The resolutions of acquired image and volume data are ever increasing. However, the resolutions of commodity display devices remain limited. This leads to an increasing gap between data and display resolutions. To bridge this gap, the standard approach is to employ output-sensitive operations on multi-resolution data representations. Output-sensitive operations facilitate interactive applications since their required computations are proportional only to the size of the data that is visible, i.e., the output, and not the full size of the input. Multi-resolution representations, such as image mipmaps, and volume octrees, are crucial in providing these operations direct access to any subset of the data at any resolution corresponding to the output. Despite its widespread use, this standard approach has some shortcomings in three important application areas, namely non-linear image operations, multi-resolution volume rendering, and large-scale image exploration. This dissertation presents new multi-resolution representations for large-scale images and volumes that address these shortcomings. Standard multi-resolution representations require low-pass pre-filtering for anti- aliasing. However, linear pre-filters do not commute with non-linear operations. This becomes problematic when applying non-linear operations directly to any coarse resolution levels in standard representations. Particularly, this leads to inaccurate output when applying non-linear image operations, e.g., color mapping and detail-aware filters, to multi-resolution images. Similarly, in multi-resolution volume rendering, this leads to inconsistency artifacts which manifest as erroneous differences in rendering outputs across resolution levels. To address these issues, we introduce the sparse pdf maps and sparse pdf volumes representations for large-scale images and volumes, respectively. These representations sparsely encode continuous probability density functions (pdfs) of multi-resolution pixel and voxel footprints in input images and volumes. We show that the continuous pdfs encoded in the sparse pdf map representation enable accurate multi-resolution non-linear image operations on gigapixel images. Similarly, we show that sparse pdf volumes enable more consistent multi-resolution volume rendering compared to standard approaches, on both artificial and real world large-scale volumes. The supplementary videos demonstrate our results. In the standard approach, users heavily rely on panning and zooming interactions to navigate the data within the limits of their display devices. However, panning across the whole spatial domain and zooming across all resolution levels of large-scale images to search for interesting regions is not practical. Assisted exploration techniques allow users to quickly narrow down millions to billions of possible regions to a more manageable number for further inspection. However, existing approaches are not fully user-driven because they typically already prescribe what being of interest means. To address this, we introduce the patch sets representation for large-scale images. Patches inside a patch set are grouped and encoded according to similarity via a permutohedral lattice (p-lattice) in a user-defined feature space. Fast set operations on p-lattices facilitate patch set queries that enable users to describe what is interesting. In addition, we introduce an exploration framework—GigaPatchExplorer—for patch set-based image exploration. We show that patch sets in our framework are useful for a variety of user-driven exploration tasks in gigapixel images and whole collections thereof.