The fundamental question concerning phase retrieval by projections in R^{m} is what is the least number of projections needed and what dimensions can be used. We will look at recent advances concerning phase retrieval by orthogonal complements and phase retrieval by hyperplanes which raise a number of problems which would give a complete answer to this fundamental problem.
KEYWORDS: Magnesium, Radon, Wavelets, Phase retrieval, Mathematics, Space operations, Molybdenum, Direct methods, Binary data, Current controlled current source
We answer a number of open problems concerning phase retrieval and phase retrieval by projections. In particular, one main theorem classifies phase retrieval by projections via collections of sequences of vectors allowing norm retrieval. Another key result computes the minimal number of vectors needed to add to a frame in order for it to possess the complement property and hence allow phase retrieval. In furthering this idea, in a third main theorem we show that when a collection of subspaces is one subspace short from allowing phase retrieval, then any partition of these subspaces spans two hyperplanes. We offer many more results in this area as well as provide a large number of examples showing the limitations of the theory.
It was shown in Ref. 1 that the unconditional constant for frame expansions is √B/A, where A and B are the frame bounds of the frame. It was also shown that a Bessel sequence is 1-unconditional if and only if it can be partitioned into an orthogonal sum of tight frames. In Ref. 2 cross-frame expansions were considered. It was shown that as long as the cross-frame expansions stay uniformly bounded away from zero, then similar results could be obtained. In this paper, we summarize these results into one concise source as well as add a few basic results that were not considered before.
The mathematical study of phase retrieval was started in 2006 in a landmark paper of Balan, Casazza and Edidin. This quickly became a heavily studied topic with implications for many areas of research in both applied mathematics and engineering. Recently there have been developments in a new area of study pertaining to phase retrieval given by projections. We give an extensive overview of the papers regarding projection phase retrieval.
A fusion frame is a collection of subspaces in a Hilbert space, generalizing the idea of a frame for signal representation.
A tool to construct fusion frames is the spectral tetris algorithm, a flexible and elementary method
to construct unit norm frames with a given frame operator having all of its eigenvalues greater than or equal to
two. We discuss how spectral tetris can be used to construct fusion frames with prescribed eigenvalues for its
fusion frame operator and with prescribed dimensions for its subspaces.
KEYWORDS: Lithium, Radon, Sensors, Sensor networks, Data processing, Signal processing, Distributed computing, Mathematics, Space operations, Detection theory
Fusion frames have become a major tool in the implementation of distributed systems. The effectiveness of fusion
frame applications in distributed systems is reflected in the efficiency of the end fusion process. This requires
the inversion of the fusion frame operator which is difficult or impossible in practice. What we want is for the
fusion frame operator to be the identity. But in most applications, especially to sensor networks, this almost
never occurs. We will solve this problem by introducing the notion of non-orthogonal fusion frames which have
the property that in most cases we can turn a family of subspaces of a Hilbert space into a non-orthogonal fusion
frame which has a fusion frame operator which is the identity.
A fusion frame is a frame-like collection of subspaces in a Hilbert space. It generalizes the concept of a frame
system for signal representation. In this paper, we study the existence and construction of fusion frames. We first
introduce two general methods, namely the spatial complement and the Naimark complement, for constructing a
new fusion frame from a given fusion frame. We then establish existence conditions for fusion frames with desired
properties. In particular, we address the following question: Given M, N, m ∈ N and {λ_{j}}^{M}_{j}
=1, does there exist
a fusion frame in R^{M} with N subspaces of dimension m for which {λ_{j}}^{M}_{j}
=1 are the eigenvalues of the associated
fusion frame operator? We address this problem by providing an algorithm which computes such a fusion frame
for almost any collection of parameters M, N, m ∈ N and {λ_{j}}^{M}_{j}
=1. Moreover, we show how this procedure can
be applied, if subspaces are to be added to a given fusion frame to force it to become tight.
One key property of frames is their resilience against erasures due to the possibility of generating stable, yet
over-complete expansions. Blind reconstruction is one common methodology to reconstruct a signal when frame
coefficients have been erased. In this paper we introduce several novel low complexity replacement schemes which
can be applied to the set of faulty frame coefficients before blind reconstruction is performed, thus serving as a
preconditioning of the received set of frame coefficients. One main idea is that frame coefficients associated with
frame vectors close to the one erased should have approximately the same value as the lost one. It is shown that
injecting such low complexity replacement schemes into blind reconstruction significantly reduce the worst-case
reconstruction error. We then apply our results to the circle frames. If we allow linear combinations of different
neighboring coefficients for the reconstruction of missing coefficients, we can even obtain perfect reconstruction
for the circle frames under certain weak conditions on the set of erasures.
We derive lower and upper bounds for the distance between a frame and the set of equal-norm Parseval frames.
The lower bound results from variational inequalities. The upper bound is obtained with a technique that uses
a family of ordinary differential equations for Parseval frames which can be shown to converge to an equal-norm
Parseval frame, if the number of vectors in a frame and the dimension of the Hilbert space they span are relatively
prime, and if the initial frame consists of vectors having sufficiently nearly equal norms.
Fusion frames are an emerging topic of frame theory, with applications to communications and distributed
processing. However, until recently, little was known about the existence of tight fusion frames, much less how
to construct them. We discuss a new method for constructing tight fusion frames which is akin to playing Tetris
with the spectrum of the frame operator. When combined with some easily obtained necessary conditions, these
Spectral Tetris constructions provide a near complete characterization of the existence of tight fusion frames.
We derive fast algorithms for doing signal reconstruction without phase. This type of problem is important in
signal processing, especially speech recognition technology, and has relevance for state tomography in quantum
theory. We show that a generic frame gives reconstruction from the absolute value of the frame coefficients in
polynomial time. An improved efficiency of reconstruction is obtained with a family of sparse frames or frames
associated with complex projective 2-designs.
The new notion of fusion frames will be presented in this article. Fusion frames provide an extensive framework
not only to model sensor networks, but also to serve as a means to improve robustness or develop efficient and
feasible reconstruction algorithms. Fusion frames can be regarded as sets of redundant subspaces each of which
contains a spanning set of local frame vectors, where the subspaces have to satisfy special overlapping properties.
Main aspects of the theory of fusion frames will be presented with a particular focus on the design of sensor
networks. New results on the construction of Parseval fusion frames will also be discussed.
We analyze a fundamental question in Hilbert space frame theory: What is the optimal decomposition of a Parseval frame? We will see that this question impacts several famous unsolved problems in different areas of mathematics. As a step towards the solution of this question, we give a new identity which holds for all Parseval frames.
We will construct new classes of Parseval frames for a Hilbert space which allow signal reconstruction from the absolute value of the frame coefficients. As a consequence, signal reconstruction can be done without using phase or its estimation.
The excess of a sequence in a Hilbert space H is the greatest number of elements that can be removed yet leave a set with the same closed span. {Forumlae available in paper}
Chirps arise in many signal processing applications, and have been extensively studied, especially in the case where chirps are regarded as functions of the real-line or of the integers. However, less attention has been paid to study of chirps over finite cyclic groups. We discuss the basic properties of such chirps, including a way in which they may be used to construct finite tight frames.
The theory of localized frames is a recently introduced concept with
broad implications to frame theory in general, as well as to the
special cases of Gabor and wavelet frames. Using the new notion of a
R-dual sequence associated with a Bessel sequence, we derive
several duality principles concerning localization in abstract frame
theory. As applications of our results we prove a duality principle
of localization of Gabor systems in the spirit of the Ron-Shen
duality principle, and obtain a Janssen representation for general
frame operators.
We show how to improve the properties of a Hilbert space frame by projecting it onto a subspace of the Hilbert space. For example, for any frame on a n-dimensional Hilbert space, there is an orthogonal projection onto a subspace of dimension n/2 (if n is even) or (n+1)/2 (if n is odd) so that the projection of the frame becomes a tight frame.
We give a physical interpretation for finite tight frames along the lines of Columb's Law in Physics. This allows us to use results from classical mechanics to anticipate results in frame theory. As a consequence, we are able to classify those frames for an N-dimensional Hilbert space which are the closest to being tight (in the sense of minimizing potential energy) while having the norms of the frame vectors prescribed in advance. This also yields a fundamental inequality that all finite tight frames must satisfy.
We examine the question of which characteristic functions yield Weyl-Heisenberg frames for various values of the parameters. We also give numerous applications of frames of characteristic functions to the general case (g, a, b).
A frame is a family {f_{i}}^{∞}_{i=1} of elements in a Hilbert space H with the property that every element in H can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coefficients. From the mathematical point of view this is gratifying, but for applications, it is a problem that the calculation requires inversion of an operator on H. The projection method is introduced to avoid this problem. The basic idea is to consider finite subfamilies {f_{i}}^{n}_{i=1} of the frame and the orthogonal projection P_{n} onto span {f_{i}}^{n}_{i=1}. For f ⊂ H, P_{n}f has a representation
as a linear combination of f_{i}, i = 1,2,...,n, and the corresponding coefficients can be calculated using finite dimensional methods. We find conditions implying that those coefficients converge to the correct frame coefficients as n→∞, in which case, we have avoided the
inversion problem. In the same spirit, we approximate the solution to a moment problem. It turns out that the class of "well-behaving
frames" are identical for the two problems we consider.
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