2,028 research outputs found
Wavelets, ridgelets and curvelets on the sphere
We present in this paper new multiscale transforms on the sphere, namely the
isotropic undecimated wavelet transform, the pyramidal wavelet transform, the
ridgelet transform and the curvelet transform. All of these transforms can be
inverted i.e. we can exactly reconstruct the original data from its
coefficients in either representation. Several applications are described. We
show how these transforms can be used in denoising and especially in a Combined
Filtering Method, which uses both the wavelet and the curvelet transforms, thus
benefiting from the advantages of both transforms. An application to component
separation from multichannel data mapped to the sphere is also described in
which we take advantage of moving to a wavelet representation.Comment: Accepted for publication in A&A. Manuscript with all figures can be
downloaded at http://jstarck.free.fr/aa_sphere05.pd
Low-l CMB Analysis and Inpainting
Reconstruction of the CMB in the Galactic plane is extremely difficult due to
the dominant foreground emissions such as Dust, Free-Free or Synchrotron. For
cosmological studies, the standard approach consists in masking this area where
the reconstruction is not good enough. This leads to difficulties for the
statistical analysis of the CMB map, especially at very large scales (to study
for e.g., the low quadrupole, ISW, axis of evil, etc). We investigate in this
paper how well some inpainting techniques can recover the low- spherical
harmonic coefficients. We introduce three new inpainting techniques based on
three different kinds of priors: sparsity, energy and isotropy, and we compare
them. We show that two of them, sparsity and energy priors, can lead to
extremely high quality reconstruction, within 1% of the cosmic variance for a
mask with Fsky larger than 80%.Comment: Submitte
True CMB Power Spectrum Estimation
The cosmic microwave background (CMB) power spectrum is a powerful
cosmological probe as it entails almost all the statistical information of the
CMB perturbations. Having access to only one sky, the CMB power spectrum
measured by our experiments is only a realization of the true underlying
angular power spectrum. In this paper we aim to recover the true underlying CMB
power spectrum from the one realization that we have without a need to know the
cosmological parameters. The sparsity of the CMB power spectrum is first
investigated in two dictionaries; Discrete Cosine Transform (DCT) and Wavelet
Transform (WT). The CMB power spectrum can be recovered with only a few
percentage of the coefficients in both of these dictionaries and hence is very
compressible in these dictionaries. We study the performance of these
dictionaries in smoothing a set of simulated power spectra. Based on this, we
develop a technique that estimates the true underlying CMB power spectrum from
data, i.e. without a need to know the cosmological parameters. This smooth
estimated spectrum can be used to simulate CMB maps with similar properties to
the true CMB simulations with the correct cosmological parameters. This allows
us to make Monte Carlo simulations in a given project, without having to know
the cosmological parameters. The developed IDL code, TOUSI, for Theoretical
pOwer spectrUm using Sparse estImation, will be released with the next version
of ISAP
Image Decomposition and Separation Using Sparse Representations: An Overview
This paper gives essential insights into the use of sparsity and morphological diversity in image decomposition and source separation by reviewing our recent work in this field. The idea to morphologically decompose a signal into its building blocks is an important problem in signal processing and has far-reaching applications in science and technology. Starck , proposed a novel decomposition method—morphological component analysis (MCA)—based on sparse representation of signals. MCA assumes that each (monochannel) signal is the linear mixture of several layers, the so-called morphological components, that are morphologically distinct, e.g., sines and bumps. The success of this method relies on two tenets: sparsity and morphological diversity. That is, each morphological component is sparsely represented in a specific transform domain, and the latter is highly inefficient in representing the other content in the mixture. Once such transforms are identified, MCA is an iterative thresholding algorithm that is capable of decoupling the signal content. Sparsity and morphological diversity have also been used as a novel and effective source of diversity for blind source separation (BSS), hence extending the MCA to multichannel data. Building on these ingredients, we will provide an overview the generalized MCA introduced by the authors in and as a fast and efficient BSS method. We will illustrate the application of these algorithms on several real examples. We conclude our tour by briefly describing our software toolboxes made available for download on the Internet for sparse signal and image decomposition and separation
Numerical Issues When Using Wavelets
International audienceWavelets and related multiscale representations pervade all areas of signal processing. The recent inclusion of wavelet algorithms in JPEG 2000 – the new still-picture compression standard– testifies to this lasting and significant impact. The reason of the success of the wavelets is due to the fact that wavelet basis represents well a large class of signals, and therefore allows us to detect roughly isotropic elements occurring at all spatial scales and locations. As the noise in the physical sciences is often not Gaussian, the modeling, in the wavelet space, of many kind of noise (Poisson noise, combination of Gaussian and Poisson noise, long-memory 1/f noise, non-stationary noise, ...) has also been a key step for the use of wavelets in scientific, medical, or industrial applications [1]. Extensive wavelet packages exist now, commercial (see for example [2]) or non commercial (see for example [3, 4]), which allows any researcher, doctor, or engineer to analyze his data using wavelets
Curvelets and Ridgelets
International audienceDespite the fact that wavelets have had a wide impact in image processing, they fail to efficiently represent objects with highly anisotropic elements such as lines or curvilinear structures (e.g. edges). The reason is that wavelets are non-geometrical and do not exploit the regularity of the edge curve. The Ridgelet and the Curvelet [3, 4] transforms were developed as an answer to the weakness of the separable wavelet transform in sparsely representing what appears to be simple building atoms in an image, that is lines, curves and edges. Curvelets and ridgelets take the form of basis elements which exhibit high directional sensitivity and are highly anisotropic [5, 6, 7, 8]. These very recent geometric image representations are built upon ideas of multiscale analysis and geometry. They have had an important success in a wide range of image processing applications including denoising [8, 9, 10], deconvolution [11, 12], contrast enhancement [13], texture analysis [14, 15], detection [16], watermarking [17], component separation [18], inpainting [19, 20] or blind source separation[21, 22]. Curvelets have also proven useful in diverse fields beyond the traditional image processing application. Let’s cite for example seismic imaging [10, 23, 24], astronomical imaging [25, 26, 27], scientific computing and analysis of partial differential equations [28, 29]. Another reason for the success of ridgelets and curvelets is the availability of fast transform algorithms which are available in non-commercial software packages following the philosophy of reproducible research, see [30, 31]
Sparse representations and bayesian image inpainting
International audienceRepresenting the image to be inpainted in an appropriate sparse dictionary, and combining elements from bayesian statistics, we introduce an expectation-maximization (EM) algorithm for image inpainting. From a statistical point of view, the inpainting can be viewed as an estimation problem with missing data. Towards this goal, we propose the idea of using the EM mechanism in a bayesian framework, where a sparsity promoting prior penalty is imposed on the reconstructed coefficients. The EM framework gives a principled way to establish formally the idea that missing samples can be recovered based on sparse representations. We first introduce an easy and efficient sparse-representation-based iterative algorithm for image inpainting. Additionally, we derive its theoretical convergence properties for a wide class of penalties. Particularly, we establish that it converges in a strong sense, and give sufficient conditions for convergence to a local or a global minimum. Compared to its competitors, this algorithms allows a high degree of flexibility to recover different structural components in the image (piece-wise smooth, curvilinear, texture, etc). We also describe some ideas to automatically find the regularization parameter
Monotone operator splitting for optimization problems in sparse recovery
International audienceThis work focuses on several optimization problems involved in recovery of sparse solutions of linear inverse problems. Such problems appear in many fields including image and signal processing, and have attracted even more interest since the emergence of the compressed sensing (CS) theory. In this paper, we formalize many of these optimization problems within a unified framework of convex optimization theory, and invoke tools from convex analysis and maximal monotone operator splitting. We characterize all these optimization problems, and to solve them, we propose fast iterative convergent algorithms using forward-backward and/or Peaceman/Douglas-Rachford splitting iterations. With non-differentiable sparsity-promoting penalties, the proposed algorithms are essentially based on iterative shrinkage. This makes them very competitive for large-scale problems. We also report some experiments on image reconstruction in CS to demonstrate the applicability of the proposed framework
- …