  We present exact formulas for both the expected number and the height
distribution of local maxima (peaks) in two distinct categories of smooth,
non-centered Gaussian fields: (i) nonstationary Gaussian processes and (ii)
stationary planar Gaussian fields. For case (i), we introduce a novel parameter
related to conditional correlation that significantly simplifies the
computation of these formulas. Notably, the peak height distribution is solely
dependent on this single parameter. In case (ii), traditional methods involving
GOE random matrices are ineffective for non-isotropic fields with mean
functions. To address this, we apply specific transformations that enable the
derivation of formulas using generalized chi-squared density functions. These
derived results provide essential tools for calculating p-values and power in
applications of signal and change point detection within environments
characterized by non-isotropic Gaussian noise.
