Backprojection of residual

An alternative method to combine the vector field of expression (15) to a scalar field is the combination of the forward and adjoint of the prediction-error operator:

The space of the output $\epsilon$ is equal the space of the input g. Unfortunately, I do not know a insightful interpretation of the procedure. If prediction-error filter is similar to a derivative operator, the combination of forward and adjoint corresponds to a sequence of two derivative operations.