Models of word alignment built as sequences of links have limited expressive power, but are easy to decode. Word aligners that model the alignment matrix can express arbitrary alignments, but are difficult to decode. We propose an alignment matrix model as a correction algorithm to an underlying sequence-based aligner. Then a greedy decoding algorithm enables the full expressive power of the alignment matrix formulation. Improved alignment performance is shown for all nine language pairs tested. The improved alignments also improved translation quality from Chinese to English and English to Italian. © 2011 Association for Computational Linguistics.