  The signal to noise ratio (SNR) fundamentally limits the information
accessible by magnetic resonance imaging (MRI). This limitation has been
addressed by a host of denoising techniques, recently including so-called
MPPCA: Principal component analysis (PCA) of the signal followed by automated
rank estimation, exploiting the Marchenko-Pastur (MP) distribution of noise
singular values. Operating on matrices comprised by data-patches, this popular
approach objectively identifies noise components and, ideally, allows noise to
be removed without introducing artifacts such as image blurring or non-local
averaging. The MPPCA rank estimation, however, relies on a large number of
noise singular values relative to the number of signal components to avoid such
ill effects. This condition is unlikely to be met when data-patches and
therefore matrices are small, for example due to spatially varying noise. Here,
we introduce tensor MPPCA (tMPPCA) for the purpose of denoising
multidimensional data, for example from multi-contrast acquisitions. Rather
than combining dimensions in matrices, tMPPCA utilizes each dimension of the
multidimensional data's inherent tensor-structure to better characterize noise,
and to recursively estimate signal components. Relative to matrix-based MPPCA,
tMPPCA requires no additional assumptions, and comparing the two in a numerical
phantom and a multi-TE diffusion MRI dataset, tMPPCA dramatically improves
denoising performance. This is particularly true for small data-patches, which
we believe will improve denoising in cases of spatially varying noise.
