Jean-Philip Royer

Computing the polyadic decomposition of nonnegative third order tensors

Computing the minimal polyadic decomposition (also often referred to as canonical decomposition or sometimes Parafac) amounts to finding the global minimum of a coercive polynomial in many variables. In the case of arrays with nonnegative entries, the low-rank approximation problem is well posed. In addition, due to the large dimension of the problem, the decomposition can be rather efficiently calculated with the help of preconditioned nonlinear conjugate gradient algorithms, as subsequently shown, if equipped with an algebraic calculation of the globally optimal stepsize in low dimension. Other algorithms are also studied (gradient and quasi-Newton approaches) for comparisons. Two versions of each algorithm are considered: the enhanced line search version (ELS), and the backtracking version alternating with ELS. Computer simulations are provided and demonstrate the good behavior of these algorithms dedicated to nonnegative arrays, compared to others put forward in the literature. Finally, applications in the context of data analysis illustrate various algorithms. The main advantage of the suggested approach is to explicitly take into account the nonnegative nature of the loading matrices in the problem parameterization, instead of enforcing positive entries by projection. According to the experiments we have run, such an approach also happens to be more robust with respect to possible modeling errors.

A regularized nonnegative canonical polyadic decomposition algorithm with preprocessing for 3D fluorescence spectroscopy

We consider blind source separation in chemical analysis focussing on the 3D fluorescence spectroscopy framework. We present an alternative method to process the Fluorescence Excitation‐Emission Matrices (FEEM): first, a preprocessing is applied to eliminate the Raman and Rayleigh scattering peaks that clutter the FEEM. To improve its robustness versus possible improper settings, we suggest to associate the classical Zepps method with a morphological image filtering technique. Then, in the second stage, the Canonical Polyadic (CP or Candecomp/Parafac) decomposition of a nonnegative three‐way array has to be computed. In the fluorescence spectroscopy context, the constituent vectors of the loading matrices should be nonnegative (since standing for spectra and concentrations). Thus, we suggest a new nonnegative third order CP decomposition algorithm (NNCP) based on a nonlinear conjugate gradient optimization algorithm with regularization terms and periodic restarts. Computer simulations performed on real experimental data are provided to enlighten the effectiveness and robustness of the whole processing chain and to validate the approach. Copyright

Computing the nonnegative 3-way tensor factorization using Tikhonov regularization

This paper deals with the minimum polyadic decomposition of a nonnegative three-way array. The main advantage of the nonnegativity constraint is that the approximation problem becomes well posed. To tackle this problem, we suggest the use of a cost function including penalty terms built with matrix exponentials. Gradient components are then derived, allowing to efficiently implement the decomposition using classical optimization algorithms. In our case ALS and conjugate gradient algorithms are studied and compared with another existing algorithm, thanks to computer simulations performed in the context of data analysis.

Nonnegative 3-way tensor factorization via conjugate gradient with globally optimal stepsize

This paper deals with the minimal polyadic decomposition (also known as canonical decomposition or Parafac) of a 3-way array, assuming each entry is positive. In this case, the low-rank approximation problem becomes well-posed. The suggested approach consists of taking into account the nonnegative nature of the loading matrices directly in the problem parameterization. Then, the three gradient components are derived allowing to efficiently implement the decomposition using classical optimization algorithms. In our case, we focus on the conjugate gradient algorithm, well matched to large problems. The good behaviour of the proposed approach is illustrated through computer simulations in the context of data analysis and compared to other existing approaches.

Fast Visual Odometry for a Low-Cost Underwater Embedded Stereo System †

This paper provides details of hardware and software conception and realization of a stereo embedded system for underwater imaging. The system provides several functions that facilitate underwater surveys and run smoothly in real-time. A first post-image acquisition module provides direct visual feedback on the quality of the taken images which helps appropriate actions to be taken regarding movement speed and lighting conditions. Our main contribution is a light visual odometry method adapted to the underwater context. The proposed method uses the captured stereo image stream to provide real-time navigation and a site coverage map which is necessary to conduct a complete underwater survey. The visual odometry uses a stochastic pose representation and semi-global optimization approach to handle large sites and provides long-term autonomy, whereas a novel stereo matching approach adapted to underwater imaging and system attached lighting allows fast processing and suitability to low computational resource systems. The system is tested in a real context and shows its robustness and promising future potential.

Cultural Heritage Resources Profiling: Ontology-based Approach.

Cultural heritage (CH) resources are very heterogeneous since the information was collected from vast diversity of cultural sites and digitally recorded in different formats. With the progress of 3D technologies, photogrammetry techniques become the adopted solution for representing CH artifacts by turning photos from small finds, to entire landscapes, into accurate 3D models. To meet knowledge representation with cultural heritage photogrammetry, this paper proposes an ontology-profiling method for modeling a real case of archaeological amphorae. The ontological profile consists of all needed information to represent a CH resource including typology attributes, geo-spatial information and photogrammetry process. An example illustrating the applicability of this profiling method to the problem of CH resources conceptualization is presented. We also outline our perspectives for using ontologies in data-driven science, in particular on modeling a complete pipeline that manages both the photogrammetric process and the archaeological knowledge.

Error Analysis of low-rank three-way tensor factorization approach to blind source separation

In tensor factorization approach to blind separation of multidimensional sources two formulas for calculating the source tensor have emerged. In practice, it is observed that these two schemes exhibit different levels of robustness against perturbations of the factors involved in the tensor model. Motivated by both practical reasons and the will to better figure this out, we present error analyses in source tensor estimation performed by low-rank factorization of three-way tensors. To that aim, computer simulations as well as the analytical calculation of the theoretical error are carried out. The conclusions drawn from these numerical and analytical error analyses are supported by the results obtained thanks to the tensor factorization based blind decomposition of an experimental multispectral image of a skin tumor.

Images fusion based on a Coupled Nonnegative Tensors Factorization approach (CNTF), application to OLCI and ETM sensors

Coastal water monitoring often requires images with good spatial and spectral resolutions. This communication describes a new method for multispectral and hyperspectral images fusion: the Coupled Nonnegative Tensor Factorization fusion approach which directly exploits the tensorial nature of multi and hyperspectral images. The CNTF fusion algorithm allows to separate both hyperspectral and multispectral images into two matrices: one of endmembers and one of abundance. For application and validation purposes, a multispectral image (ETM) and a Sentinel 3/OLCI image was simulated from a Hyperspectral Imaging for Coastal Waters image (HICO). The resulting fusion image was compared with a reference image which was generated from the HICO image too. Statistical parameters such as RASE, ERGAS, SAM, correlation and bias, are computed for the spectral and spatial assessment of the fused products.