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Parallel Magnetic Resonance Imaging as Approximation in a Reproducing Kernel Hilbert Space
In Magnetic Resonance Imaging (MRI) data samples are collected in the spatial
frequency domain (k-space), typically by time-consuming line-by-line scanning
on a Cartesian grid. Scans can be accelerated by simultaneous acquisition of
data using multiple receivers (parallel imaging), and by using more efficient
non-Cartesian sampling schemes. As shown here, reconstruction from samples at
arbitrary locations can be understood as approximation of vector-valued
functions from the acquired samples and formulated using a Reproducing Kernel
Hilbert Space (RKHS) with a matrix-valued kernel defined by the spatial
sensitivities of the receive coils. This establishes a formal connection
between approximation theory and parallel imaging. Theoretical tools from
approximation theory can then be used to understand reconstruction in k-space
and to extend the analysis of the effects of samples selection beyond the
traditional g-factor noise analysis to both noise amplification and
approximation errors. This is demonstrated with numerical examples.Comment: 28 pages, 7 figure
Monotone thematic factorizations of matrix functions
We continue the study of the so-called thematic factorizations of admissible
very badly approximable matrix functions. These factorizations were introduced
by V.V. Peller and N.J. Young for studying superoptimal approximation by
bounded analytic matrix functions. Even though thematic indices associated with
a thematic factorization of an admissible very badly approximable matrix
function are not uniquely determined by the function itself, R.B. Alexeev and
V.V. Peller showed that the thematic indices of any monotone non-increasing
thematic factorization of an admissible very badly approximable matrix function
are uniquely determined. In this paper, we prove the existence of monotone
non-decreasing thematic factorizations for admissible very badly approximable
matrix functions. It is also shown that the thematic indices appearing in a
monotone non-decreasing thematic factorization are not uniquely determined by
the matrix function itself. Furthermore, we show that the monotone
non-increasing thematic factorization gives rise to a great number of other
thematic factorizations.Comment: To appear in Journal of Approximation Theor
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