  In high-dimensional magnetic resonance imaging applications, time-consuming,
sequential acquisition of data samples in the spatial frequency domain
($k$-space) can often be accelerated by accounting for dependencies along
imaging dimensions other than space in linear reconstruction, at the cost of
noise amplification that depends on the sampling pattern. Examples are
support-constrained, parallel, and dynamic MRI, and $k$-space sampling
strategies are primarily driven by image-domain metrics that are expensive to
compute for arbitrary sampling patterns. It remains challenging to provide
systematic and computationally efficient automatic designs of arbitrary
multidimensional Cartesian sampling patterns that mitigate noise amplification,
given the subspace to which the object is confined. To address this problem,
this work introduces a theoretical framework that describes local geometric
properties of the sampling pattern and relates these properties to a measure of
the spread in the eigenvalues of the information matrix described by its first
two spectral moments. This new criterion is then used for very efficient
optimization of complex multidimensional sampling patterns that does not
require reconstructing images or explicitly mapping noise amplification.
Experiments with in vivo data show strong agreement between this criterion and
traditional, comprehensive image-domain- and $k$-space-based metrics,
indicating the potential of the approach for computationally efficient
(on-the-fly), automatic, and adaptive design of sampling patterns.
