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This article constructs simultaneous confidence bands (SCBs) for functionalparameters 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 precisecovering rates even for non-Gaussian processes. As a proof of concept we studythe functional signal-plus-noise model and derive a CLT for an estimator of theLipschitz-Killing curvatures, the only data dependent quantities in the tGKFSCBs. Extensions to discrete sampling with additive observation noise arediscussed using scale space ideas from regression analysis. Here we providesufficient conditions on the processes and kernels to obtain convergence of thefunctional scale space surface.  The theoretical work is accompanied by a simulation study comparing differentmethods to construct SCBs for the population mean. We show that the tGKF workswell even for small sample sizes and only a Rademacher multiplier-$t$ bootstrapperforms similarily well. For larger sample sizes the tGKF often outperformsthe bootstrap methods and is computational faster. We apply the method todiffusion tensor imaging (DTI) fibers using a scale space approach for thedifference of population means. R code is available in our Rpackage SCBfda.

Simultaneous Confidence Bands for Functional Data Using the Gaussian Kinematic Formula