7,478 research outputs found
Sampling and Super-resolution of Sparse Signals Beyond the Fourier Domain
Recovering a sparse signal from its low-pass projections in the Fourier
domain is a problem of broad interest in science and engineering and is
commonly referred to as super-resolution. In many cases, however, Fourier
domain may not be the natural choice. For example, in holography, low-pass
projections of sparse signals are obtained in the Fresnel domain. Similarly,
time-varying system identification relies on low-pass projections on the space
of linear frequency modulated signals. In this paper, we study the recovery of
sparse signals from low-pass projections in the Special Affine Fourier
Transform domain (SAFT). The SAFT parametrically generalizes a number of well
known unitary transformations that are used in signal processing and optics. In
analogy to the Shannon's sampling framework, we specify sampling theorems for
recovery of sparse signals considering three specific cases: (1) sampling with
arbitrary, bandlimited kernels, (2) sampling with smooth, time-limited kernels
and, (3) recovery from Gabor transform measurements linked with the SAFT
domain. Our work offers a unifying perspective on the sparse sampling problem
which is compatible with the Fourier, Fresnel and Fractional Fourier domain
based results. In deriving our results, we introduce the SAFT series (analogous
to the Fourier series) and the short time SAFT, and study convolution theorems
that establish a convolution--multiplication property in the SAFT domain.Comment: 42 pages, 3 figures, manuscript under revie
Learning SO(3) Equivariant Representations with Spherical CNNs
We address the problem of 3D rotation equivariance in convolutional neural
networks. 3D rotations have been a challenging nuisance in 3D classification
tasks requiring higher capacity and extended data augmentation in order to
tackle it. We model 3D data with multi-valued spherical functions and we
propose a novel spherical convolutional network that implements exact
convolutions on the sphere by realizing them in the spherical harmonic domain.
Resulting filters have local symmetry and are localized by enforcing smooth
spectra. We apply a novel pooling on the spectral domain and our operations are
independent of the underlying spherical resolution throughout the network. We
show that networks with much lower capacity and without requiring data
augmentation can exhibit performance comparable to the state of the art in
standard retrieval and classification benchmarks.Comment: Camera-ready. Accepted to ECCV'18 as oral presentatio
Exact positivity of the Wigner and P-functions of a Markovian open system
We discuss the case of a Markovian master equation for an open system, as it
is frequently found from environmental decoherence. We prove two theorems for
the evolution of the quantum state. The first one states that for a generic
initial state the corresponding Wigner function becomes strictly positive after
a finite time has elapsed. The second one states that also the P-function
becomes exactly positive after a decoherence time of the same order. Therefore
the density matrix becomes exactly decomposable into a mixture of Gaussian
pointer states.Comment: 11 pages, references added, typo corrected, to appear in J. Phys.
Dynamical versus diffraction spectrum for structures with finite local complexity
It is well-known that the dynamical spectrum of an ergodic measure dynamical
system is related to the diffraction measure of a typical element of the
system. This situation includes ergodic subshifts from symbolic dynamics as
well as ergodic Delone dynamical systems, both via suitable embeddings. The
connection is rather well understood when the spectrum is pure point, where the
two spectral notions are essentially equivalent. In general, however, the
dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of
finite local complexity and establish the equivalence of the dynamical spectrum
with a collection of diffraction spectra of the system and certain factors.
This equivalence gives access to the dynamical spectrum via these diffraction
spectra. It is particularly useful as the diffraction spectra are often simpler
to determine and, in many cases, only very few of them need to be calculated.Comment: 27 pages; some minor revisions and improvement
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