Julien Marot

Lower-Rank Tensor Approximation and Multiway Filtering

This paper presents some recent filtering methods based on the lower-rank tensor approximation approach for denoising tensor signals. In this approach, multicomponent data are represented by tensors, that is, multiway arrays, and the presented tensor filtering methods rely on multilinear algebra. First, the classical channel-by-channel SVD-based filtering method is overviewed. Then, an extension of the classical matrix filtering method is presented. It is based on the lower rank-

Subspace-Based and DIRECT Algorithms for Distorted Circular Contour Estimation

(K_1,\dots,K_N)

Hand posture classification by means of a new contour signature

truncation of the higher order SVD which performs a multimode principal component analysis (PCA) and is implicitly developed for an additive white Gaussian noise. Two tensor filtering methods recently developed by the authors are also overviewed. The first method consists of an improvement of the multimode PCA-based tensor filtering in the case of an additive correlated Gaussian noise. This improvement is specially done thanks to the fourth order cumulant slice matrix. The second method consists of an extension of Wiener filtering for data tensors. The performances and comparative results between all these tensor filtering methods are presented for the cases of noise reduction in color images, multispectral images, and multicomponent seismic data.

Fast subspace-based source localization methods

Circular features are commonly sought in digital image processing. The subspace-based line detection (SLIDE) method proposed to estimate the center and the radius of a single circle. In this paper, we introduce a novel method for estimating several radii while extending the circle estimation to retrieve circular-like distorted contours. Particularly, we develop and validate a new model for virtual signal generation by simulating a circular antenna. The circle center is estimated by the SLIDE method. A variable speed propagation scheme toward the circular antenna yields a linear phase signal. Therefore, a high-resolution method provides the radius. Either the gradient method or the more robust combination of dividing rectangles and spline interpolation can extend this method for free form object segmentation. The retrieval of multiple non concentric circles and rotated ellipses is also considered. To evaluate the performance of the proposed methods, we compare them with a least-squares method, Hough transform, and gradient vector flow. We apply the proposed method to hand-made images while considering some real-world images.

Optimization And Interpolation For Distorted Contour Estimation

This paper deals with hand posture recognition. Thanks to an adequate setup, we afford a database of hand photographs. We propose a novel contour signature, obtained by transforming the image content into several signals. The proposed signature is invariant to translation, rotation, and scaling. It can be used for posture classification purposes. We generate this signature out of photographs of hands: experiments show that the proposed signature provides good recognition results, compared to Hu moments and Fourier descriptors.

Multidimensional Signal Processing and Applications

Source localization is based on the spectral matrix algebraic properties. Propagator, and Ermolaev-Gershman (EG) noneigenvector algorithms exhibit a low computational load. Propagator is based on spectral matrix partitioning. EG algorithm obtains an approximation of noise subspace using an adjustable power parameter of the spectral matrix and choosing a threshold value. In this paper, we aim at demonstrating the usefulness of QR and LU factorizations of the spectral matrix to improve these methods. Experiments show that the modified propagator and EG algorithms based on factorized spectral matrix lead to better localization results, compared to the existing methods.

Fast and improved hand classification using dimensionality reduction and test set reduction

Distorted curves retrieval is faced for robotic way screening, particle trajectory characterization, aerial and satellite image analysis. This image processing problem has been transposed to an array processing problem by adopting specific conventions. Some solutions for wavefront distortions canceling have already been proposed. In this paper we aim at improving an existing method for distorted curves retrieval, making use of a global optimization algorithm. We show that it is possible to combine an optimization method to an interpolation method in order to obtain a reliable and fast algorithm

Fast tensor signal filtering using fixed point algorithm

In our daily lives and almost unconsciously, we deal with multidimensional data. From color images converted to the luminance and chrominance format to magnetic resonance images commonly acquired for health purposes, from different fashions to write an alphabet to array processing signals underlying any telecommunication system, we deal with multidimensional data. In this special issue, we tried to show the variety of the topics which are currently investigated with multidimensional signal processing tools. The mathematical tools presented in this issue are as diverse as adaptive detectors, wavelet processing, principal component analysis, and improved classical image processing tools such as histogram equalization. In the array processing paradigm, a two-dimensional matrix containing the data depends on the polarization properties of the sources, their number, and the number of sensors in the receiving antenna. Hence the interest of a multidimensional representation, including a polarization variable with two or three possible values, and a real and a complex part for the source amplitudes. In the image processing paradigm, data are as various as magnetic resonance or color images, whose representation can be transferred from the RGB (red green blue) format to other spaces emphasizing for instance the luminance or the chrominance. It is shown how magnetic resonance brain images are classified with support vector machine. To avoid problems related to high dimensionality, which is current in big data processing, adequate features are extracted from the data by discrete wavelet transform and principal component analysis. Color spaces, which are useful for skin detection, for instance, are also further investigated: whatever the representation space is, a color image is a third order tensor, in other words, a three-dimensional data. It is shown how to detect image splicing with the help of merged features in the chrominance space: the relationships between pixels in a neighborhood are studied with a Markov process and the extraction of DCT features from the chrominance channel. Then, with the help of new color spaces, it is shown how evolved versions of neural networks called extreme learning machines can fuse multiple information such as color and local spatial information from face images. The “multi” aspect can also appear in the image processing paradigm when multiple images are obtained from several parameters. In images provided by synthetic aperture radar exploited for flood detection, contrast enhancement is achieved by an adjustable histogram equalization technique. For such an application where the visual aspect of the results are much important much, a color image, that is, a multidimensional signal, can be built from several two-dimensional result images, to get an informative map, where the color informs on the nature of the imaged scene, flooded or not, for instance. Starting from images, a set of multidimensional data is extracted from Serbian texts: the Serbian alphabet, made of 30 letters, can be expressed in a Latin or in a Cyrillic fashion. All letters can be classified into four sets. By studying the frequencies of occurrence of each type of letter in a text, one can deduce that this text is written in the Latin or the Cyrillic fashion. In this application, matrices describing the cooccurrence in the distribution of the four types of letters are built out of any text, to make use of the classical texture features. Adapting the texture features to such a text recognition application, introducing a parameter which is the writing fashion, is a brand new idea. The “multi” aspect can also relate to multiresolution. Histogram of oriented gradients and hue descriptors can be merged to combine information related to the shape of an object and its color. By computing the merged data at several resolution levels, an innovative multidimensional descriptor is obtained. An application considered in this special issue is aircraft characterization and detection of images. In a nutshell, the “multi” representation attracts the interest of researchers from very diverse application fields. Hopefully, this special issue will contribute in diffusing the models and tools of multidimensional signal processing to various application fields. Salah Bourennane Julien Marot Caroline Fossati Ahmed Bouridane Klaus Spinnler

Propagator method for an application to contour estimation

In this paper, we consider an issue of hand posture classification. We improve a recently proposed signature, a matrix containing the distance of all contour pixels to an arbitrary reference point. Adequate pre-processings ensure the invariance properties of the signature. Candidate postures are pre-selected with a surface criterion, and Principal Component Analysis (PCA) reduces the dimensionality of the data, which improves the classification process.

Phase Distortion Estimation by DIRECT and Spline Interpolation Algorithms

Subspace-based methods rely on the selection of leading eigenvectors, associated with dominant eigenvalues. They have been extended to tensor data processing, such as denoising. Usually EVD (eigenvalue decomposition) is performed and data projection on leading eigenvectors results in noise reduction. Tensor processing methods, in particular multiway Wiener filtering algorithm, include an ALS (alternating least squares) loop, which involves several EVDs. Fixed point algorithm is a faster method than EVD to estimate a fixed number of eigenvectors. In this paper, we adapt fixed point algorithm to the estimation of only the required leading eigenvectors in a tensor processing framework. We adapt inverse power method to estimate the required noise variance. We provide a comparative study in terms of speed through an application to hyperspectral image denoising.