# ILRS Reference Point Determination Using Close Range Photogrammetry
## 2 Mathematical Background
### Spherical Simplex Unscented Transformation  
Both the functional model of the bundle adjustment described in Section 2.1 and the functional model of the reference point determination model given in Section 2.2 are nonlinear. In these cases, the observations \(\mathbf{L}\) can be expressed as a function \(\mathbf{f}\) of the unknown parameters \(\mathbf{X}\), which motivates the use of the nonlinear Gauss-Markov model. Let \(\mathbf{\hat{x}}=\mathbf{\hat{X}}-\mathbf{X}^{0}\) the estimated correction of the unknown parameters w.r.t. the approximation values \(\mathbf{X}^{0},\mathbf{l}=\mathbf{L}-\mathbf{f}(\mathbf{X}^{0})\) the so-called reduced observation vector, \(\mathbf{A}\) the Jacobian matrix that contains the partial derivations of the linearized function w.r.t. \(\mathbf{X}^{0}\), and vector \(\mathbf{e}\sim\mathcal{N}(\mathbf{0},\mathbf{\Sigma})\) be the normally distributed observational errors. Matrix \(\mathbf{\Sigma}\) is the known positive-definite dispersion matrix, also denoted as the stochastic model. The solution of the nonlinear problem \(\mathbf{x}=\mathbf{g}(\mathbf{l})\) is derived using the first-order truncated Taylor series of \(\mathbf{g}\) as a linear substitute problem within the GMM, i.e.,  
\[\mathbf{\hat{x}}=\mathbf{J}\mathbf{l}=\left(\mathbf{A}^{\mathrm{T}}\mathbf{ \Sigma}^{-1}\mathbf{A}\right)^{-}\mathbf{A}^{\mathrm{T}}\mathbf{\Sigma}^{-1} \mathbf{l}. \tag{20}\]  
Substituting \(\mathbf{\hat{x}}=\mathbf{\tilde{x}}+\mathbf{\tilde{e}}_{\mathbf{x}}\) and \(\mathbf{l}=\mathbf{\tilde{l}}+\mathbf{\tilde{e}}\) yields  
\[\mathbf{\hat{x}}=\mathbf{\tilde{x}}+\mathbf{\tilde{e}}_{\mathbf{x}}=\mathbf{ J}(\mathbf{\tilde{l}}+\mathbf{\tilde{e}})=\mathbf{\tilde{x}}+\mathbf{J} \mathbf{\tilde{e}}, \tag{21}\]  
and the well-known property of the expectation value of linear function \(\mathrm{E}\{\mathbf{\hat{x}}\}=\mathbf{\tilde{x}}\) is obtained, i.e.,  
\[\mathrm{E}\{\mathbf{\tilde{e}}_{\mathbf{x}}\}=\mathrm{E}\{\mathbf{J}\mathbf{ \tilde{e}}\}=\mathbf{J}\,\mathrm{E}\{\mathbf{\tilde{e}}\}=\mathbf{0}. \tag{22}\]  
Here, tilde indicates true values. However, introducing the second-order truncated Taylor series of \(\mathbf{g}\) as substitute problem yields  
\[\mathbf{\tilde{e}}_{\mathbf{x}}=\mathbf{J}\mathbf{\tilde{e}}+\frac{1}{2} \Big{[}\mathbf{\tilde{e}}^{\mathrm{T}}\mathbf{H}_{\mathrm{i}}\mathbf{\tilde{ e}}\Big{]}_{i}, \tag{23}\]  
and the expectation value reads  
\[\mathrm{E}\{\mathbf{\tilde{e}}_{\mathbf{x}}\}=\frac{1}{2}[\mathrm{tr}( \mathbf{H}_{\mathrm{i}}\mathbf{\Sigma})]_{i}, \tag{24}\]  
where \(\mathbf{H}_{i}\) is the Hessian of the \(i\)th function in \(\mathbf{g}\) evaluated at \(\mathbf{\tilde{e}}\)[28; 29; 30]. For that reason, the linear substitute problem in Equation (20) is generally biased, if \(\mathbf{g}\) is nonlinear, and Equation (24) provides a second-order correction term. Analogically, one finds the expectation value for the dispersion [28; 29; 30], i.e.,  
\[\mathrm{E}\Big{\{}\mathbf{\tilde{e}}_{\mathbf{x}}\mathbf{\tilde{e}}_{\mathbf{ x}}^{\mathrm{T}}\Big{\}}=\mathbf{J}\mathbf{\Sigma}\mathbf{J}^{\mathrm{T}}+\frac{1}{2} \big{[}\mathrm{tr}(\mathbf{H}_{\mathrm{i}}\mathbf{\Sigma}\mathbf{H}_{\mathrm{i}} \mathbf{\Sigma})\big{]}_{i,j}, \tag{25}\]  
where the first term corresponds to the well-known linear propagation of errors, and the second term is the second-order correction of the dispersion. The use of the second or higher-order Taylor series expansion becomes challenging, if the nonlinear function of the application results from a multidimensional and complex analysis procedure, such as for the bundle adjustment and the reference point determination. To evaluate the influence of the nonlinear function onto the estimates, usually a Monte Carlo simulation (MCS) is alternatively applied ([16] p. 654 ff). In contrast to the truncated Taylor series, which approximates the nonlinear problem by a simplified substitute problem, the MCS approximates the probability distribution [31].