Linear data compression methods, such as principal component analysis (PCA) or proper orthogonal decomposition (POD), have long been used to extract low-dimensional bases for manifolds describing the solutions of elliptic partial differential equations (PDEs). These methods cannot learn low-dimensional representations for systems with a large Kolmogorov N-width (KNW), including equations as benign as linear advection. This has catalyzed a surge in interest in nonlinear data compression techniques suitable for transport-dominated systems. Methods that make use of procedures from image registration, wherein distinct fields are aligned via smooth diffeomorphisms, have proven to effectively compress solution fields exhibiting localized, advecting features. Evidence of this can be seen both in studies on canonical systems, and by their successful application to larger-scale problems. This talk will explore new developments in registration methods for the compression of simulation data with a large KNW. We will present the application of both sparse and dense image registration techniques to systems under explored in existing literature. In particular, we will focus on time-dependent conservation laws with both moving shocks and traveling waves, as well as steady hypersonic aerodynamics of complex geometries. The methods implemented here build on existing works, but the modifications needed to make them more robust and scalable are novel. A variety of methods will be presented, and the talk will conclude with guidance on the suitability of each method for different applications.