Precise FWER Control for Gaussian Related Fields: Riding the SuRF to continuous land -- Part 1

The Gaussian Kinematic Formula (GKF) is a powerful and computationally efficient tool to perform statistical inference on random fields and became a well-established tool in the analysis of neuroimaging data. Using realistic error models, recent articles show that GKF based methods for \emph{voxelwise inference} lead to conservative control of the familywise error rate (FWER) and for cluster-size inference lead to inflated false positive rates. In this series of articles we identify and resolve the main causes of these shortcomings in the traditional usage of the GKF for voxelwise inference. This first part removes the \textit{good lattice assumption} and allows the data to be non-stationary, yet still assumes the data to be Gaussian. The latter assumption is resolved in part 2, where we also demonstrate that our GKF based methodology is non-conservative under realistic error models.