This paper is concerned with achieving optimal coherence for highly redundant real unit-norm frames. As the redundancy grows, the number of vectors in the frame becomes too large to admit equiangular arrangements. In this case, other geometric optimality criteria need to be identified. To this end, we use an iteration of the embedding technique by Conway, Hardin and Sloane. As a consequence of their work, a quadratic mapping embeds equiangular lines into a simplex in a real Euclidean space. Here, higher degree polynomial maps embed highly redundant unit-norm frames to simplices in high-dimensional Euclidean spaces. We focus on the lowest degree case in which the embedding is quartic.
In this paper we use Euclidean embeddings and random hyperplane tessellations to construct binary block codes. The construction proceeds in two stages. First, an auxiliary ternary code is chosen which consists of vectors in the union of coordinate subspaces. The subspaces are selected so that any two vectors of different support have a sufficiently large distance. In addition, any two ternary vectors from the auxiliary codebook that have common support are at a guaranteed minimum distance. In the second stage, the auxiliary ternary code is converted to a binary code by an additional random hyperplane tessellation.
The objective of this paper is to study the performance of fusion frames for packet encodings in the presence of erasures. These frames encode a vector in a Hilbert space in terms of its components in subspaces, which can be identified with packets of linear coefficients. We evaluate the fusion frame performance under some statistical assumption on the vector to be transmitted, when part of the packets is transmitted perfectly and another part is lost in an adversarial, deterministic manner. The performance is measured by the mean-squared Euclidean norm of the reconstruction error when averaged over the transmission of all unit vectors. Our main result is that a random selection of fusion frames performs nearly as well as previously known optimal bounds for the error, characterized by optimal packings of subspaces, which are known not to exist in all dimensions.
This paper presents a method to recover a bandlimited signal, up to an overall multiplicative constant, from the
roots of its short-time Fourier transform. We assume that only finitely many sample values are non-zero. To
generate the number of roots needed for recovery, we use a type of aliasing, a time-frequency quasi-periodization
of the transform. We investigate the stability of the recovery algorithm under perturbations of the signal,
in particular under low-pass filtering, and verify the stability results with numerical experiments. In these
experiments we implement a deconvolution strategy for sparse bandlimited signals, whose non-zero sample values
are interspersed with vanishing ones. The recovery from roots of such signals is insensitive to the effect of random
echoes. In addition, we study the effect of aliasing by the time-frequency quasi-periodization on such sparse
signals. If the signal is convolved with white noise, then the number of roots generated with the quasi-periodized
short-time Fourier transform can be adjusted to be proportional to the number of non-vanishing samples to give
recoverability with overwhelming probability.
The fast digital shearlet transform (FDST) was recently introduced as a means to analyze natural images
efficiently, owing to the fact that those are typically governed by cartoon-like structures. In this paper, we
introduce and discuss a first-order hybrid sigma-delta quantization algorithm for coarsely quantizing the shearlet
coefficients generated by the FDST. Radial oversampling in the frequency domain together with our choice for
the quantization helps suppress the reconstruction error in a similar way as first-order sigma-delta quantization
for finite frames. We provide a theoretical bound for the reconstruction error and confirm numerically that the
error is in accordance with this theoretical decay.
In this paper we analyze the use of frames for the transmission and error-correction of analog signals via a
memoryless erasure-channel. We measure performance in terms of the mean-square error remaining after error
correction and reconstruction. Our results continue earlier works on frames as codes which were mostly concerned
with the smallest number of erased coefficients. To extend these works we borrow some ideas from binary coding
theory and realize them with a novel class of frames, which carry a particular fusion frame architecture. We show
that a family of frames from this class achieves a mean-square reconstruction error remaining after corrections
which decays faster than any inverse power in the number of frame coefficients.
We present steerlets, a new class of wavelets which allow us to define wavelet transforms that are covariant with
respect to rigid motions in d dimensions. The construction of steerlets is derived from an Isotropic Multiresolution
Analysis, a variant of a Multiresolution Analysis whose core subspace is closed under translations by integers
and under all rotations. Steerlets admit a wide variety of design characteristics ranging from isotropy, that is the
full insensitivity to orientations, to directional and orientational selectivity for local oscillations and singularities.
The associated 2D or 3D-steerlet transforms are fast MRA-type of transforms suitable for processing of discrete
data. The subband decompositions obtained with 2D or 3D-steerlets behave covariantly under the action of the
respective rotation group on an image, so that each rotated steerlet is the linear combination of other steerlets
in the same subband.
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.
The general objective in this paper is the loss-insensitive transmission of vectors by linear, redundant
encoding with the help of frames. Our specific goal is to find frames which minimize the mean-square
reconstruction error for cyclic burst erasures with known
burst-length statistics. Finding the best
frames among the cyclic ones is reduced to a discrete optimization problem. We provide an upper and
lower bound for the mean-square error and discuss a family of frames for which both bounds coincide
while the upper bound is minimized.
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.
This paper investigates the performance of randomly dithered first and higher-order sigma-delta quantization
applied to the frame coefficients of a vector in a
infinite-dimensional Hilbert space. We compute
the mean square error resulting from linear reconstruction with the quantized frame coefficients. When
properly dithered, this computation simplifies in the same way as under the assumption of the white-noise
hypothesis. The results presented here are valid for a uniform
mid-tread quantizer operating in
the no-overload regime. We estimate the large-redundancy asymptotics of the error for each family of
tight frames obtained from regular sampling of a bounded, differentiable path in the Hilbert space. In
order to achieve error asymptotics that are comparable to the quantization of oversampled band-limited
functions, we require the use of smoothly terminated frame paths.
We analyze localized textural consistencies in high-resolution Micro CT scans of coronary arteries to identify the appearance of diagnostically relevant changes in tissue. For the efficient and accurate processing of CT volume data, we use fast algorithms associated with three-dimensional so-called isotropic multiresolution wavelets that implement a redundant, frame-based image encoding without directional preference. Our algorithm identifies textural consistencies by correlating coefficients in the wavelet representation.
The central topic of this paper is the linear, redundant encoding of vectors using frames for the purpose of loss-insensitive data transmission. Our goal is to minimize the reconstruction error when frame coefficients are accidentally erased. Two-uniform frames are known to be optimal for handling up to two erasures, in the sense that they minimize the largest Euclidean error norm when up to two frame coefficients are set to zero. Here, we consider the case when an arbitrary number of the frame coefficients of a vector is lost. We derive general error bounds and apply these to concrete examples. We show that among the 227 known equivalence classes of two-uniform (36,15)-frames arising from Hadamard matrices, there are 5 that give smallest error bounds for up to 8 erasures.
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