4,351 research outputs found
Duality in reconstruction systems
We consider reconstruction systems (RS's), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS's. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS's, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS's the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results.Fil: Massey, Pedro Gustavo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; ArgentinaFil: Ruiz, Mariano Andres. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Stojanoff, Demetrio. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
Filter Bank Fusion Frames
In this paper we characterize and construct novel oversampled filter banks
implementing fusion frames. A fusion frame is a sequence of orthogonal
projection operators whose sum can be inverted in a numerically stable way.
When properly designed, fusion frames can provide redundant encodings of
signals which are optimally robust against certain types of noise and erasures.
However, up to this point, few implementable constructions of such frames were
known; we show how to construct them using oversampled filter banks. In this
work, we first provide polyphase domain characterizations of filter bank fusion
frames. We then use these characterizations to construct filter bank fusion
frame versions of discrete wavelet and Gabor transforms, emphasizing those
specific finite impulse response filters whose frequency responses are
well-behaved.Comment: keywords: filter banks, frames, tight, fusion, erasures, polyphas
Robust dual reconstruction systems and fusion frames
We study the duality of reconstruction systems, which are g-frames in a finite dimensional setting. These systems allow redundant linear encoding-decoding schemes implemented by the so-called dual reconstruction systems. We are particularly interested in the projective reconstruction systems that are the analogue of fusion frames in this context. Thus, we focus on dual systems of a fixed projective system that are optimal with respect to erasures of the reconstruction system coefficients involved in the decoding process. We consider two different measures of the reconstruction error in a blind reconstruction algorithm. We also study the projective reconstruction system that best approximate an arbitrary reconstruction system, based on some well known results in matrix theory. Finally, we present a family of examples in which the problem of existence of a dual projective system of a reconstruction system of this type is considered.Fil: Massey, Pedro Gustavo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Ruiz, Mariano Andres. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Stojanoff, Demetrio. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
Optimal reconstruction systems for erasures and for the q-potential
We introduce the -potential as an extension of the Benedetto-Fickus frame
potential, defined on general reconstruction systems and we show that protocols
are the minimizers of this potential under certain restrictions. We extend
recent results of B.G. Bodmann on the structure of optimal protocols with
respect to 1 and 2 lost packets where the worst (normalized) reconstruction
error is computed with respect to a compatible unitarily invariant norm. We
finally describe necessary and sufficient (spectral) conditions, that we call
-fundamental inequalities, for the existence of protocols with prescribed
properties by relating this problem to Klyachko's and Fulton's theory on sums
of hermitian operators
A combinatorial characterization of tight fusion frames
In this paper we give a combinatorial characterization of tight fusion frame
(TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case
has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our
characterization does not have this limitation. We also develop some methods
for generating TFF sequences. The basic technique is a majorization principle
for TFF sequences combined with spatial and Naimark dualities. We use these
methods and our characterization to give necessary and sufficient conditions
which are satisfied by the first three highest ranks. We also give a
combinatorial interpretation of spatial and Naimark dualities in terms of
Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences
which have unique maximal elements with respect to majorization partial order.
Finally, we give several examples illustrating our techniques including an
example of tight fusion frame which can not be constructed by the existing
spectral tetris techniques. We end the paper by giving a complete list of
maximal TFF sequences in dimensions less than ten.Comment: 31 page
Sparse Recovery from Combined Fusion Frame Measurements
Sparse representations have emerged as a powerful tool in signal and
information processing, culminated by the success of new acquisition and
processing techniques such as Compressed Sensing (CS). Fusion frames are very
rich new signal representation methods that use collections of subspaces
instead of vectors to represent signals. This work combines these exciting
fields to introduce a new sparsity model for fusion frames. Signals that are
sparse under the new model can be compressively sampled and uniquely
reconstructed in ways similar to sparse signals using standard CS. The
combination provides a promising new set of mathematical tools and signal
models useful in a variety of applications. With the new model, a sparse signal
has energy in very few of the subspaces of the fusion frame, although it does
not need to be sparse within each of the subspaces it occupies. This sparsity
model is captured using a mixed l1/l2 norm for fusion frames.
A signal sparse in a fusion frame can be sampled using very few random
projections and exactly reconstructed using a convex optimization that
minimizes this mixed l1/l2 norm. The provided sampling conditions generalize
coherence and RIP conditions used in standard CS theory. It is demonstrated
that they are sufficient to guarantee sparse recovery of any signal sparse in
our model. Moreover, a probabilistic analysis is provided using a stochastic
model on the sparse signal that shows that under very mild conditions the
probability of recovery failure decays exponentially with increasing dimension
of the subspaces
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