Linear-quadratic Mixture Research Articles

From separability/identifiability properties of bilinear and linear-quadratic mixture matrix factorization to factorization algorithms

Blind source separation (BSS) and Blind Mixture Identification (BMI) methods typically concern unknown source signals, transferred through a given class of functions with unknown parameter values, which yields mixed observations. Using only these observations, BSS/BMI aims at estimating the source signals and/or mixing parameters. Most investigations concern linear instantaneous mixing functions. They contain two aspects. The first one consists in proposing general BSS/BMI principles, e.g. Independent Component Analysis, Sparse Component Analysis or Nonnegative Matrix Factorization (NMF), and/or deriving associated practical algorithms. The second aspect consists in analyzing the properties resulting from these principles. This is of utmost importance, to determine if the proposed BSS/BMI principles are guaranteed to separate the source signals and to identify the considered mixing model up to acceptable indeterminacies. These separability/identifiability analyses are even more important for nonlinear mixtures, that were shown to potentially yield higher indeterminacies. Among them, bilinear and linear-quadratic mixtures are receiving increasing attention, e.g. due to their application to remote sensing. Especially, extensions of NMF were recently proposed for them, but the resulting separability/identifiability properties were not analyzed. We here address this topic, moreover proceeding further by investigating Bilinear and Linear-Quadratic Mixture Matrix Factorization (BMMF and LQMMF) approaches without nonnegativity constraints. We especially show that, whereas nonlinearity is often considered to be a burden, it yields an essentially unique decomposition under mild conditions for BMMF. On the contrary, full LQMMF is shown to yield spurious solutions, which increases the usefulness of combining it with nonnegativity constraints in applications where data meet these constraints. Algorithms based on this framework are also defined in this paper and their performance is reported.

A Linear-Quadratic Model for the Quantification of a Mixture of Two Diluted Gases with a Single Metal Oxide Sensor.

The aim of our work is to quantify two gases (acetone and ethanol) diluted in an air buffer using only a single metal oxide (MOX) sensor. We took advantage of the low selectivity of the MOX sensor, exploiting a dual-temperature mode. Working at two temperatures of the MOX sensitive layer allowed us to obtain diversity in the measures. Two virtual sensors were created to characterize our gas mixture. We presented a linear-quadratic mixture sensing model which was closer to the experimental data. To validate this model and the experimental protocol, we inverted the system of quadratic equations to quantify a mixture of the two gases. The linear-quadratic model was compared to the bilinear model proposed in the literature. We presented an experimental evaluation on mixtures made of a few ppm of acetone and ethanol, and we obtained a precision close to the ppm. This is an important step towards medical applications, particularly in terms of diabetes, to deliver a non-invasive measure with a low-cost device.

Recurrent networks for separating extractable-target nonlinear mixtures. Part II. Blind configurations

While most reported blind source separation methods concern linear mixtures, we here address the nonlinear case. In the first part of this paper, we introduced a general class of nonlinear mixtures which can be inverted using recurrent networks. That part was focused on separating structures themselves and therefore on the non-blind configuration, whereas the current paper addresses the estimation of the parameters of this large class of structures in a blind context. We propose a maximum likelihood approach to this end. The main advantage of this approach is that it exploits the knowledge of the parametric model of mixing transformation in the separation procedure while its implementation does not require the knowledge of the explicit inverse of the model because the separating structure can be designed using a recurrent network. In particular, we illustrate in detail the proposed approach for a linear-quadratic mixture by using an extended recurrent network with self-feedback parameters which guarantee its local stability. Simulation results show the very good performance of the proposed algorithm.

Bayesian Source Separation of Linear and Linear-quadratic Mixtures Using Truncated Priors

In this work, we propose a Bayesian source separation method of linear-quadratic (LQ) and linear mixtures. Since our method relies on truncated prior distributions, it is particularly useful when the bounds of the sources and of the mixing coefficients are known in advance; this is the case, for instance, in non-negative matrix factorization. To implement our idea, we consider a Gibbs’ sampler equipped with latent variables, which are set to simplify the sampling steps. Experiments with synthetic data point out that the new proposal performs well in situations where classical ICA-based solutions fail to separate the sources. Moreover, in order to illustrate the application of our method to actual data, we consider the problem of separating scanned images.

Recurrent networks for separating extractable-target nonlinear mixtures. Part I: Non-blind configurations

While most reported source separation methods concern linear mixtures, we here address the nonlinear case. Even for a known nonlinear mixing model, creating a system which implements the exact inverse of this model is not straightforward for most nonlinear models. We first define a large class of possibly nonlinear models, i.e. “additive-target mixtures” (ATM), for which this inversion may be achieved thanks to the nonlinear recurrent networks that we propose to this end. We then further extend this approach to the “extractable-target mixtures” (ETM) that we also introduce in this paper. We illustrate these general approaches for two specific classes of mixtures, i.e. linear-quadratic mixtures, and quadratic ones. We then focus on our networks suited to linear-quadratic mixtures and we provide a detailed analysis of their equilibrium points and their stability. This allows us to introduce an automated procedure for selecting their free weights so as to guarantee the stability of a separating point for any source signals. Test results show the effectiveness of this approach for various types of source signals.

Late functional response of mouse urinary bladder to fractionated X-irradiation

Purpose: Late functional changes in mouse urinary bladder were studied after fractionated irradiation in order to define parameters of repair of sublethal radiation damage. Materials and methods: In the repair capacity study, 1 to 10 fractions were applied within 5 days. To assess the kinetics of repair, 4 fractions were applied with intervals of 0, 0.5, 1, 2, 4, or 8h. The endpoint used to establish dose-effect relationships was a reduction in the cystometrically measured bladder capacity by 50% of the individual pre-treatment value. Data analysis was performed with a linear-quadratic mixture model, which includes latent-time analysis in an effect-dependent manner. Results: The alpha/beta values were found to be 4.4Gy (95% confidence interval [2.0-8.4]) for the repair capacity data alone and 3.7Gy (1.7-6.7) in combination with the repair kinetics data. The halftime of repair, assuming mono-exponential repair kinetics, was 0.39h (95% CI [0.23-1.3]). The mean latent time decreased from around 40 weeks to 12 weeks, inversely dependent on the biologically effective dose. In animals experiencing an acute response during the first month after treatment, the incidence of severe late responses was significantly increased (p=0.0025). Conclusions: The present results confirm that the sparing effect in the mouse urinary bladder by dose fractionation is in the lower range of chronically responding tissues. The half-time of repair is in the lower range reported for mouse tissues.

Relationship between the in vitro radiosensitivity of skin fibroblasts and the expression of subcutaneous fibrosis, telangiectasia, and skin erythema after radiotherapy

Objective: To investigate if the occurrence of subcutaneous fibrosis after radiotherapy in an unselected group of breast cancer patients is related to cellular radiosensitivity of skin fibroblasts as measured in a clonogenic assay. Materials and methods: An in vitro colony-forming assay of normal fibroblast radiosensitivity was applied to primary skin biopsies from 31 breast cancer patients who received post-mastectomy radiotherapy with large doses per fraction (2.7–3.9 Gy) more than 10 years earlier. Three clinical normal-tissue endpoints were assessed. Two late endpoints, subcutaneous fibrosis and telangiectasia, were evaluated in three treatment fields by a single experienced clinician. In addition, skin erythema had been assessed at the end of treatment by members of the staff and junior staff. From previous analyses of normal tissue response, individual clinical radiosensitivity could be assessed as ‘excess risk’ of each of the three reactions. This was defined as the difference between the actual observed response in the patient and the expected response estimated from individual treatment characteristics in a linear quadratic (LQ) mixture model and, for the two late endpoints, with correction for the follow-up time. This clinical radioresponsiveness was compared with the in vitro radiosensitivity of the skin fibroblasts. To this end, the fractions of colony-forming cells after graded single doses were fitted by an LQ survival curve using non-linear and linear regression from which the surviving fraction at 3.5 Gy (SF 3.5) was estimated. Assessment at 3.5 Gy was chosen to reflect the fraction size during clinical radiotherapy. Results: A statistically significant variability of in vitro radiosensitivity between patients could be detected for both SF 2 ( P = 0.0095) and SF 3.5 ( P = 0.0008). A significant correlation was observed between SF 3.5 and excess risk of fibrosis ( r s = −0.46, P = 0.009) while no association was found between fibroblast radiosensitivity and either the occurrence of severe skin telangiectasia or the acute endpoint skin erythema. Conclusion: These results suggest that variability in the occurrence of subcutaneous fibrosis, but not telangiectasia or erythema, after radiotherapy is partly accounted for by differences in cellular radiosensitivity of normal skin fibroblasts.