[afwaller]

Sometimes losing that information is very useful however, to arrive at a representation that has a specific meaning.

For example, an x-ray computed tomography (CT) image volume in 3D may be projected into various 2D synthetic planar x-ray projections (digitally reconstructed radiographs).

There are countless situations where projections are very useful.

[unsrsly]

In fact, the 3D CT image itself is reconstructed from projections! See https://en.wikipedia.org/wiki/Tomographic_reconstruction

[thaumasiotes]

Yes, but this is a case where you have three dimensions in the input and also three dimensions in the output. (You're assembling the 3D image from different 2D slices with different depth coordinates.)

The third dimension is discretized while the other two are continuous; the reconstruction consists of smoothing out the third dimension.

[unsrsly]

In the real world, all of the dimensions are discretized. The input dimensions are theta (gantry angle), n (detector channel), and z (table position). The output dimensions are x, y, z. For a fixed z, the plot of projection intensity as a function of theta and n is called a "sinogram" (which is indeed a 2D space) which gets reconstructed to form an image (also 2D). It is true that there are three dimensions of raw data and three dimensions of reconstructed image. However, due to various tricks with reconstruction models, the total number of samples does not have to be the same in the raw data and in the output. As a result you can see recon models employing nonsquare matrices. For more information, you can read about methods like iterative reconstruction, compressed sensing, and differentiated backprojection. This description is adequate for axial tomography, whereas other geometries like cone beam tomography are more complicated (see FDK method).

[thaumasiotes]

I don't disagree, but I think my comment helps to illustrate a reason why square matrices really are more special than other matrices.