Nonlinear method for determining external orientation elements of digital images obtained from drone

We have suggested the method of application of a direct solving of the systems of nonlinear equations for finding the elements of external orientation (EEO) to perform aerial photography by unmanned aerial vehicles (UAVs). 

The elements of external orientation functions search for the minimum of the function $F$, which is the sum of squares of coordinate differences on the image and is calculated by the measured coordinates on the ground, or the minimum of the function $G$, which is constructed using co-linearity and is the sum of squares of differences of the given coordinates $X_i$, $Y_i$, $Z_i$ $(i=1,2,\ldots,n)$ on the ground and those which were calculated by the values $x_i$, $y_i$ $(i=1,2,\ldots,n)$ measured on the image.  In contrast to the classical approach, the choice of such a type of function is due to the possibility of implementing the algorithm using mathematical packages.  Since some of the unknown coordinates $X_i$, $Y_i$, $Z_i$ (the origin of the coordinate system is the center of projection) are included in the function $G$ as arguments linearly, fulfillment of the conditions of the minimum of this function (equality of partial zero derivatives) in this case is simpler.  This allows us to determine them through the angular elements of the EEO, which reduces the system of six equations to the system of three equations, being dependent on the angular elements.  The function $G$ is differentiated with respect to the variables dependent on the angular elements to obtain the three other equations.  The obtained in this way system of equations is solved by the parameter variation method and gives us the solution of the required EEOs with a given accuracy. 

The proposed algorithm gives us a real opportunity to clarify the values of EEO, moreover, the linear EEOs are determined with maximum accuracy, that makes it possible to increase the accuracy of the spatial coordinates of the points of the terrain. 

The application of digital image processing from UAVs will significantly extend the range of implementation of aerial photography from UAVs to solve a variety of topographic, cadastral and engineering problems.

The proposed technique was tested on the relevant materials of aerial photography from UAVs at control points, which made it possible to confirm the optimality of the technique.

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Mathematical Modeling and Computing, Vol. 9, No. 3, pp. 627–636 (2022)