In this thesis, we take a signal-processing approach to two research areas outside of the core of the signal processing research: geometry reconstruction and light propagation through non-uniform media.

In the first area, we consider new sampling schemes inspired by problems of surface recovery and continuous localisation. We analyse sampling at unknown locations and show signal recovery uniqueness under constraints. We also study non-synchronised sampling of vector-valued signals or fields, that arise in the problem of localisation of a smoothly moving device. We provide sufficient conditions for device localisation. To this end, we formulate the problem of rank-one sensing of full-rank matrices and develop sufficient conditions for matrix recovery when some measurements are missing.

In the second area, we study Lippmann photography, a century-old colour photography technique. We model recording and viewing of a Lippmann photograph as analysis and synthesis operators and study the invertibility of their composition. We extend our analysis to a potential method of creating digital photographs, namely internal modification of glass with femto-second lasers.