10,295 research outputs found
Conformal Mapping on Rough Boundaries II: Applications to bi-harmonic problems
We use a conformal mapping method introduced in a companion paper to study
the properties of bi-harmonic fields in the vicinity of rough boundaries. We
focus our analysis on two different situations where such bi-harmonic problems
are encountered: a Stokes flow near a rough wall and the stress distribution on
the rough interface of a material in uni-axial tension. We perform a complete
numerical solution of these two-dimensional problems for any univalued rough
surfaces. We present results for sinusoidal and self-affine surface whose slope
can locally reach 2.5. Beyond the numerical solution we present perturbative
solutions of these problems. We show in particular that at first order in
roughness amplitude, the surface stress of a material in uni-axial tension can
be directly obtained from the Hilbert transform of the local slope. In case of
self-affine surfaces, we show that the stress distribution presents, for large
stresses, a power law tail whose exponent continuously depends on the roughness
amplitude
Nearly Optimal Deterministic Algorithm for Sparse Walsh-Hadamard Transform
For every fixed constant , we design an algorithm for computing
the -sparse Walsh-Hadamard transform of an -dimensional vector in time . Specifically, the
algorithm is given query access to and computes a -sparse satisfying , for an absolute constant , where is the
transform of and is its best -sparse approximation. Our
algorithm is fully deterministic and only uses non-adaptive queries to
(i.e., all queries are determined and performed in parallel when the algorithm
starts).
An important technical tool that we use is a construction of nearly optimal
and linear lossless condensers which is a careful instantiation of the GUV
condenser (Guruswami, Umans, Vadhan, JACM 2009). Moreover, we design a
deterministic and non-adaptive compressed sensing scheme based
on general lossless condensers that is equipped with a fast reconstruction
algorithm running in time (for the GUV-based
condenser) and is of independent interest. Our scheme significantly simplifies
and improves an earlier expander-based construction due to Berinde, Gilbert,
Indyk, Karloff, Strauss (Allerton 2008).
Our methods use linear lossless condensers in a black box fashion; therefore,
any future improvement on explicit constructions of such condensers would
immediately translate to improved parameters in our framework (potentially
leading to reconstruction time with a reduced exponent in
the poly-logarithmic factor, and eliminating the extra parameter ).
Finally, by allowing the algorithm to use randomness, while still using
non-adaptive queries, the running time of the algorithm can be improved to
HYDRA: Hybrid Deep Magnetic Resonance Fingerprinting
Purpose: Magnetic resonance fingerprinting (MRF) methods typically rely on
dictio-nary matching to map the temporal MRF signals to quantitative tissue
parameters. Such approaches suffer from inherent discretization errors, as well
as high computational complexity as the dictionary size grows. To alleviate
these issues, we propose a HYbrid Deep magnetic ResonAnce fingerprinting
approach, referred to as HYDRA.
Methods: HYDRA involves two stages: a model-based signature restoration phase
and a learning-based parameter restoration phase. Signal restoration is
implemented using low-rank based de-aliasing techniques while parameter
restoration is performed using a deep nonlocal residual convolutional neural
network. The designed network is trained on synthesized MRF data simulated with
the Bloch equations and fast imaging with steady state precession (FISP)
sequences. In test mode, it takes a temporal MRF signal as input and produces
the corresponding tissue parameters.
Results: We validated our approach on both synthetic data and anatomical data
generated from a healthy subject. The results demonstrate that, in contrast to
conventional dictionary-matching based MRF techniques, our approach
significantly improves inference speed by eliminating the time-consuming
dictionary matching operation, and alleviates discretization errors by
outputting continuous-valued parameters. We further avoid the need to store a
large dictionary, thus reducing memory requirements.
Conclusions: Our approach demonstrates advantages in terms of inference
speed, accuracy and storage requirements over competing MRF method
Conformal Mapping on Rough Boundaries I: Applications to harmonic problems
The aim of this study is to analyze the properties of harmonic fields in the
vicinity of rough boundaries where either a constant potential or a zero flux
is imposed, while a constant field is prescribed at an infinite distance from
this boundary. We introduce a conformal mapping technique that is tailored to
this problem in two dimensions. An efficient algorithm is introduced to compute
the conformal map for arbitrarily chosen boundaries. Harmonic fields can then
simply be read from the conformal map. We discuss applications to "equivalent"
smooth interfaces. We study the correlations between the topography and the
field at the surface. Finally we apply the conformal map to the computation of
inhomogeneous harmonic fields such as the derivation of Green function for
localized flux on the surface of a rough boundary
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