870 research outputs found
3D weak lensing with spin wavelets on the ball
We construct the spin flaglet transform, a wavelet transform to analyze spin
signals in three dimensions. Spin flaglets can probe signal content localized
simultaneously in space and frequency and, moreover, are separable so that
their angular and radial properties can be controlled independently. They are
particularly suited to analyzing of cosmological observations such as the weak
gravitational lensing of galaxies. Such observations have a unique 3D
geometrical setting since they are natively made on the sky, have spin angular
symmetries, and are extended in the radial direction by additional distance or
redshift information. Flaglets are constructed in the harmonic space defined by
the Fourier-Laguerre transform, previously defined for scalar functions and
extended here to signals with spin symmetries. Thanks to various sampling
theorems, both the Fourier-Laguerre and flaglet transforms are theoretically
exact when applied to bandlimited signals. In other words, in numerical
computations the only loss of information is due to the finite representation
of floating point numbers. We develop a 3D framework relating the weak lensing
power spectrum to covariances of flaglet coefficients. We suggest that the
resulting novel flaglet weak lensing estimator offers a powerful alternative to
common 2D and 3D approaches to accurately capture cosmological information.
While standard weak lensing analyses focus on either real or harmonic space
representations (i.e., correlation functions or Fourier-Bessel power spectra,
respectively), a wavelet approach inherits the advantages of both techniques,
where both complicated sky coverage and uncertainties associated with the
physical modeling of small scales can be handled effectively. Our codes to
compute the Fourier-Laguerre and flaglet transforms are made publicly
available.Comment: 24 pages, 4 figures, version accepted for publication in PR
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
Sampling Sparse Signals on the Sphere: Algorithms and Applications
We propose a sampling scheme that can perfectly reconstruct a collection of
spikes on the sphere from samples of their lowpass-filtered observations.
Central to our algorithm is a generalization of the annihilating filter method,
a tool widely used in array signal processing and finite-rate-of-innovation
(FRI) sampling. The proposed algorithm can reconstruct spikes from
spatial samples. This sampling requirement improves over
previously known FRI sampling schemes on the sphere by a factor of four for
large . We showcase the versatility of the proposed algorithm by applying it
to three different problems: 1) sampling diffusion processes induced by
localized sources on the sphere, 2) shot noise removal, and 3) sound source
localization (SSL) by a spherical microphone array. In particular, we show how
SSL can be reformulated as a spherical sparse sampling problem.Comment: 14 pages, 8 figures, submitted to IEEE Transactions on Signal
Processin
Efficient analysis and representation of geophysical processes using localized spherical basis functions
While many geological and geophysical processes such as the melting of
icecaps, the magnetic expression of bodies emplaced in the Earth's crust, or
the surface displacement remaining after large earthquakes are spatially
localized, many of these naturally admit spectral representations, or they may
need to be extracted from data collected globally, e.g. by satellites that
circumnavigate the Earth. Wavelets are often used to study such nonstationary
processes. On the sphere, however, many of the known constructions are somewhat
limited. And in particular, the notion of `dilation' is hard to reconcile with
the concept of a geological region with fixed boundaries being responsible for
generating the signals to be analyzed. Here, we build on our previous work on
localized spherical analysis using an approach that is firmly rooted in
spherical harmonics. We construct, by quadratic optimization, a set of
bandlimited functions that have the majority of their energy concentrated in an
arbitrary subdomain of the unit sphere. The `spherical Slepian basis' that
results provides a convenient way for the analysis and representation of
geophysical signals, as we show by example. We highlight the connections to
sparsity by showing that many geophysical processes are sparse in the Slepian
basis.Comment: To appear in the Proceedings of the SPIE, as part of the Wavelets
XIII conference in San Diego, August 200
Wavelet/shearlet hybridized neural networks for biomedical image restoration
Recently, new programming paradigms have emerged that combine parallelism and numerical computations with algorithmic differentiation. This approach allows for the hybridization of neural network techniques for inverse imaging problems with more traditional methods such as wavelet-based sparsity modelling techniques. The benefits are twofold: on the one hand traditional methods with well-known properties can be integrated in neural networks, either as separate layers or tightly integrated in the network, on the other hand, parameters in traditional methods can be trained end-to-end from datasets in a neural network "fashion" (e.g., using Adagrad or Adam optimizers). In this paper, we explore these hybrid neural networks in the context of shearlet-based regularization for the purpose of biomedical image restoration. Due to the reduced number of parameters, this approach seems a promising strategy especially when dealing with small training data sets
Continuous Curvelet Transform: I. Resolution of the Wavefront Set
We discuss a Continuous Curvelet Transform (CCT), a transform f â Îf (a, b, θ) of functions f(x1, x2) on R^2, into a transform domain with continuous scale a > 0, location b â R^2, and orientation θ â [0, 2Ď). The transform is defined by Îf (a, b, θ) = {f, Îłabθ} where
the inner products project f onto analyzing elements called curvelets Îł_(abθ) which are smooth and of rapid decay away from an a by âa rectangle with minor axis pointing in direction θ. We call them curvelets because this anisotropic behavior allows them to âtrackâ the
behavior of singularities along curves. They are continuum scale/space/orientation analogs of the discrete frame of curvelets discussed in Candès and Donoho (2002).
We use the CCT to analyze several objects having singularities at points, along lines, and along smooth curves. These examples show that for fixed (x0, θ0), Îf (a, x0, θ0) decays rapidly as a â 0 if f is smooth near x0, or if the singularity of f at x0 is oriented in a different direction than θ_0. Generalizing these examples, we state general theorems showing that decay properties of
Îf (a, x0, θ0) for fixed (x0, θ0), as a â 0 can precisely identify the wavefront set and the H^m- wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0, θ0) near which Îf (a, x, θ) is not of rapid decay as a â 0; the H^m-wavefront set is the closure of those points (x0, θ0) where the âdirectional parabolic square functionâ
S^m(x, θ) = ( Ę|Îf (a, x, θ)|^2 ^(da) _a^3+^(2m))^(1/2)
is not locally integrable. The CCT is closely related to a continuous transform used by Hart Smith in his study
of Fourier Integral Operators. Smithâs transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (Fourier-Bros-Iagolnitzer) and Wave Packets (Cordoba-Fefferman) transforms. We describe their
similarities and differences in resolving the wavefront set
Slepian Wavelets for the Analysis of Incomplete Data on Manifolds
Many fields in science and engineering measure data that inherently live on non-Euclidean geometries, such as the sphere. Techniques developed in the Euclidean setting must be extended to other geometries. Due to recent interest in geometric deep learning, analogues of Euclidean techniques must also handle general manifolds or graphs. Often, data are only observed over partial regions of manifolds, and thus standard whole-manifold techniques may not yield accurate predictions. In this thesis, a new wavelet basis is designed for datasets like these.
Although many definitions of spherical convolutions exist, none fully emulate the Euclidean definition. A novel spherical convolution is developed, designed to tackle the shortcomings of existing methods. The so-called sifting convolution exploits the sifting property of the Dirac delta and follows by the inner product of a function with the translated version of another. This translation operator is analogous to the Euclidean translation in harmonic space and exhibits some useful properties. In particular, the sifting convolution supports directional kernels; has an output that remains on the sphere; and is efficient to compute. The convolution is entirely generic and thus may be used with any set of basis functions. An application of the sifting convolution with a topographic map of the Earth demonstrates that it supports directional kernels to perform anisotropic filtering.
Slepian wavelets are built upon the eigenfunctions of the Slepian concentration problem of the manifold - a set of bandlimited functions which are maximally concentrated within a given region. Wavelets are constructed through a tiling of the Slepian harmonic line by leveraging the existing scale-discretised framework. A straightforward denoising formalism demonstrates a boost in signal-to-noise for both a spherical and general manifold example. Whilst these wavelets were inspired by spherical datasets, like in cosmology, the wavelet construction may be utilised for manifold or graph data
Scalable and equivariant spherical CNNs by discrete-continuous (DISCO) convolutions
No existing spherical convolutional neural network (CNN) framework is both
computationally scalable and rotationally equivariant. Continuous approaches
capture rotational equivariance but are often prohibitively computationally demanding. Discrete approaches offer more favorable computational performance
but at the cost of equivariance. We develop a hybrid discrete-continuous (DISCO)
group convolution that is simultaneously equivariant and computationally scalable
to high-resolution. While our framework can be applied to any compact group, we
specialize to the sphere. Our DISCO spherical convolutions exhibit SO(3) rotational equivariance, where SO(n) is the special orthogonal group representing
rotations in n-dimensions. When restricting rotations of the convolution to the
quotient space SO(3)/SO(2) for further computational enhancements, we recover
a form of asymptotic SO(3) rotational equivariance. Through a sparse tensor implementation we achieve linear scaling in number of pixels on the sphere for both
computational cost and memory usage. For 4k spherical images we realize a saving of 109
in computational cost and 104
in memory usage when compared to the
most efficient alternative equivariant spherical convolution. We apply the DISCO
spherical CNN framework to a number of benchmark dense-prediction problems
on the sphere, such as semantic segmentation and depth estimation, on all of which
we achieve the state-of-the-art performance
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