Calibration of a Terrestrial Laser Scanner - Institute of Geodesy and ...

xMfr2r2cAdjustment of SphereA sphere can be described mathematically by [Bronstein and Semendjajew, 1999]x2 + y2 + z2-=0,(C.l)where x, y, z are the 3D coordinates and r is the radius. Equation (C.l) requires that the center of the sphereequals the origin of the coordinate system. Implementing an arbitrary oriented sphere,center point, the following equationis obtainedwith an undefined(x-+ (y- yMf + (z- zMf-=0. (C.2)The unknowns of the sphere in Equation (C.2) are the three coordinates of the center point xm,Vm,zmand, depending on whether the sphere is calibrated or not, the radius.The resulting equation system isunambiguous if at least three points are given by their coordinates x, y, z.If more than three 3D points aregiven, an adjustment problem can be formulated, which is either based on• an observation model: radius is known (Gauss-Markov Model), or• a mixed model: radius is unknown (Gauss-Helmert Model).A detailed mathematical description of the adjustment models is given in [Leick, 2004]. Since in surveyingthe number of observations and thus, the redundancy is often far below 1000, adjustment problemscan beeasily solved without computational problems. Considering laser scanning, the number of observations,i.e. the number of points of a point cloud, increases significantly. The number of pointscan reach severalthousands up to hundreds of thousands. Such huge observations lead to both computational and storageproblems since the matrices for solving the adjustment become too large. This necessitates the adjustmentequations to be formed in such a way that they can be solved by a computer without any difficulties.The unknowns of the adjustment problems introduced are based on the following relations [Leick, 2004]f(x,l): x= (ATPA)-1ATPl (observation model) (C.3)f(x,l): x= [AT{BTp-1B)-1A]-1AT{BTp-1B)w (mixed model) (C.4)where x is the vector of unknowns, /is the vector of observations, P denotes the weight matrix and w isthe discrepancies vector.The matrices A and B are based on linearization of the nonlinear mathematicalmodels, cf. Equation (C.2), around the known point of expansion (x0) [Leick, 2004]: