9,598 research outputs found
Harmonic Wavelet Transform and Image Approximation
In 2006, Saito and Remy proposed a new transform called the Laplace Local Sine Transform (LLST) in image processing as follows. Let f be a twice continuously differentiable function on a domain Ω. First we approximate f by a harmonic function u such that the residual component v=f−u vanishes on the boundary of Ω. Next, we do the odd extension for v, and then do the periodic extension, i.e. we obtain a periodic odd function v
*. Finally, we expand v
* into Fourier sine series. In this paper, we propose to expand v
* into a periodic wavelet series with respect to biorthonormal periodic wavelet bases with the symmetric filter banks. We call this the Harmonic Wavelet Transform (HWT). HWT has an advantage over both the LLST and the conventional wavelet transforms. On the one hand, it removes the boundary mismatches as LLST does. On the other hand, the HWT coefficients reflect the local smoothness of f in the interior of Ω. So the HWT algorithm approximates data more efficiently than LLST, periodic wavelet transform, folded wavelet transform, and wavelets on interval. We demonstrate the superiority of HWT over the other transforms using several standard images
Phase Harmonic Correlations and Convolutional Neural Networks
A major issue in harmonic analysis is to capture the phase dependence of
frequency representations, which carries important signal properties. It seems
that convolutional neural networks have found a way. Over time-series and
images, convolutional networks often learn a first layer of filters which are
well localized in the frequency domain, with different phases. We show that a
rectifier then acts as a filter on the phase of the resulting coefficients. It
computes signal descriptors which are local in space, frequency and phase. The
non-linear phase filter becomes a multiplicative operator over phase harmonics
computed with a Fourier transform along the phase. We prove that it defines a
bi-Lipschitz and invertible representation. The correlations of phase harmonics
coefficients characterise coherent structures from their phase dependence
across frequencies. For wavelet filters, we show numerically that signals
having sparse wavelet coefficients can be recovered from few phase harmonic
correlations, which provide a compressive representationComment: 26 pages, 8 figure
Simulating full-sky interferometric observations
Aperture array interferometers, such as that proposed for the Square
Kilometre Array (SKA), will see the entire sky, hence the standard approach to
simulating visibilities will not be applicable since it relies on a tangent
plane approximation that is valid only for small fields of view. We derive
interferometric formulations in real, spherical harmonic and wavelet space that
include contributions over the entire sky and do not rely on any tangent plane
approximations. A fast wavelet method is developed to simulate the visibilities
observed by an interferometer in the full-sky setting. Computing visibilities
using the fast wavelet method adapts to the sparse representation of the
primary beam and sky intensity in the wavelet basis. Consequently, the fast
wavelet method exhibits superior computational complexity to the real and
spherical harmonic space methods and may be performed at substantially lower
computational cost, while introducing only negligible error to simulated
visibilities. Low-resolution interferometric observations are simulated using
all of the methods to compare their performance, demonstrating that the fast
wavelet method is approximately three times faster that the other methods for
these low-resolution simulations. The computational burden of the real and
spherical harmonic space methods renders these techniques computationally
infeasible for higher resolution simulations. High-resolution interferometric
observations are simulated using the fast wavelet method only, demonstrating
and validating the application of this method to realistic simulations. The
fast wavelet method is estimated to provide a greater than ten-fold reduction
in execution time compared to the other methods for these high-resolution
simulations.Comment: 16 pages, 9 figures, replaced to match version accepted by MNRAS
(major additions to previous version including new fast wavelet method
The Real and Complex Techniques in Harmonic Analysis from the Point of View of Covariant Transform
This note reviews complex and real techniques in harmonic analysis. We
describe a common source of both approaches rooted in the covariant transform
generated by the affine group.
Keywords: wavelet, coherent state, covariant transform, reconstruction
formula, the affine group, ax+b-group, square integrable representations,
admissible vectors, Hardy space, fiducial operator, approximation of the
identity, maximal functions, atom, nucleus, atomic decomposition, Cauchy
integral, Poisson integral, Hardy--Littlewood maximal functions, grand maximal
function, vertical maximal functions, non-tangential maximal functions,
intertwining operator, Cauchy-Riemann operator, Laplace operator, singular
integral operator, SIO, boundary behaviour, Carleson measure.Comment: 31 pages, AMS-LaTeX, no figures; v2: a major revision, sections on
representations of the ax+b group and transported norms are added; v3: major
revision: an outline section on complex and real variables techniques are
added, numerous smaller improvements; v4: minor correction
Sparse image reconstruction on the sphere: analysis and synthesis
We develop techniques to solve ill-posed inverse problems on the sphere by
sparse regularisation, exploiting sparsity in both axisymmetric and directional
scale-discretised wavelet space. Denoising, inpainting, and deconvolution
problems, and combinations thereof, are considered as examples. Inverse
problems are solved in both the analysis and synthesis settings, with a number
of different sampling schemes. The most effective approach is that with the
most restricted solution-space, which depends on the interplay between the
adopted sampling scheme, the selection of the analysis/synthesis problem, and
any weighting of the l1 norm appearing in the regularisation problem. More
efficient sampling schemes on the sphere improve reconstruction fidelity by
restricting the solution-space and also by improving sparsity in wavelet space.
We apply the technique to denoise Planck 353 GHz observations, improving the
ability to extract the structure of Galactic dust emission, which is important
for studying Galactic magnetism.Comment: 11 pages, 6 Figure
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