Characterising and identifying hidden metallic objects has important applications in: 1. Security screening at border controls, public events and transport hubs and detecting threats such as knives and firearms. Knife crime is at its highest recorded level in 10 years in the UK and firearm offences are seeing an increase, many being smuggled into the UK from Europe via ferry routes. 2. The identification of the 110 million current unexploded anti-personal landmines, unexplored ordnance (UXO) such as mortar and cluster bombs in areas of former conflicts. Using current techniques, it has been estimated to take over 1000 years to clear them so that the land can be returned to normal use. 3. Improvements to scrap sorting, which has the potential to improve the recycling of valuable metals. Going beyond the planned work in the EPSRC funded project EP/R002134/1, a new form of object charactization called a generalised magnetic polarizabilty tensor (GMPT) has been established for highly conducting objects, offering significant advantages over existing small object characterisations. Specifically: 1. Offering the possibility to better discriminate between objects and, hence, the potential to achieve better classification and identification of objects. 2. Being able to use the non-uniform fields generated by a metal detector in a creative way, overcoming the assumption that field is uniform over the object in current small object characterisations, and, hence, achieving better 1. There is much still to learn about GMPTs and there are open mathematical questions about the appropriate choices of invariants (required for object classification) and using representation theory to determine symmetry groups for symmetrical metallic objects (such as mortar bombs). This project will seek to address these questions, which are required before a revolutionary new type of metal detector offering substantial improvements in the above applications can be built.