  This article constructs simultaneous confidence bands (SCBs) for functional
parameters using the Gaussian Kinematic formula of $t$-processes (tGKF).
Although the tGKF relies on Gaussianity, we show that a central limit theorem
(CLT) for the parameter of interest is enough to obtain asymptotically precise
covering rates even for non-Gaussian processes. As a proof of concept we study
the functional signal-plus-noise model and derive a CLT for an estimator of the
Lipschitz-Killing curvatures, the only data dependent quantities in the tGKF
SCBs. Extensions to discrete sampling with additive observation noise are
discussed using scale space ideas from regression analysis. Here we provide
sufficient conditions on the processes and kernels to obtain convergence of the
functional scale space surface.
  The theoretical work is accompanied by a simulation study comparing different
methods to construct SCBs for the population mean. We show that the tGKF works
well even for small sample sizes and only a Rademacher multiplier-$t$ bootstrap
performs similarily well. For larger sample sizes the tGKF often outperforms
the bootstrap methods and is computational faster. We apply the method to
diffusion tensor imaging (DTI) fibers using a scale space approach for the
difference of population means. R code is available in our Rpackage SCBfda.
