As lots of people already pointed out: there is more going on than just a smart upsampling technique.

See section 4 of the paper - they train on a set of 3d models from some 'ShapeNetCore' dataset, from which they generate sample inputs (renderings of the model with randomized viewpoint and lighting) and corresponding target outputs (voxelized model).

They train specialized networks for different classes of objects 'aeroplanes, chairs and car', so reconstructing on all classes at the same time probably still has some issues.

An interesting point about their use of this coarse-to-fine progression that the article omits: they do the same trick for training their net - first train to predict the coarse voxels, and when those work start predicting the next level.