  We investigate the asymptotic behavior of minimizers for the singularly
perturbed Perona-Malik functional in one dimension. In a previous study, we
have shown that blow-ups of these minimizers at a suitable scale converge to
staircase-like piecewise constant functions.
  Building upon these findings, we delve into finer scales, revealing that both
the vertical and horizontal regions of the staircase steps display cubic
polynomial behavior after appropriate rescaling.
  Our analysis hinges on identifying the dominant terms of the functional
within each regime, elucidating the mechanisms driving the observed asymptotic
behavior.
