40,025 research outputs found
Phase Retrieval with Application to Optical Imaging
This review article provides a contemporary overview of phase retrieval in
optical imaging, linking the relevant optical physics to the information
processing methods and algorithms. Its purpose is to describe the current state
of the art in this area, identify challenges, and suggest vision and areas
where signal processing methods can have a large impact on optical imaging and
on the world of imaging at large, with applications in a variety of fields
ranging from biology and chemistry to physics and engineering
Critical Parameter Values and Reconstruction Properties of Discrete Tomography: Application to Experimental Fluid Dynamics
We analyze representative ill-posed scenarios of tomographic PIV with a focus
on conditions for unique volume reconstruction. Based on sparse random seedings
of a region of interest with small particles, the corresponding systems of
linear projection equations are probabilistically analyzed in order to
determine (i) the ability of unique reconstruction in terms of the imaging
geometry and the critical sparsity parameter, and (ii) sharpness of the
transition to non-unique reconstruction with ghost particles when choosing the
sparsity parameter improperly. The sparsity parameter directly relates to the
seeding density used for PIV in experimental fluids dynamics that is chosen
empirically to date. Our results provide a basic mathematical characterization
of the PIV volume reconstruction problem that is an essential prerequisite for
any algorithm used to actually compute the reconstruction. Moreover, we connect
the sparse volume function reconstruction problem from few tomographic
projections to major developments in compressed sensing.Comment: 22 pages, submitted to Fundamenta Informaticae. arXiv admin note:
text overlap with arXiv:1208.589
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
Compressive auto-indexing in femtosecond nanocrystallography
Ultrafast nanocrystallography has the potential to revolutionize biology by
enabling structural elucidation of proteins for which it is possible to grow
crystals with 10 or fewer unit cells on the side. The success of
nanocrystallography depends on robust orientation-determination procedures that
allow us to average diffraction data from multiple nanocrystals to produce a
three dimensional (3D) diffraction data volume with a high signal-to-noise
ratio. Such a 3D diffraction volume can then be phased using standard
crystallographic techniques. "Indexing" algorithms used in crystallography
enable orientation determination of diffraction data from a single crystal when
a relatively large number of reflections are recorded. Here we show that it is
possible to obtain the exact lattice geometry from a smaller number of
measurements than standard approaches using a basis pursuit solver.Comment: Spence Festschrift on Ultramicroscop
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