Frederic Rue

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.

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.

A multiscale vision model applied to astronomical images

Abstract We have implemented a multiscale vision model based on the wavelet transform to analyse field astronomical images. The discrete transform is performed by the a trous algorithm. The vision model is based on the notion of the significant structures. We identify the pixels of the wavelet transform space (WTS) associated with the objects. At 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. So, 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 astronomical images which shows that complex structures are better analysed than using usual astronomical vision models.

Wavelets and the study of the distant Universe

The large-scale distribution of galaxies in the Universe exhibits structures at various scales, these so-called groups, clusters, and superclusters of galaxies being more or less hierarchically organized. A specific vision model is needed in order to detect, describe, and classify each component of this hierarchy. To do so, we have developed a multiscale vision model based on an unfolding into a scale space allowing us to detect structures of different sizes. A discrete wavelet transform is done by the a trous algorithm. The algorithm is implemented for astronomical images and also for lists of object positions, currently called catalogues in astronomical literature. Some applications on astrophysical data of cosmological interest are briefly described: (1) inventory procedures for galaxy counts on wide-field images, (2) processing of X-ray cluster images lending to the analyses of the total matter distribution, and (3) detection of large-scale structures from galaxy counts, From the analyses of N-body simulations we show that the vision model from the wavelet transform provides a new statistical indicator on cosmological scenarios.

The analysis of SAR images by multiscale methods

The analysis of SAR images requires to reduce the speckle noise due to the coherent character of the radar signal. This noise is multiplicative, which often leads to logarithmically transform the modulus. A bias, which depends on the local homogeneity is introduced, and the noise is amplified. The minimum variance estimator leads to process the energy image instead of the modulus one for reducing this multiplicative noise. The proposed methods are based on a multiscale vision model for which the image is only described by its significant structural features on a set of scales. The multiscale analysis is performed by a redundant discrete wavelet transform, the a trous algorithm. We have determined the distribution law of the wavelet coefficient of the energy in the case of a statistically uniform image. This allows us to extract the significant wavelet coefficients, according to the noise. A filtering algorithm is derived by taking into account only these coefficients. This method is extended to a co-addition of SAR images. Taking into account the multiresolution support obtained from the thresholding in the wavelet transform space (WTS), the image is decomposed into a set of objects, by a 3D segmentation in WTS.

Wavelet Transform and Multiscale Vision Models

We have implemented multiscale vision models based on the wavelet transform to analyze field astronomical images. The discrete transform is performed by the a trous or the pyramidal algorithms. The vision models are based on the notion of the significant structures. Different kind of noises have beeen taken into account. We identify the pixels of the wavelet transform space (WTS) associated with the objects. At 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. So, 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 astronomical images which shows that complex structures are better analyzed than using usual astronomical vision models.

Multiscale methods applied to the analysis of SAR images

The analysis of SAR images requires in a first step to reduce the speckle noise which is due to the coherent character of the RADAR signal. The application of the minimum variance bound estimator leads to process the energy image instead of the amplitude one for the reduction of this multiplicative noise. The proposed analyzing methods are based on a multiscale vision model for which the image is only described by its significant structural features at a set of dyadic scales. The multiscale analysis is performed by a redundant discrete wavelet transform, the a trous algorithm. The filtering algorithm is interactive. At each step we compute the ratio between the observed energy image and the restored one. We detect at each scale the significant structures, by taking into account the exponential probability distribution function of the energy for the determination of the significant wavelet coefficients. The ratio is restored from its significant coefficients, and the restored image is updated. The iterations are stopped when any significant structure is detected in the ratio. Then, we are interested to extract and to analyze the contained objects. The multiscale analysis allows us an approach well adapted to diffused objects, without contrasted edges. An object is defined as a local maximum in the wavelet transform space (WTS). All the structures form a 3D connected set which is hierarchically organized. This set gives the description of an object in the WTS. The image of each object is restored y an inverse algorithm. The comparison between images taken at different epochs is done using the multiscale vision model. THat allows us to enhance the features at a given scale which have significantly varied. The correlation coefficients between the structures detected at each scale are far form the ones obtained between the pixel energy. For example, this method is very suitable to detect and to describe faint large scale variations.

Pyramidal vision model applied to astronomical images

A multiscale vision model based on a pyramidal wavelet transform is described in the present paper. The pyramidal wavelet algorithm is modified in order to satisfy a correct sampling at each scale. Objects are defined by trees of statistically significant coefficients in the wavelet transform space. The object images are then restored using the conjugate gradient method. By comparing with the model based on the a trous algorithm, tests on simulated and real images show that this model presents a good compromise between analysis quality and the memory space and computation time needed.