  The relation between two Morse functions defined on a common domain can be
studied in terms of their Jacobi set. The Jacobi set contains points in the
domain where the gradients of the functions are aligned. Both the Jacobi set
itself as well as the segmentation of the domain it induces have shown to be
useful in various applications. Unfortunately, in practice functions often
contain noise and discretization artifacts causing their Jacobi set to become
unmanageably large and complex. While there exist techniques to simplify Jacobi
sets, these are unsuitable for most applications as they lack fine-grained
control over the process and heavily restrict the type of simplifications
possible.
  In this paper, we introduce a new framework that generalizes critical point
cancellations in scalar functions to Jacobi sets in two dimensions. We focus on
simplifications that can be realized by smooth approximations of the
corresponding functions and show how this implies simultaneously simplifying
contiguous subsets of the Jacobi set. These extended cancellations form the
atomic operations in our framework, and we introduce an algorithm to
successively cancel subsets of the Jacobi set with minimal modifications
according to some user-defined metric. We prove that the algorithm is correct
and terminates only once no more local, smooth and consistent simplifications
are possible. We disprove a previous claim on the minimal Jacobi set for
manifolds with arbitrary genus and show that for simply connected domains, our
algorithm reduces a given Jacobi set to its simplest configuration.
