The curvelet transform for image denoising

We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement

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

Wavelet domain Bayesian denoising of string signal in the cosmic microwave background

An algorithm is proposed for denoising the signal induced by cosmic strings in the cosmic microwave background (CMB). A Bayesian approach is taken, based on modeling the string signal in the wavelet domain with generalized Gaussian distributions. Good performance of the algorithm is demonstrated by simulated experiments at arcminute resolution under noise conditions including primary and secondary CMB anisotropies, as well as instrumental noise.Comment: 16 pages, 11 figures. Version 2 matches version accepted for publication in MNRAS. Changes include substantial clarifications on our approach and a significant reduction of manuscript lengt

Analysis of CMB maps with 2D wavelets

We consider the 2D wavelet transform with two scales to study sky maps of temperature anisotropies in the cosmic microwave background radiation (CMB). We apply this technique to simulated maps of small sky patches of size 12.8 \times 12.8 square degrees and 1.5' \times 1.5' pixels. The relation to the standard approach, based on the cl's is established through the introduction of the scalogram. We consider temperature fluctuations derived from standard, open and flat-Lambda CDM models. We analyze CMB anisotropies maps plus uncorrelated Gaussian noise (uniform and non-uniform) at idfferent S/N levels. We explore in detail the denoising of such maps and compare the results with other techniques already proposed in the literature. Wavelet methods provide a good reconstruction of the image and power spectrum. Moreover, they are faster than previously proposed methods.Comment: latex file 7 pages + 5 postscript files + 1 gif file; accepted for publication in A&A

A multiscale regularized restoration algorithm for XMM-Newton data

We introduce a new multiscale restoration algorithm for images with few photons counts and its use for denoising XMM data. We use a thresholding of the wavelet space so as to remove the noise contribution at each scale while preserving the multiscale information of the signal. Contrary to other algorithms the signal restoration process is the same whatever the signal to noise ratio is. Thresholds according to a Poisson noise process are indeed computed analytically at each scale thanks to the use of the unnormalized Haar wavelet transform. Promising preliminary results are obtained on X-ray data for Abell 2163 with the computation of a temperature map.Comment: To appear in the Proceedings of `Galaxy Clusters and the High Redshift Universe Observed in X-rays', XXIth Moriond Astrophysics Meeting (March 2001), Eds. Doris Neumann et a

Source detection using a 3D sparse representation: application to the Fermi gamma-ray space telescope

The multiscale variance stabilization Transform (MSVST) has recently been proposed for Poisson data denoising. This procedure, which is nonparametric, is based on thresholding wavelet coefficients. We present in this paper an extension of the MSVST to 3D data (in fact 2D-1D data) when the third dimension is not a spatial dimension, but the wavelength, the energy, or the time. We show that the MSVST can be used for detecting and characterizing astrophysical sources of high-energy gamma rays, using realistic simulated observations with the Large Area Telescope (LAT). The LAT was launched in June 2008 on the Fermi Gamma-ray Space Telescope mission. The MSVST algorithm is very fast relative to traditional likelihood model fitting, and permits efficient detection across the time dimension and immediate estimation of spectral properties. Astrophysical sources of gamma rays, especially active galaxies, are typically quite variable, and our current work may lead to a reliable method to quickly characterize the flaring properties of newly-detected sources.Comment: Accepted. Full paper will figures available at http://jstarck.free.fr/aa08_msvst.pd

Wavelets Applied to CMB Maps: a Multiresolution Analysis for Denoising

Analysis and denoising of Cosmic Microwave Background (CMB) maps are performed using wavelet multiresolution techniques. The method is tested on $12^{\circ}.8\times 12^{\circ}.8$ maps with resolution resembling the experimental one expected for future high resolution space observations. Semianalytic formulae of the variance of wavelet coefficients are given for the Haar and Mexican Hat wavelet bases. Results are presented for the standard Cold Dark Matter (CDM) model. Denoising of simulated maps is carried out by removal of wavelet coefficients dominated by instrumental noise. CMB maps with a signal-to-noise, $S/N \sim 1$, are denoised with an error improvement factor between 3 and 5. Moreover we have also tested how well the CMB temperature power spectrum is recovered after denoising. We are able to reconstruct the $C_{\ell}$'s up to $l\sim 1500$ with errors always below $20$ in cases with $S/N \ge 1$.Comment: latex file 9 pages + 5 postscript figures + 1 gif figure (figure 6), to be published in MNRA

Polarized wavelets and curvelets on the sphere

The statistics of the temperature anisotropies in the primordial cosmic microwave background radiation field provide a wealth of information for cosmology and for estimating cosmological parameters. An even more acute inference should stem from the study of maps of the polarization state of the CMB radiation. Measuring the extremely weak CMB polarization signal requires very sensitive instruments. The full-sky maps of both temperature and polarization anisotropies of the CMB to be delivered by the upcoming Planck Surveyor satellite experiment are hence being awaited with excitement. Multiscale methods, such as isotropic wavelets, steerable wavelets, or curvelets, have been proposed in the past to analyze the CMB temperature map. In this paper, we contribute to enlarging the set of available transforms for polarized data on the sphere. We describe a set of new multiscale decompositions for polarized data on the sphere, including decimated and undecimated Q-U or E-B wavelet transforms and Q-U or E-B curvelets. The proposed transforms are invertible and so allow for applications in data restoration and denoising.Comment: Accepted. Full paper will figures available at http://jstarck.free.fr/aa08_pola.pd

Fast Poisson Noise Removal by Biorthogonal Haar Domain Hypothesis Testing

Methods based on hypothesis tests (HTs) in the Haar domain are widely used to denoise Poisson count data. Facing large datasets or real-time applications, Haar-based denoisers have to use the decimated transform to meet limited-memory or computation-time constraints. Unfortunately, for regular underlying intensities, decimation yields discontinuous estimates and strong "staircase" artifacts. In this paper, we propose to combine the HT framework with the decimated biorthogonal Haar (Bi-Haar) transform instead of the classical Haar. The Bi-Haar filter bank is normalized such that the p-values of Bi-Haar coefficients (pBH) provide good approximation to those of Haar (pH) for high-intensity settings or large scales; for low-intensity settings and small scales, we show that pBH are essentially upper-bounded by pH. Thus, we may apply the Haar-based HTs to Bi-Haar coefficients to control a prefixed false positive rate. By doing so, we benefit from the regular Bi-Haar filter bank to gain a smooth estimate while always maintaining a low computational complexity. A Fisher-approximation-based threshold imple- menting the HTs is also established. The efficiency of this method is illustrated on an example of hyperspectral-source-flux estimation