What you describe sounds a bit like demand paging of a memory-mapped file. The problem with implementing it for a 2d image is that a given rectangular region doesn't map to a contiguous area in memory. It's easy to construct a long thin image that would cause problems for a line-based demand paging strategy. For example, ten pixels high by a billion pixels wide.
Edit: skipping sections of the line to get to the region of interest is fine, I suppose, but what's really needed is a hierarchical quadtree-like organization of the storage, surely...
It is a bit like demand paging, yes. There are formats for which rectangular regions map to contiguous block of memory.
TIFF has specs for tiles, strips, subfiles, layers, etc. AFIAK, hardly anyone uses those, but they certainly exist.
JPEG is also composed of 8x8 squares and is easy to stream. The API has per-scanline read & write callbacks IIRC. But you're right; a very very wide one might be a case it can't deal with.
Edit: skipping sections of the line to get to the region of interest is fine, I suppose, but what's really needed is a hierarchical quadtree-like organization of the storage, surely...