7,531 research outputs found
Continuous Curvelet Transform: II. Discretization and Frames
We develop a unifying perspective on several decompositions exhibiting directional parabolic scaling. In each decomposition, the individual atoms are highly anisotropic at fine scales, with effective support obeying the parabolic scaling principle length ≈ width^2. Our comparisons allow to extend Theorems known for one decomposition to others. We start from a Continuous Curvelet Transform f → Γ_f (a, b, θ) of functions f(x_1, x_2) on R^2, with parameter space indexed by scale a > 0, location b ∈ R^2, and orientation θ. The transform projects f onto a curvelet γ_(abθ), yielding coefficient Γ_f (a, b, θ) = f, _(γabθ); the corresponding curvelet γ_(abθ) is defined by parabolic dilation in polar frequency domain coordinates. We establish a reproducing formula and Parseval relation for the transform, showing that these curvelets provide a continuous tight frame. The CCT is closely related to a continuous transform introduced by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on true affine parabolic
scaling of a single mother wavelet, while the CCT can only be viewed as true affine parabolic scaling in euclidean coordinates by taking a slightly different mother wavelet at each scale. Smith’s transform, unlike the CCT, does not provide a continuous tight frame. We show that, with the right underlying wavelet in Smith’s transform, the analyzing elements of the two transforms become increasingly similar at increasingly fine scales.
We derive a discrete tight frame essentially by sampling the CCT at dyadic intervals in scale a_j = 2^−j, at equispaced intervals in direction, θ_(jℓ), = 2π2^(−j/2)ℓ, and equispaced sampling on a rotated anisotropic grid in space. This frame is a complexification of the ‘Curvelets 2002’ frame constructed by Emmanuel Candès et al. [1, 2, 3]. We compare this discrete frame with a composite system which at coarse scales is the same as this frame but
at fine scales is based on sampling Smith’s transform rather than the CCT. We are able to show a very close approximation of the two systems at fine scales, in a strong operator norm sense. Smith’s continuous transform was intended for use in forming molecular decompositions
of Fourier Integral Operators (FIO’s). Our results showing close approximation of the curvelet frame by a composite frame using true affine paraboblic scaling at fine scales allow us to cross-apply Smith’s results, proving that the discrete curvelet transform gives sparse representations of FIO’s of order zero. This yields an alternate proof of a recent result of Candès and Demanet about the sparsity of FIO representations in discrete curvelet frames
Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds
The Euclidean scattering transform was introduced nearly a decade ago to
improve the mathematical understanding of convolutional neural networks.
Inspired by recent interest in geometric deep learning, which aims to
generalize convolutional neural networks to manifold and graph-structured
domains, we define a geometric scattering transform on manifolds. Similar to
the Euclidean scattering transform, the geometric scattering transform is based
on a cascade of wavelet filters and pointwise nonlinearities. It is invariant
to local isometries and stable to certain types of diffeomorphisms. Empirical
results demonstrate its utility on several geometric learning tasks. Our
results generalize the deformation stability and local translation invariance
of Euclidean scattering, and demonstrate the importance of linking the used
filter structures to the underlying geometry of the data.Comment: 35 pages; 3 figures; 2 tables; v3: Revisions based on reviewer
comment
Multiscale Representations for Manifold-Valued Data
We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere , the special orthogonal group , the positive definite matrices , and the Grassmann manifolds . The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the and maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as , , , where the and maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper
Real-World Repetition Estimation by Div, Grad and Curl
We consider the problem of estimating repetition in video, such as performing
push-ups, cutting a melon or playing violin. Existing work shows good results
under the assumption of static and stationary periodicity. As realistic video
is rarely perfectly static and stationary, the often preferred Fourier-based
measurements is inapt. Instead, we adopt the wavelet transform to better handle
non-static and non-stationary video dynamics. From the flow field and its
differentials, we derive three fundamental motion types and three motion
continuities of intrinsic periodicity in 3D. On top of this, the 2D perception
of 3D periodicity considers two extreme viewpoints. What follows are 18
fundamental cases of recurrent perception in 2D. In practice, to deal with the
variety of repetitive appearance, our theory implies measuring time-varying
flow and its differentials (gradient, divergence and curl) over segmented
foreground motion. For experiments, we introduce the new QUVA Repetition
dataset, reflecting reality by including non-static and non-stationary videos.
On the task of counting repetitions in video, we obtain favorable results
compared to a deep learning alternative
Multiperiodicity, modulations and flip-flops in variable star light curves I. Carrier fit method
The light curves of variable stars are commonly described using simple
trigonometric models, that make use of the assumption that the model parameters
are constant in time. This assumption, however, is often violated, and
consequently, time series models with components that vary slowly in time are
of great interest. In this paper we introduce a class of data analysis and
visualization methods which can be applied in many different contexts of
variable star research, for example spotted stars, variables showing the
Blazhko effect, and the spin-down of rapid rotators. The methods proposed are
of explorative type, and can be of significant aid when performing a more
thorough data analysis and interpretation with a more conventional method.Our
methods are based on a straightforward decomposition of the input time series
into a fast "clocking" periodicity and smooth modulating curves. The fast
frequency, referred to as the carrier frequency, can be obtained from earlier
observations (for instance in the case of photometric data the period can be
obtained from independently measured radial velocities), postulated using some
simple physical principles (Keplerian rotation laws in accretion disks), or
estimated from the data as a certain mean frequency. The smooth modulating
curves are described by trigonometric polynomials or splines. The data
approximation procedures are based on standard computational packages
implementing simple or constrained least-squares fit-type algorithms.Comment: 14 pages, 23 figures, submitted to Astronomy and Astrophysic
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