21,158 research outputs found
Artifacts in incomplete data tomography - with applications to photoacoustic tomography and sonar
We develop a paradigm using microlocal analysis that allows one to
characterize the visible and added singularities in a broad range of incomplete
data tomography problems. We give precise characterizations for photo- and
thermoacoustic tomography and Sonar, and provide artifact reduction strategies.
In particular, our theorems show that it is better to arrange Sonar detectors
so that the boundary of the set of detectors does not have corners and is
smooth. To illustrate our results, we provide reconstructions from synthetic
spherical mean data as well as from experimental photoacoustic data
Uniformity, Universality, and Computability Theory
We prove a number of results motivated by global questions of uniformity in
computability theory, and universality of countable Borel equivalence
relations. Our main technical tool is a game for constructing functions on free
products of countable groups.
We begin by investigating the notion of uniform universality, first proposed
by Montalb\'an, Reimann and Slaman. This notion is a strengthened form of a
countable Borel equivalence relation being universal, which we conjecture is
equivalent to the usual notion. With this additional uniformity hypothesis, we
can answer many questions concerning how countable groups, probability
measures, the subset relation, and increasing unions interact with
universality. For many natural classes of countable Borel equivalence
relations, we can also classify exactly which are uniformly universal.
We also show the existence of refinements of Martin's ultrafilter on Turing
invariant Borel sets to the invariant Borel sets of equivalence relations that
are much finer than Turing equivalence. For example, we construct such an
ultrafilter for the orbit equivalence relation of the shift action of the free
group on countably many generators. These ultrafilters imply a number of
structural properties for these equivalence relations.Comment: 61 Page
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