  The Image-Computing Spectral Sequence computes the homology of the image of a
finite map from the alternating homology of the multiple point spaces of the
map. A related spectral sequence was obtained by Gabrielov, Vorobjob and Zell
which computes the homology of the image of a closed map from the homology of
$k$-fold fibred products of the map. We give new proofs of these results, in
case the map can be triangulated. Thanks to work of Hardt, this holds for a
very wide range of maps, and in particular for most of the finite maps of
interest in singularity theory. The proof seems conceptually simpler and more
canonical than earlier proofs.
