14,075 research outputs found
Discrete Ridgelet Transforms for Image Representation
The ridgelet transform (Candes and Donoho, 1999) was introduced as a new multiscale representation for functions on continuous spaces that are smooth away from discontinuities along lines. In this paper, we present several discrete versions of the ridgelet transform that lead to algorithmic implementations. The resulting transforms are invertible, non-redundant and computed via fast algorithms. Furthermore, this construction leads to a family of directional orthonormal bases for digital images. Numerical results show that the new transforms are more effective than the wavelet transform in approximating and denoising images with straight edges. Keywords: wavelets, ridgelets, Radon transform, directional bases, discrete transforms, non-linear approximation, image denoising
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
On the computation of directional scale-discretized wavelet transforms on the sphere
We review scale-discretized wavelets on the sphere, which are directional and
allow one to probe oriented structure in data defined on the sphere.
Furthermore, scale-discretized wavelets allow in practice the exact synthesis
of a signal from its wavelet coefficients. We present exact and efficient
algorithms to compute the scale-discretized wavelet transform of band-limited
signals on the sphere. These algorithms are implemented in the publicly
available S2DW code. We release a new version of S2DW that is parallelized and
contains additional code optimizations. Note that scale-discretized wavelets
can be viewed as a directional generalization of needlets. Finally, we outline
future improvements to the algorithms presented, which can be achieved by
exploiting a new sampling theorem on the sphere developed recently by some of
the authors.Comment: 13 pages, 3 figures, Proceedings of Wavelets and Sparsity XV, SPIE
Optics and Photonics 2013, Code is publicly available at http://www.s2dw.org
Fast directional continuous spherical wavelet transform algorithms
We describe the construction of a spherical wavelet analysis through the
inverse stereographic projection of the Euclidean planar wavelet framework,
introduced originally by Antoine and Vandergheynst and developed further by
Wiaux et al. Fast algorithms for performing the directional continuous wavelet
analysis on the unit sphere are presented. The fast directional algorithm,
based on the fast spherical convolution algorithm developed by Wandelt and
Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature
implementation for Npix pixels on the sphere, and allows one to perform a
directional spherical wavelet analysis of a 10^6 pixel map on a personal
computer.Comment: 10 pages, 3 figures, replaced to match version accepted by IEEE
Trans. Sig. Pro
ENO-wavelet transforms for piecewise smooth functions
We have designed an adaptive essentially nonoscillatory (ENO)-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENO-wavelet transforms we adaptively change the function and use the same uniform stencils. The ENO-wavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate
- …