New non-linear adjustment methods for application in geodesy and related fields

• Applicant: Professor Dr.-Ing. Frank Neitzel
• Subject Area: Geodesy, Photogrammetry, Remote Sensing, Geoinformatics, Cartography
• Term: from 2013 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 242160557

A common problem in engineering or other technical sciences is that oftentimes desired target val-ues cannot be determined directly. Hence the unknown parameters need to be connected to corre-sponding measurements by applying a functional model. An example is the determination of 3D coordinates of an object based on observed distances and directions.In order to ensure accuracy and reliability always more measurements are conducted than needed for the unique determination of the unknown parameters. In such cases an adjustment problem arises which leads to the best possible result under consideration of inevitable random measurement errors.Based upon scientific efforts that have been carried out from 1980 onwards within the field of mathematical statistics not only random measurement errors but also errors in the parameters of the functional model have been considered, a case which is referred to as "Errors-In-Variables (EIV) Model".In order to solve the resulting non-linear adjustment problem the "Total Least-Squares (TLS)" approach has been developed. This approach transfers, under certain conditions, the determination of desired parameters into an eigenvalue problem for which efficient solution methods are available.This approach, however, fails in case of special structures regarding the functional and/or stochastic model. Within the proposed project new non-linear adjustment methods should be developed that enable a generalised solution of the EIV-model.The general methodology is based on setting up appropriate non-linear normal equations for gen-eralised problems in order to obtain the best possible solution regarding numerical efficiency and convergence behaviour. The new methods should also be applied for regularisation of ill-posed problems, robust parameter estimation in presence of outliers and consideration of dispersion matrices of arbitrary structure. A problem which previously has not been treated is the EIV-model with prior information which should yield to TLS collocation.These novel adjustment methods should be applied in Geodesy (e.g. for coordinate transformation), geostatistics (e.g. kriging) and computer vision (e.g. 3D surface matching). The adaptation to challenges in other subject areas should be possible due the generalised formulation of the new non-linear adjustment methods.

• DFG Programme: Research Grants