1,928 research outputs found
Image reconstruction from incomplete information
Imperial Users onl
Phase Retrieval for Sparse Signals: Uniqueness Conditions
In a variety of fields, in particular those involving imaging and optics, we
often measure signals whose phase is missing or has been irremediably
distorted. Phase retrieval attempts the recovery of the phase information of a
signal from the magnitude of its Fourier transform to enable the reconstruction
of the original signal. A fundamental question then is: "Under which conditions
can we uniquely recover the signal of interest from its measured magnitudes?"
In this paper, we assume the measured signal to be sparse. This is a natural
assumption in many applications, such as X-ray crystallography, speckle imaging
and blind channel estimation. In this work, we derive a sufficient condition
for the uniqueness of the solution of the phase retrieval (PR) problem for both
discrete and continuous domains, and for one and multi-dimensional domains.
More precisely, we show that there is a strong connection between PR and the
turnpike problem, a classic combinatorial problem. We also prove that the
existence of collisions in the autocorrelation function of the signal may
preclude the uniqueness of the solution of PR. Then, assuming the absence of
collisions, we prove that the solution is almost surely unique on 1-dimensional
domains. Finally, we extend this result to multi-dimensional signals by solving
a set of 1-dimensional problems. We show that the solution of the
multi-dimensional problem is unique when the autocorrelation function has no
collisions, significantly improving upon a previously known result.Comment: submitted to IEEE TI
Reconstruction of two-dimensional signals from the Fourier transform magnitude
Originally presented as author's thesis (Sc. D.--Massachusetts Institute of Technology), 1986.Bibliography: p. 155-158.Supported in part by the Advanced Research Projects Agency monitored by ONR under contract no. N00014-81-K-0742 Supported in part by the National Science Foundation under grant ECS-8407285David Izraelevitz
Signal reconstruction from phase or magnitude
Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1981.MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING.Vita.Bibliography: leaves 146-150.by Monson H. Hayes III.Sc.D
Numerical methods for phase retrieval
In this work we consider the problem of reconstruction of a signal from the
magnitude of its Fourier transform, also known as phase retrieval. The problem
arises in many areas of astronomy, crystallography, optics, and coherent
diffraction imaging (CDI). Our main goal is to develop an efficient
reconstruction method based on continuous optimization techniques. Unlike
current reconstruction methods, which are based on alternating projections, our
approach leads to a much faster and more robust method. However, all previous
attempts to employ continuous optimization methods, such as Newton-type
algorithms, to the phase retrieval problem failed. In this work we provide an
explanation for this failure, and based on this explanation we devise a
sufficient condition that allows development of new reconstruction
methods---approximately known Fourier phase. We demonstrate that a rough (up to
radians) Fourier phase estimate practically guarantees successful
reconstruction by any reasonable method. We also present a new reconstruction
method whose reconstruction time is orders of magnitude faster than that of the
current method-of-choice in phase retrieval---Hybrid Input-Output (HIO).
Moreover, our method is capable of successful reconstruction even in the
situations where HIO is known to fail. We also extended our method to other
applications: Fourier domain holography, and interferometry. Additionally we
developed a new sparsity-based method for sub-wavelength CDI. Using this method
we demonstrated experimental resolution exceeding several times the physical
limit imposed by the diffraction light properties (so called diffraction
limit).Comment: PhD. Thesi
Toward single particle reconstruction without particle picking: Breaking the detection limit
Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray
crystallography and NMR spectroscopy as a high-resolution structural method for
biological macromolecules. In a cryo-EM experiment, the microscope produces
images called micrographs. Projections of the molecule of interest are embedded
in the micrographs at unknown locations, and under unknown viewing directions.
Standard imaging techniques first locate these projections (detection) and then
reconstruct the 3-D structure from them. Unfortunately, high noise levels
hinder detection. When reliable detection is rendered impossible, the standard
techniques fail. This is a problem especially for small molecules, which can be
particularly hard to detect. In this paper, we propose a radically different
approach: we contend that the structure could, in principle, be reconstructed
directly from the micrographs, without intermediate detection. As a result,
even small molecules should be within reach for cryo-EM. To support this claim,
we setup a simplified mathematical model and demonstrate how our
autocorrelation analysis technique allows to go directly from the micrographs
to the sought signals. This involves only one pass over the micrographs, which
is desirable for large experiments. We show numerical results and discuss
challenges that lay ahead to turn this proof-of-concept into a competitive
alternative to state-of-the-art algorithms
The Data Big Bang and the Expanding Digital Universe: High-Dimensional, Complex and Massive Data Sets in an Inflationary Epoch
Recent and forthcoming advances in instrumentation, and giant new surveys,
are creating astronomical data sets that are not amenable to the methods of
analysis familiar to astronomers. Traditional methods are often inadequate not
merely because of the size in bytes of the data sets, but also because of the
complexity of modern data sets. Mathematical limitations of familiar algorithms
and techniques in dealing with such data sets create a critical need for new
paradigms for the representation, analysis and scientific visualization (as
opposed to illustrative visualization) of heterogeneous, multiresolution data
across application domains. Some of the problems presented by the new data sets
have been addressed by other disciplines such as applied mathematics,
statistics and machine learning and have been utilized by other sciences such
as space-based geosciences. Unfortunately, valuable results pertaining to these
problems are mostly to be found only in publications outside of astronomy. Here
we offer brief overviews of a number of concepts, techniques and developments,
some "old" and some new. These are generally unknown to most of the
astronomical community, but are vital to the analysis and visualization of
complex datasets and images. In order for astronomers to take advantage of the
richness and complexity of the new era of data, and to be able to identify,
adopt, and apply new solutions, the astronomical community needs a certain
degree of awareness and understanding of the new concepts. One of the goals of
this paper is to help bridge the gap between applied mathematics, artificial
intelligence and computer science on the one side and astronomy on the other.Comment: 24 pages, 8 Figures, 1 Table. Accepted for publication: "Advances in
Astronomy, special issue "Robotic Astronomy
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