Figure 1 – The principal of the laser scanner operation

         There are two methods of determining distances. The first one is called Time-of-Flight (TOF). It is based on the measuring of the laser impulse action time. This impulse is generated by the sensor, reflected by the object and received by the sensing element.
In this case the distance is calculated under the formula:

(1)

         The second method is based on the phase difference measuring. The sensor radiates the harmonic vibrations of the set wave length, at this moment the initial phase becomes determined, then the signal is reflected by the object and received by the sensing element, at this moment the final phase is determined. The distance is calculated under the formula:

(2)

          By the angle of the mirrors turn at the observation period as well as by the distance measured the processor calculates each point coordinates under the formula:

(3)

Urgency and the purpose

In the most cases in the result of the laser scanning the object data are presented in the view of the rectangular Cartesian coordinates, whose system forms the so-called “cloud” of points.
         The “cloud” processing presupposes a description of the real objects via mathematical models. These geometric data are the basic material for further processing. With this purpose there are required different algorithms allowing to transform certain objects into the geometric primitives from the “clouds” of points.
         At the present time there is no algorithm that lets determine geometric primitives definitely and automatically, i.e. without human being participation. That is why the purpose of my research is study and development of the algorithms allowing to do it.
         The simplest geometric element that most frequently occurs and relatively easy models is a flat surface. More difficult geometric objects can be described further on basis of planes.

The review of conditions of researches and workings out on the theme

         There are different algorithms of the planes determination. In particular, Andreas Rietdorf in his paper “Automatisierte Auswertung und Kalibrierung von scannenden Messsystemen mit tachzmetrischem Messprinzip” (Munhen 2005) [1] as a criterion for the decision, if the points lie on the plane, fulfilled the determinant calculation. There are correspondingly analyzed two adjacent triangles that rather precisely determine the plane.
         According to his paper, to each face of the triangle there should be found two further faces that have a common point with the first face. Thus, there exist 4 points determining the parallelepiped. If the mixed product of three vectors made by four points equals zero, then these four points lie on the one and the same plane. Thus, belonging of any point to the plane is determined from the equality to the zero of the determinant made by the three vectors:

         Let us examine another algorithm.
         Let us assume that there is an X matrix, consisting of the points coordinates in the XYZ space. Let us check if these points belong to the plane. For this we transform them in such a way that they wholly lie on the XY plane. Then for all points belonging to the plane, a z coordinate will be equal 0. In the event that any point does not belong to the plane, then the X matrix will be inclined onto a some angle. The point with the maximum z coordinate modulus will be false and needs to be excluded from further calculations. The calculations are made until

Figure 2 - Transformation of the plane points coordinates, here: – coordinates system – this is a system where the plane lies; x, ó, z coordinates system – this is a system where the points lies in the xy plane.

         Let us deduct from each coordinate of the X matrix the average values () of each coordinates. We will receive an A matrix, all points of which will be spread towards the center of gravity:

(4)

         By multiplying the A matrix by AT, we will receive a symmetric N matrix. This is required to define eigenvectors (U) of the A matrix.
         Let us explain the necessity of the proper vectors determination. The task of the coordinates transformation can be reduced to the basis transformation. For this purpose each point of the A matrix should be considered as a vector. Then, according to the definition, a set of the space vectors is called an operator. The basis transformation results into two different matrixes corresponding to one and the same operator, i.e. similar ones. These similar matrixes are connected between each other by the following ratio:

(5)

If the eigenvalues of the A matrix are pairwise different, then eigenvectors are orthogonal and linearly independent and, consequently, they can be used as a basis. Then formula (6) is as follows:

(6)

Planned results

         The result of this research will be the creation of the program in Delphi environment, allowing to sort out geometric primitives. The final conclusions concerning the research results are planned to be drawn by December, 2008. At the present moment it is possible to say that the above described algorithm with the use of the transformed coordinates analysis gives a greater reliability of the results received than a similar algorithm with the use of the determinate.

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