Nonnegatively Constrained Least Squares Research Articles

Confocal Raman Spectroscopic Imaging for Evaluation of Distribution of Nano-Formulated Hydrophobic Active Cosmetic Ingredients in Hydrophilic Films.

Film-forming systems are highly relevant to the topical administration of active ingredients (AI) to the body. Enhanced contact with the skin can increase the efficacy of delivery and penetration during prolonged exposure. However, after the evaporation of volatile solvents to form a thin film, the distribution of the ingredient should remain homogenous in order to ensure the effectiveness of the formula. This is especially critical for the use of hydrophobic molecules that have poor solubility in hydrophilic films. In order to address this concern, hydroxyphenethyl esters (PHE) of Punica granatum seed oil were prepared as a nanosuspension stabilised by poloxamers (NanoPHE). NanoPHE was then added to a formulation containing polyvinyl alcohol (PVA) as a film forming agent, Glycerol as a plasticiser and an antimicrobial agent, SepicideTM HB. Despite their reliability, reference methods such as high-performance liquid chromatography are increasingly challenged due to the need for consumables and solvents, which is contrary to current concerns about green industry in the cosmetics field. Moreover, such methods fail to provide spatially resolved chemical information. In order to investigate the distribution of ingredients in the dried film, Confocal Raman imaging (CRI) coupled to Non-negatively Constrained Least Squares (NCLS) analysis was used. The reconstructed heat maps from a range of films containing systematically varying PHE concentrations highlighted the changes in spectral contribution from each of the ingredients. First, using NCLS scores it was demonstrated that the distributions of PVA, Glycerol, SepicideTM HB and PHE were homogenous, with respective relative standard deviations (RSD) of 3.33%, 2.48%, 2.72% and 6.27%. Second, the respective relationships between ingredient concentrations in the films and their Raman responses, and the spectral abundance were established. Finally, a model for absolute quantification for PHE was be constructed using the percentage of spectral abundance. The prepared %w/w concentrations regressed against predicted %w/w concentrations, displaying high correlation (R2 = 0.995), while the Root Mean Squared Error (0.0869% w/w PHE) confirmed the precision of the analysis. The mean percent relative error of 3.75% indicates the accuracy to which the concentration in dried films could be determined, further supporting the suitability of CRI for analysis of composite solid film matrix. Ultimately, it was demonstrated that nanoformulation of hydrophobic PHE provides homogenous distribution in PVA based film-forming systems independent of the concentration of NanoPHE used in the formula.

An Alternating Rank-k Nonnegative Least Squares Framework (ARkNLS) for Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is a prominent technique for data dimensionality reduction that has been widely used for text mining, computer vision, pattern discovery, and bioinformatics. In this paper, a framework called ARkNLS (Alternating Rank-k Nonnegativity constrained Least Squares) is proposed for computing NMF. First, a recursive formula for the solution of the rank-k nonnegativity-constrained least squares (NLS) is established. This recursive formula can be used to derive the closed-form solution for the Rank-k NLS problem for any integer k $\ge$ 1. As a result, each subproblem for an alternating rank-k nonnegative least squares framework can be obtained based on this closed form solution. Assuming that all matrices involved in rank-k NLS in the context of NMF computation are of full rank, two of the currently best NMF algorithms HALS (hierarchical alternating least squares) and ANLS-BPP (Alternating NLS based on Block Principal Pivoting) can be considered as special cases of ARkNLS with k = 1 and k = r for rank r NMF, respectively. This paper is then focused on the framework with k = 3, which leads to a new algorithm for NMF via the closed-form solution of the rank-3 NLS problem. Furthermore, a new strategy that efficiently overcomes the potential singularity problem in rank-3 NLS within the context of NMF computation is also presented. Extensive numerical comparisons using real and synthetic data sets demonstrate that the proposed algorithm provides state-of-the-art performance in terms of computational accuracy and cpu time.

Kernel-Based Weighted Abundance Constrained Linear Spectral Mixture Analysis for Remotely Sensed Images

Linear spectral mixture analysis (LSMA) is a theory that can be used to perform spectral unmixing where three major LSMA techniques, least squares orthogonal subspace projection (LSOSP), non-negativity constrained least squares (NCLS) and fully constrained least squares (FCLS) have been developed for this purpose. Subsequently, these three techniques were further extended to Fisher's LSMA (FLSMA), weighted abundance constrained LSMA (WAC-LSMA) and kernel-based LSMA (K-LSMA). This paper combines both approaches of KLSMA and WAC-LSMA to derive a most general version of LSMA, kernel-based WACLSMA (KWAC-LSMA), which includes all the above-mentioned LSMA as its special cases. In particular, a new version of kernelizing FLSMA, referred to as kernel FLSMA (K-FLSMA) can be also developed to enhance the FLSMA performance by replacing the weighting matrix used in WAC-LSMA with a matrix specified by the within-class scatter matrix. The utility of the KWAC-LSMA is further demonstrated by multispectral and hyperspectral experiments for performance analysis.

A Fully Constrained Optimization Method for Time-Resolved Multispectral Fluorescence Lifetime Imaging Microscopy Data Unmixing

This paper presents a new unmixing methodology of multispectral fluorescence lifetime imaging microscopy (m-FLIM) data, in which the spectrum is defined as the combination of time-domain fluorescence decays at multiple emission wavelengths. The method is based on a quadratic constrained optimization (CO) algorithm that provides a closed-form solution under equality and inequality restrictions. In this paper, it is assumed that the time-resolved fluorescence spectrum profiles of the constituent components are linearly independent and known a priori. For comparison purposes, the standard least squares (LS) solution and two constrained versions nonnegativity constrained least squares (NCLS) and fully constrained least squares (FCLS) (Heinz and Chang, 2001) are also tested. Their performance was evaluated by using synthetic simulations, as well as imaged samples from fluorescent dyes and ex vivo tissue. In all the synthetic evaluations, the CO obtained the best accuracy in the estimations of the proportional contributions. CO could achieve an improvement ranging between 41% and 59% in the relative error compared to LS, NCLS, and FCLS at different signal-to-noise ratios. A liquid mixture of fluorescent dyes was also prepared and imaged in order to provide a controlled scenario with real data, where CO and FCLS obtained the best performance. The CO and FCLS were also tested with 20 ex vivo samples of human coronary arteries, where the expected concentrations are qualitatively known. A certainty measure was employed to assess the confidence in the estimations made by each algorithm. The experiments confirmed a better performance of CO, since this method is optimal with respect to equality and inequality restrictions in the linear unmixing formulation. Thus, the evaluation showed that CO achieves an accurate characterization of the samples. Furthermore, CO is a computational efficient alternative to estimate the abundance of components in m-FLIM data, since a global optimal solution is always guaranteed in a closed form.

Digital Dewaxing of Raman Signals: Discrimination between Nevi and Melanoma Spectra Obtained from Paraffin-Embedded Skin Biopsies

Malignant melanoma (MM) is the most severe tumor affecting the skin and accounts for three quarters of all skin cancer deaths. Raman spectroscopy is a promising nondestructive tool that has been increasingly used for characterization of the molecular features of cancerous tissues. Different multivariate statistical analysis techniques are used in order to extract relevant information that can be considered as functional spectroscopic descriptors of a particular pathology. Paraffin embedding (waxing) is a highly efficient process used to conserve biopsies in tumor banks for several years. However, the use of non-dewaxed formalin-fixed paraffin-embedded tissues for Raman spectroscopic investigations remains very restricted, limiting the development of the technique as a routine analytical tool for biomedical purposes. This is due to the highly intense signal of paraffin, which masks important vibrations of the biological tissues. In addition to being time consuming and chemical intensive, chemical dewaxing methods are not efficient and they leave traces of the paraffin in tissues, which affects the Raman signal. In the present study, we use independent component analysis (ICA) on Raman spectral images collected on melanoma and nevus samples. The sources obtained from these images are then used to eliminate, using non-negativity constrained least squares (NCLS), the paraffin contribution from each individual spectrum of the spectral images of nevi and melanomas. Corrected spectra of both types of lesion are then compared and classified into dendrograms using hierarchical cluster analysis (HCA).

A novel approach for spectral unmixing, classification, and concentration estimation of chemical and biological agents

In this paper, spectral unmixing methods, which are extensively used in hyperspectral imaging area, are proposed for classification and abundance fraction (concentration) estimation of chemical and biological agents that exist in the mixture form. Several government-furnished datasets, which were collected through the infrared spectrum method, were thoroughly analyzed. Two similarity measures-the spectral angle mapper and spectral information divergence-were investigated in order to provide a quantitative comparison basis with respect to the performance of the applied spectral unmixing methods in the existence of similar and distinct agents. The use of the similarity measures provided valuable information about the signature characteristics of the agents, which led to a better understanding about the capabilities of the investigated methods. The orthogonal subspace projection (OSP) method was investigated as the first unmixing, classification, and abundance estimation technique. It was observed that the OSP method provided good results when the number of agents in the database was small and was composed of distinct agents. However, when the number of agents was incremented by adding agents that share similar characteristics, the abundance estimation accuracy gradually degraded in addition to generating negative abundance fraction estimates. The second investigated unmixing method was called nonnegatively constrained least squares (NCLS). The results and analyses indicated that the NCLS method outperformed the OSP approach by providing considerably more accurate fraction estimates while at the same time not generating any negative fraction estimates; thus, the use of the NCLS method was found to be promising in detection and abundance fraction estimation of chemical and biological agents that exist in the form of mixtures. In addition, efficient implementation of NCLS has resulted in much lower computations than the conventional OSP implementation.

Supramolecular Fluorophores for Biological Studies: Phenylene Vinylene-Amino Acid Amphiphiles

We report here on a family of self-assembling fluorescent organic amphiphiles with a biomolecular L-lysine hydrophile and a photonically active phenylene vinylene hydrophobe. Unlike conventional amphiphiles, these segmented dendrimers feature a rigid, branched hydrophobe, and have packing characteristics controlled by the ratio of cross-sectional areas of the hydrophobe and hydrophile. In dilute solution, the amphiphiles form supramolecular aggregates, which are easily taken in by cells through an endocytic pathway, and have no discernible effect on cell proliferation or morphology. An analogous pyrene-based amphiphile was cytotoxic, suggesting that cell survival may be linked either to the self-assembling nature of the amphiphiles, or to the specific properties of the phenylene vinylene segment. The combination of photonic and biological components in these amphiphiles provides great potential for applications in sensing or delivery of molecules to intracellular targets.

Fast algorithm for the solution of large‐scale non‐negativity‐constrained least squares problems

AbstractAlgorithms for multivariate image analysis and other large‐scale applications of multivariate curve resolution (MCR) typically employ constrained alternating least squares (ALS) procedures in their solution. The solution to a least squares problem under general linear equality and inequality constraints can be reduced to the solution of a non‐negativity‐constrained least squares (NNLS) problem. Thus the efficiency of the solution to any constrained least square problem rests heavily on the underlying NNLS algorithm. We present a new NNLS solution algorithm that is appropriate to large‐scale MCR and other ALS applications. Our new algorithm rearranges the calculations in the standard active set NNLS method on the basis of combinatorial reasoning. This rearrangement serves to reduce substantially the computational burden required for NNLS problems having large numbers of observation vectors. Copyright © 2005 John Wiley & Sons, Ltd.

Constrained subpixel target detection for remotely sensed imagery

Target detection in remotely sensed images can be conducted spatially, spectrally or both. The difficulty of detecting targets in remotely sensed images with spatial image analysis arises from the fact that the ground sampling distance is generally larger than the size of targets of interest in which case targets are embedded in a single pixel and cannot be detected spatially. Under this circumstance target detection must be carried out at subpixel level and spectral analysis offers a valuable alternative. In this paper, the problem of subpixel spectral detection of targets in remote sensing images is considered, where two constrained target detection approaches are studied and compared. One is a target abundance-constrained approach, referred to as nonnegatively constrained least squares (NCLS) method. It is a constrained least squares spectral mixture analysis method which implements a nonnegativity constraint on the abundance fractions of targets of interest. Another is a target signature-constrained approach, called constrained energy minimization (CEM) method. It constrains the desired target signature with a specific gain while minimizing effects caused by other unknown signatures. A quantitative study is conducted to analyze the advantages and disadvantages of both methods. Some suggestions are further proposed to mitigate their disadvantages.

A NMR technique for the analysis of pore structure: Numerical inversion of relaxation measurements

NMR spin—lattice relaxation measurements of fluids contained in porous solids could be a powerful tool for pore size/structure analysis. It does not suffer from many of the inherent disadvantages of mercury porosimetry/nitrogen condensation such as the necessity of a pore shape assumption, measuring only the smallest constriction in a pore, sample compression (porosimetry), and limited pore size range (condensation). However, the successful extraction of the desired pore volume distribution from experimental spin—lattice relaxation measurements requires the solution of an “ill-posed” problem. A recently developed nonnegatively constrained least-squares (NNLS) algorithm was used to approach this problem. Two methods were used including matching relaxation data in real time using the NNLS algorithm and transforming both the data and the governing equations into the frequency time domain and subsequent matching with NNLS. Numerical experiments were conducted using artificial pore size distribution data generated for a variety of distribution types. The two methods were used to analyze both the exact generated data and the generated data containing 5% random error. Direct analysis of the data was more accurate in reproducing the artificial pore structures than the Fourier transform analysis except in the limit of high experimental noise. Deconvolution of the artificial pore structure from exact data gave nearly exact results. Analysis of data after introducing 5% random error to the data yielded less accurate, yet stable results. Small oscillations in the calculated distributions were noted but these are on the same order of or smaller than those normally encountered in porosimetry/condensation experiments.