This response has just been overwhelming. Thank you all!
Very briefly, the multires issues raised by jsampsonPC and others basically revolve around the question of how an image can be represented in a way that allows the whole thing to be accessed at very low resolution, or a small part to be accessed at high resolution,
or anything in between. Also, each such access needs to involve limited server disk IO and limited processing on the client side. The ideas first used to implement this efficiently and elegantly were developed in part by my old thesis advisor, Ingrid Daubechies.
Do searches on "wavelets", "wavelet image compression", and "JPEG2000" to learn more. We don't use JPEG2000 (or, in fact, wavelets) in Photosynth, although when Seadragon was an independent company we did use JPEG2000. The basic ideas are similar, though.
The Seadragon collection model generalizes these multiresolution concepts to collections of images (or other, non-image visual content, like text and vector maps), not just single images. Again, the design requirement was to allow massive collections of massive
images to be opened remotely without too much work either for the server or the client.
Astronomy and microscopy applications: YES. Among many others. We hope to get to a point where there's an open platform to develop on, so that these ideas and applications can be developed the right way by the people who really understand what should be made