1,843 research outputs found
On Ambiguity in Linear Inverse Problems: Entrywise Bounds on Nearly Data-Consistent Solutions and Entrywise Condition Numbers
Ill-posed linear inverse problems appear frequently in various signal
processing applications. It can be very useful to have theoretical
characterizations that quantify the level of ill-posedness for a given inverse
problem and the degree of ambiguity that may exist about its solution.
Traditional measures of ill-posedness, such as the condition number of a
matrix, provide characterizations that are global in nature. While such
characterizations can be powerful, they can also fail to provide full insight
into situations where certain entries of the solution vector are more or less
ambiguous than others. In this work, we derive novel theoretical lower- and
upper-bounds that apply to individual entries of the solution vector, and are
valid for all potential solution vectors that are nearly data-consistent. These
bounds are agnostic to the noise statistics and the specific method used to
solve the inverse problem, and are also shown to be tight. In addition, our
results also lead us to introduce an entrywise version of the traditional
condition number, which provides a substantially more nuanced characterization
of scenarios where certain elements of the solution vector are less sensitive
to perturbations than others. Our results are illustrated in an application to
magnetic resonance imaging reconstruction, and we include discussions of
practical computation methods for large-scale inverse problems, connections
between our new theory and the traditional Cram\'{e}r-Rao bound under
statistical modeling assumptions, and potential extensions to cases involving
constraints beyond just data-consistency
On Optimality in ROVir
We recently published an approach named ROVir (Region-Optimized Virtual
coils) that uses the beamforming capabilities of a multichannel magnetic
resonance imaging (MRI) receiver array to achieve coil compression (reducing an
original set of receiver channels into a much smaller number of virtual
channels for the purposes of dimensionality reduction), while simultaneously
preserving the MRI signal from desired spatial regions and suppressing the MRI
signal arising from unwanted spatial regions. The original ROVir procedure is
computationally-simple to implement (involving just a single small generalized
eigendecomposition), and its signal-suppression capabilities have proven useful
in an increasingly wide range of MRI applications. Our original paper made
claims about the theoretical optimality of this generalized eigendecomposition
procedure, but did not present the details. The purpose of this write-up is to
elaborate on these mathematical details, and to introduce a new greedy
iterative ROVir algorithm that enjoys certain advantages over the original
ROVir calculation approach. This discussion is largely academic, with
implications that we suspect will be minor for practical applications -- we
have only observed small improvements to ROVir performance in the cases we have
tried, and it would have been safe in these cases to still use the simpler
original calculation procedure with negligible practical impact on the final
imaging results.Comment: 7 pages, 4 figure
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