Doctor of Philosophy(PhD) / This thesis proposes the use of the augmented system form - a generalisation of the information form representing both observations and states. In conjunction with this, this thesis proposes a novel graph representation for the estimation problem together with a graph based linear direct solving algorithm. The augmented system form is a mathematical description of the estimation problem showing the states and observations. The augmented system form allows a more general range of factorisation orders among the observations and states, which is essential for constraints and is beneficial for sparsity and numerical reasons. The proposed graph structure is a novel sparse data structure providing more symmetric access and faster traversal and modification operations than the compressed-sparse-column (CSC) sparse matrix format. The graph structure was developed as a fundamental underlying structure for the formulation of sparse estimation problems. This graph-theoretic representation replaces conventional sparse matrix representations for the estimation states, observations and their interconnections. This thesis contributes a new implementation of the indefinite LDL factorisation algorithm based entirely in the graph structure. This direct solving algorithm was developed in order to exploit the above new approaches of this thesis. The factorisation operations consist of accessing adjacencies and modifying the graph edges. The developed solving algorithm demonstrates the significant differences in the form and approach of the graph-embedded algorithm compared to a conventional matrix implementation. The contributions proposed in this thesis improve estimation methods by providing novel mathematical data structures used to represent states, observations and the sparse links between them. These offer improved flexibility and capabilities which are exploited in the solving algorithm. The contributions constitute a new framework for the development of future online and incremental solving, data association and analysis algorithms for online, large scale localisation and mapping.

Doctor of Philosophy(PhD) / This thesis proposes the use of the augmented system form - a generalisation of the information form representing both observations and states. In conjunction with this, this thesis proposes a novel graph representation for the estimation problem together with a graph based linear direct solving algorithm. The augmented system form is a mathematical description of the estimation problem showing the states and observations. The augmented system form allows a more general range of factorisation orders among the observations and states, which is essential for constraints and is beneficial for sparsity and numerical reasons. The proposed graph structure is a novel sparse data structure providing more symmetric access and faster traversal and modification operations than the compressed-sparse-column (CSC) sparse matrix format. The graph structure was developed as a fundamental underlying structure for the formulation of sparse estimation problems. This graph-theoretic representation replaces conventional sparse matrix representations for the estimation states, observations and their interconnections. This thesis contributes a new implementation of the indefinite LDL factorisation algorithm based entirely in the graph structure. This direct solving algorithm was developed in order to exploit the above new approaches of this thesis. The factorisation operations consist of accessing adjacencies and modifying the graph edges. The developed solving algorithm demonstrates the significant differences in the form and approach of the graph-embedded algorithm compared to a conventional matrix implementation. The contributions proposed in this thesis improve estimation methods by providing novel mathematical data structures used to represent states, observations and the sparse links between them. These offer improved flexibility and capabilities which are exploited in the solving algorithm. The contributions constitute a new framework for the development of future online and incremental solving, data association and analysis algorithms for online, large scale localisation and mapping.