Albert Bijaoui

Multiresolution support applied to image filtering and restoration

The notion of a multiresolution support is introduced. This is a sequence of Boolean images related to significant pixels at each of a number of resolution levels. The multiresolution support is then used for noise suppression, in the context of image filtering, or iterative image restoration. Algorithmic details, and a range of practical examples, illustrate this approach.

A multiscale vision model adapted to the astronomical images

Abstract The analysis of the sky shows many kinds of hierarchically distributed objects. We have introduced a multiscale vision model based on the wavelet transform. The discrete transform is performed by the a trous algorithm which furnishes an isotropic vision, with a unique wavelet function. The vision model is based on the notion of the significant structures. We identify the pixels of the wavelet transform space (WTS) we can attribute to the objects. At each scale a region labelling is done. An interscale connectivity graph is then established. Connected trees are identified from the preceding graph. An object is generally associated to a subtree built from this graph. The identification of WTS pixels related to a given object leads to reconstructing an image by partial restoration algorithms. The object properties are extracted from the restored image. The main difficulty lies in the object reconstruction knowing the wavelet coefficients in the volume where the object is defined. It is a classical inverse problem. We choose to solve it using iterative algorithms. These algorithms give correct restored images, as we show on different examples, without or with adding a Gaussian noise. The influence of close objects can be partially removed.

Density estimation with non–parametric methods

One key issue in several astrophysical prob- lems is the evaluation of the density probability function underlying an observational discrete data set. We here review two non-parametric density estimators which re- cently appeared in the astrophysical literature, namely the adaptive kernel density estimator and the Maximum Penalized Likelihood technique, and describe another method based on the wavelet transform. The eciency of these estimators is tested by using ex- tensive numerical simulations in the one-dimensional case. The results are in good agreement with theoretical func- tions and the three methods appear to yield consistent es- timates. However, the Maximum Penalized Likelihood suf- fers from a lack of resolution and high computational cost due to its dependency on a minimization algorithm. The small dierences between kernel and wavelet estimates are mainly explained by the ability of the wavelet method to take into account local gaps in the data distribution. This new approach is very promising, since smaller struc- tures superimposed onto a larger one are detected only by this technique, especially when small samples are investi- gated. Thus, wavelet solutions appear to be better suited for subclustering studies. Nevertheless, kernel estimates seem more robust and are reliable solutions although some small-scale details can be missed. In order to check these estimators with respect to pre- vious studies, two galaxy redshift samples, related to the galaxy cluster A3526 and to the Corona Borealis region, have been analyzed. In both these cases claims for bi- modality are conrmed at a high condence level.

A Multiscale Vision Model to Analyse Field Astronomical Images

We have implemented a multiscale vision model based on the wavelet transform to analyse field astronomical images. The discrete transform is performed by the à trous algorithm. The vision model is based on the notion of significant structures. We identify the pixels of the associated wavelet transform space (WTS) with the objects. For each scale a region labelling is carried out. An interscale connectivity graph is then established. In accordance with some rules that permit false detections to be removed, the objects and their sub-objects are identified. They define respectively trees and sub-trees in the graph. In this way, the identification of the WTS pixels of the tree related to a given object leads to the reconstruction of its image by the conjugate gradient method. The model has been tested successfully on simulated images of stars and galaxies which allow us to show the capabilities of the detection and restoration procedures of the model. Finally, tests on real images show that one can analyse complex structures better than with classical astronomical vision models.

Temperature map computation for X-ray clusters of galaxies

Recent numerical simulations have shown that the variations of the gas temperature in clusters of galaxies are in- dicative of the dynamical state of these clusters. Maps of the temperature variation show complex structures with different shapes at different spatial scales, such as hot compression regions, filaments, cooling flows, or large-scale temperature pro- files. A new multiscale spectro-imagery algorithm for restoring the spatial temperature variations within clusters of galaxies is presented here. It has been especially developed to work with the EPIC MOS1, MOS2 and PN spectro-imagers on board the XMM-Newton satellite. The temperature values are fitted to an emission model that includes the source, the cosmic X-ray background and cosmic-ray induced particle background. The spatial temperature variations are coded at different scales in the wavelet space using the Haar wavelet and denoised by thresholding the wavelet coefficients. Our local temperature estimator behaves asymptotically like an optimal mininum variance bound estimator. But it is highly sensitive to the instrumental and astrophysical backgrounds, so that a good knowledge of each component of the emission model is required. Our algorithm has been applied to a simulated 60 ks observation of a merging cluster at z = 0.1. The cluster at different stages of merging has been provided by 3-D hydrodynamical simulations of structure formation (AMR). The multiscale approach has enabled us to restore the faint structures within the core of the merging subgroups where the gas emissivity is high, but also the temperature decrease at large scale in their external regions.

Wavelets, Gaussian mixtures and Wiener filtering

In this short communication a non-linear Wiener filtering is obtained from a Bayesian approach of the probability density function modelled as a Gaussian mixture. Its application on the wavelet denoising leads to excellent results.

DeQuant: a flexible multiresolution restoration framework

The contribution of this paper is to develop and analyze a new wavelet-based regularized restoration method for noise removal in photon limited imagery. This new framework presents the two main following advantages: (1) it assigns a new value to the non-significant wavelet coefficients which reduces artifacts and enables the incorporation of realistic prior information into the estimation process and (2) it is based on a local detection process with a measure of the significance of the detected structures. Its potential to improve nuclear medicine imaging is examined.

Disparity analysis: a wavelet transform approach

Describes a new method for the computation of a disparity map between a couple of stereo images. The disparities are computed along the x and y axes, respectively at each point of the image. In order to compute the disparity field, first a set of ground control points is detected in both images. Next, a mapping of the disparities over the entire image is done using the kriging method. Finally, the stereo couple of images is registered using the disparity maps. >

Astronomical image data compression by morphological skeleton transformation

Astronomical instruments currently provide a large amount of data. Nowadays, a large part of these data are image frames obtained with receivers of increasing size. The scan of large astronomical plates using fast microdensitometers gives image frames of over 30000×30000 pixels. More and more often, images are transmitted over a network in order to control the observations, to process the data, and to examine or to fill a data bank. The time taken for archiving, the cost of communication, the available memory given by magnetic tapes, and the limited bandwidth of transmission lines are reasons which lead us to examine the data compression of astronomical images.The astronomical image has the characteristic of being a set of astronomical sources in the sky background whose values are not zero. We are, in fact, only interested in the astronomical sources. Once a suitable detection is made, we generally want a compression without any distorsion. In this paper, we present a method which can be adapted for this purpose. It is based on morphological skeleton transformations. The experimental results show that it can give us an efficient compression. Moreover, the flexibility of choosing a structure element adapted to different images and the simplicity of implementation are other advantages of this method. Because of these characteristics, different compression applications may be treated.

On the distribution of the wavelet coefficient for a Poisson noise

We determine in this paper the general expression of the probability density function of the wavelet coefficient for a Poisson noise. We then examine the special case of the Haar wavelet for which we give threshold tables enabling the reader to process his own data.