15,063 research outputs found
Phase retrieval in frame theory
This dissertation is the study of phase retrieval in frame theory. The first part is concerned with the analysis of phase retrieval and the complete classification of norm retrieval. Norm retrieval is essential to transfer the properties of phase retrieval to the complement space. The first section includes the results regarding projections and also the characterization of phase retrieval and norm retrieval for subspaces. The second part is the study of weak phase retrieval which was motivated by the idea of reducing the number of vectors satisfying the properties close to phase retrieval. The last section provides the correlation between weak phase retrieval and phase retrieval properties along with the examples illustrating the relationship between weak phase retrieval and the related concepts.Includes bibliographical reference
Random projections and the optimization of an algorithm for phase retrieval
Iterative phase retrieval algorithms typically employ projections onto
constraint subspaces to recover the unknown phases in the Fourier transform of
an image, or, in the case of x-ray crystallography, the electron density of a
molecule. For a general class of algorithms, where the basic iteration is
specified by the difference map, solutions are associated with fixed points of
the map, the attractive character of which determines the effectiveness of the
algorithm. The behavior of the difference map near fixed points is controlled
by the relative orientation of the tangent spaces of the two constraint
subspaces employed by the map. Since the dimensionalities involved are always
large in practical applications, it is appropriate to use random matrix theory
ideas to analyze the average-case convergence at fixed points. Optimal values
of the gamma parameters of the difference map are found which differ somewhat
from the values previously obtained on the assumption of orthogonal tangent
spaces.Comment: 15 page
Phase retrieval by hyperplanes
We show that a scalable frame does phase retrieval if and only if the
hyperplanes of its orthogonal complements do phase retrieval. We then show this
result fails in general by giving an example of a frame for which
does phase retrieval but its induced hyperplanes fail phase retrieval.
Moreover, we show that such frames always exist in for any
dimension . We also give an example of a frame in which fails
phase retrieval but its perps do phase retrieval. We will also see that a
family of hyperplanes doing phase retrieval in must contain at
least hyperplanes. Finally, we provide an example of six hyperplanes in
which do phase retrieval
Common pulse retrieval algorithm: a fast and universal method to retrieve ultrashort pulses
We present a common pulse retrieval algorithm (COPRA) that can be used for a
broad category of ultrashort laser pulse measurement schemes including
frequency-resolved optical gating (FROG), interferometric FROG, dispersion
scan, time domain ptychography, and pulse shaper assisted techniques such as
multiphoton intrapulse interference phase scan (MIIPS). We demonstrate its
properties in comprehensive numerical tests and show that it is fast, reliable
and accurate in the presence of Gaussian noise. For FROG it outperforms
retrieval algorithms based on generalized projections and ptychography.
Furthermore, we discuss the pulse retrieval problem as a nonlinear
least-squares problem and demonstrate the importance of obtaining a
least-squares solution for noisy data. These results improve and extend the
possibilities of numerical pulse retrieval. COPRA is faster and provides more
accurate results in comparison to existing retrieval algorithms. Furthermore,
it enables full pulse retrieval from measurements for which no retrieval
algorithm was known before, e.g., MIIPS measurements
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