of material points in three-dimensional Euclidean space, orthogonal coordinate systems are usually used. The remainder of this paper propose an algorithm based on the use of three (or more) centroids and the generation of the new space of points in which the following elements will be calculated: change scale coefficient (checking the similarity transformation), exterior orientation of sensor: rotation matrix and coordinates of sensor expressed in external coordinates system. The method is a further development of the Total Free Station (TFS) transformation [15] to several centroids in Hilbert type space It can be applied for local, precise surveying as well as for geodynamic, close range photogrammetry and engineering purposes, especially for non-stable places (e.g., on floating vessels) where levelling of an instrument may not be possible and it is necessary to determine the exterior orientation of sensors. This method is suitable for sensors which can measure angles and/or distances like Total Station, laser scanners (e.g., TLS—Terrestrial Laser Scanner, ALS—Airborne Laser Scanner), echo-sounders and can be applied to metric and non-metric cameras for photogrammetric purposes
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