The estimation of a PEF with conventional methods for this example was impossible. Single scale PEF estimation can provide a good solution, but small changes in scale can create very different final solutions, especially when used as the starting guess for a nonlinear problem. Multiscale PEF estimation overcomes the variability in scales, and produces a result which is not perfect, but gives a robust result that requires less prior knowledge.
At this point, there are several avenues that can still be explored, such as examining the differences between normalized linear interpolation and binning while regridding, using weighting functions during both the multiscale PEF estimation and the nonlinear estimation, and developing better automated methods of choosing appropriate scales for the estimation.