  We generalize the bridge between analysis and synthesis estimators by Elad,
Milanfar and Rubinstein (2007) to rank deficient cases. This is a starting
point for the study of the connection between analysis and synthesis for total
variation regularized estimators. In particular, the case of first order total
variation regularized estimators over general graphs and their synthesis form
are studied.
  We give a definition of the discrete graph derivative operator based on the
notion of line graph and provide examples of the synthesis form of
$k^{\text{th}}$ order total variation regularized estimators over a range of
graphs.
