Optimality Conditions for D.C. Vector Optimization Problems under D.C. Constraints

2000 Mathematics Subject Classification: Primary 90C29; Secondary 49K30.

COMMIT: Convex Optimization Modeling for Microstructure Informed Tractography.

Tractography is a class of algorithms aiming at in vivo mapping the major neuronal pathways in the white matter from diffusion magnetic resonance imaging (MRI) data. These techniques offer a powerful tool to noninvasively investigate at the macroscopic scale the architecture of the neuronal connections of the brain. However, unfortunately, the reconstructions recovered with existing tractography algorithms are not really quantitative even though diffusion MRI is a quantitative modality by nature. As a matter of fact, several techniques have been proposed in recent years to estimate, at the voxel level, intrinsic microstructural features of the tissue, such as axonal density and diameter, by using multicompartment models. In this paper, we present a novel framework to reestablish the link between tractography and tissue microstructure. Starting from an input set of candidate fiber-tracts, which are estimated from the data using standard fiber-tracking techniques, we model the diffusion MRI signal in each voxel of the image as a linear combination of the restricted and hindered contributions generated in every location of the brain by these candidate tracts. Then, we seek for the global weight of each of them, i.e., the effective contribution or volume, such that they globally fit the measured signal at best. We demonstrate that these weights can be easily recovered by solving a global convex optimization problem and using efficient algorithms. The effectiveness of our approach has been evaluated both on a realistic phantom with known ground-truth and in vivo brain data. Results clearly demonstrate the benefits of the proposed formulation, opening new perspectives for a more quantitative and biologically plausible assessment of the structural connectivity of the brain.

Kernel-based Mapping of Meteorological Fields in Complex Orography

Due to the advances in sensor networks and remote sensing technologies, the acquisition and storage rates of meteorological and climatological data increases every day and ask for novel and efficient processing algorithms. A fundamental problem of data analysis and modeling is the spatial prediction of meteorological variables in complex orography, which serves among others to extended climatological analyses, for the assimilation of data into numerical weather prediction models, for preparing inputs to hydrological models and for real time monitoring and short-term forecasting of weather.In this thesis, a new framework for spatial estimation is proposed by taking advantage of a class of algorithms emerging from the statistical learning theory. Nonparametric kernel-based methods for nonlinear data classification, regression and target detection, known as support vector machines (SVM), are adapted for mapping of meteorological variables in complex orography.With the advent of high resolution digital elevation models, the field of spatial prediction met new horizons. In fact, by exploiting image processing tools along with physical heuristics, an incredible number of terrain features which account for the topographic conditions at multiple spatial scales can be extracted. Such features are highly relevant for the mapping of meteorological variables because they control a considerable part of the spatial variability of meteorological fields in the complex Alpine orography. For instance, patterns of orographic rainfall, wind speed and cold air pools are known to be correlated with particular terrain forms, e.g. convex/concave surfaces and upwind sides of mountain slopes.Kernel-based methods are employed to learn the nonlinear statistical dependence which links the multidimensional space of geographical and topographic explanatory variables to the variable of interest, that is the wind speed as measured at the weather stations or the occurrence of orographic rainfall patterns as extracted from sequences of radar images. Compared to low dimensional models integrating only the geographical coordinates, the proposed framework opens a way to regionalize meteorological variables which are multidimensional in nature and rarely show spatial auto-correlation in the original space making the use of classical geostatistics tangled.The challenges which are explored during the thesis are manifolds. First, the complexity of models is optimized to impose appropriate smoothness properties and reduce the impact of noisy measurements. Secondly, a multiple kernel extension of SVM is considered to select the multiscale features which explain most of the spatial variability of wind speed. Then, SVM target detection methods are implemented to describe the orographic conditions which cause persistent and stationary rainfall patterns. Finally, the optimal splitting of the data is studied to estimate realistic performances and confidence intervals characterizing the uncertainty of predictions.The resulting maps of average wind speeds find applications within renewable resources assessment and opens a route to decrease the temporal scale of analysis to meet hydrological requirements. Furthermore, the maps depicting the susceptibility to orographic rainfall enhancement can be used to improve current radar-based quantitative precipitation estimation and forecasting systems and to generate stochastic ensembles of precipitation fields conditioned upon the orography.

Accelerated Microstructure Imaging via Convex Optimization (AMICO) in crossing fibers

Mapping the microstructure properties of the local tissues in the brain is crucial to understand any pathological condition from a biological perspective. Most of the existing techniques to estimate the microstructure of the white matter assume a single axon orientation whereas numerous regions of the brain actually present a fiber-crossing configuration. The purpose of the present study is to extend a recent convex optimization framework to recover microstructure parameters in regions with multiple fibers.

Accelerated Microstructure Imaging via Convex Optimisation for regions with multiple fibres (AMICOx)

This paper reviews and extends our previous work to enable fast axonal diameter mapping from diffusion MRI data in the presence of multiple fibre populations within a voxel. Most of the existing mi-crostructure imaging techniques use non-linear algorithms to fit their data models and consequently, they are computationally expensive and usually slow. Moreover, most of them assume a single axon orientation while numerous regions of the brain actually present more complex configurations, e.g. fiber crossing. We present a flexible framework, based on convex optimisation, that enables fast and accurate reconstructions of the microstructure organisation, not limited to areas where the white matter is coherently oriented. We show through numerical simulations the ability of our method to correctly estimate the microstructure features (mean axon diameter and intra-cellular volume fraction) in crossing regions.

Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H−convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H−convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples.

Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l ∞(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel–Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system’s data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504–1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system’s coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.

On the stability of the Motzkin representation of closed convex sets

A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. This paper analyzes the continuity properties of the set-valued mapping associating to each couple (C,D) formed by a compact convex set C and a closed convex cone D its Minkowski sum C + D. The continuity properties of other related mappings are also analyzed.

Preference Mapping Of Dulce De Leche Commercialized In Brazilian Markets.

Dulce de leche samples available in the Brazilian market were submitted to sensory profiling by quantitative descriptive analysis and acceptance test, as well sensory evaluation using the just-about-right scale and purchase intent. External preference mapping and the ideal sensory characteristics of dulce de leche were determined. The results were also evaluated by principal component analysis, hierarchical cluster analysis, partial least squares regression, artificial neural networks, and logistic regression. Overall, significant product acceptance was related to intermediate scores of the sensory attributes in the descriptive test, and this trend was observed even after consumer segmentation. The results obtained by sensometric techniques showed that optimizing an ideal dulce de leche from the sensory standpoint is a multidimensional process, with necessary adjustments on the appearance, aroma, taste, and texture attributes of the product for better consumer acceptance and purchase. The optimum dulce de leche was characterized by high scores for the attributes sweet taste, caramel taste, brightness, color, and caramel aroma in accordance with the preference mapping findings. In industrial terms, this means changing the parameters used in the thermal treatment and quantitative changes in the ingredients used in formulations.

Phylogenetic And Phylogeographic Mapping Of The Avian Coronavirus Spike Protein-encoding Gene In Wild And Synanthropic Birds.

The evolution and population dynamics of avian coronaviruses (AvCoVs) remain underexplored. In the present study, in-depth phylogenetic and Bayesian phylogeographic studies were conducted to investigate the evolutionary dynamics of AvCoVs detected in wild and synanthropic birds. A total of 500 samples, including tracheal and cloacal swabs collected from 312 wild birds belonging to 42 species, were analysed using molecular assays. A total of 65 samples (13%) from 22 bird species were positive for AvCoV. Molecular evolution analyses revealed that the sequences from samples collected in Brazil did not cluster with any of the AvCoV S1 gene sequences deposited in the GenBank database. Bayesian framework analysis estimated an AvCoV strain from Sweden (1999) as the most recent common ancestor of the AvCoVs detected in this study. Furthermore, the analysis inferred an increase in the AvCoV dynamic demographic population in different wild and synanthropic bird species, suggesting that birds may be potential new hosts responsible for spreading this virus.

Application Of Laser Microdissection Icp-ms For High Resolution Elemental Mapping In Mouse Brain Tissue: A Comparative Study With Laser Ablation Icp-ms.

Mapping of elements in biological tissue by laser induced mass spectrometry is a fast growing analytical methodology in life sciences. This method provides a multitude of useful information of metal, nonmetal, metalloid and isotopic distribution at major, minor and trace concentration ranges, usually with a lateral resolution of 12-160 µm. Selected applications in medical research require an improved lateral resolution of laser induced mass spectrometric technique at the low micrometre scale and below. The present work demonstrates the applicability of a recently developed analytical methodology - laser microdissection associated to inductively coupled plasma mass spectrometry (LMD ICP-MS) - to obtain elemental images of different solid biological samples at high lateral resolution. LMD ICP-MS images of mouse brain tissue samples stained with uranium and native are shown, and a direct comparison of LMD and laser ablation (LA) ICP-MS imaging methodologies, in terms of elemental quantification, is performed.

Using sentinels to detect intersections of convex and nonconvex polygons

We describe finite sets of points, called sentinels, which allow us to decide if isometric copies of polygons, convex or not, intersect. As an example of the applicability of the concept of sentinel, we explain how they can be used to formulate an algorithm based on the optimization of differentiable models to pack polygons in convex sets. Mathematical subject classification: 90C53, 65K05.

QTL mapping for yield components in a tropical maize population using microsatellite markers

QTL mapping provides usefull information for breeding programs since it allows the estimation of genomic locations and genetic effects of chromossomal regions related to the expression of quantitative traits. The objective of this study was to map QTL related to several agronomic important traits associated with grain yield: ear weight (EW), prolificacy (PROL), ear number (NE), ear length (EL) and diameter (ED), number of rows on the ear (NRE) and number of kernels per row on the ear (NKPR). Four hundred F-2:3 tropical maize progenies were evaluated in five environments in Piracicaba, Sao Paulo, Brazil. The genetic map was previously estimated and had 117 microssatelite loci with average distance of 14 cM. Data was analysed using Composite Interval Mapping for each trait. Thirty six QTL were mapped and related to the expression of EW (2), PROL (3), NE (2), EL (5), ED (5), NRE (10), NKPR (5). Few QTL were mapped since there was high GxE interaction. Traits EW, PROL and EN showed high genetic correlation with grain yield and several QTL mapped to similar genomic regions, which could cause the observed correlation. However, further analysis using apropriate statistical models are required to separate linked versus pleiotropic QTL. Five QTL (named Ew1, Ne1, Ed3, Nre3 and Nre10) had high genetic effects, explaining from 10.8% (Nre3) to 16.9% (Nre10) of the phenotypic variance, and could be considered in further studies.

Heteroduplex formation and S1 digestion for mapping alternative splicing sites

The identification of alternatively spliced transcripts has contributed to a better comprehension of developmental mechanisms, tissue-specific physiological processes and human diseases. Polymerase chain reaction amplification of alternatively spliced variants commonly leads to the formation of heteroduplexes as a result of base pairing involving exons common between the two variants. S1 nuclease cleaves single-stranded loops of heteroduplexes and also nicks the opposite DNA strand. In order to establish a strategy for mapping alternative splice-prone sites in the whole transcriptome, we developed a method combining the formation of heteroduplexes between 2 distinct splicing variants and S1 nuclease digestion. For 20 consensuses identified here using this methodology, 5 revealed a conserved splice site after inspection of the cDNA alignment against the human genome (exact splice sites). For 8 other consensuses, conserved splice sites were mapped at 2 to 30 bp from the border, called proximal splice sites; for the other 7 consensuses, conserved splice sites were mapped at 40 to 800 bp, called distal splice sites. These latter cases showed a nonspecific activity of S1 nuclease in digesting double-strand DNA. From the 20 consensuses identified here, 5 were selected for reverse transcription-polymerase chain reaction validation, confirming the splice sites. These data showed the potential of the strategy in mapping splice sites. However, the lack of specificity of the S1 nuclease enzyme is a significant obstacle that impedes the use of this strategy in large-scale studies.

Effect of spatial inhomogeneity on the mapping between strongly interacting fermions and weakly interacting spins

A combined analytical and numerical study is performed of the mapping between strongly interacting fermions and weakly interacting spins, in the framework of the Hubbard, t-J, and Heisenberg models. While for spatially homogeneous models in the thermodynamic limit the mapping is thoroughly understood, we here focus on aspects that become relevant in spatially inhomogeneous situations, such as the effect of boundaries, impurities, superlattices, and interfaces. We consider parameter regimes that are relevant for traditional applications of these models, such as electrons in cuprates and manganites, and for more recent applications to atoms in optical lattices. The rate of the mapping as a function of the interaction strength is determined from the Bethe-Ansatz for infinite systems and from numerical diagonalization for finite systems. We show analytically that if translational symmetry is broken through the presence of impurities, the mapping persists and is, in a certain sense, as local as possible, provided the spin-spin interaction between two sites of the Heisenberg model is calculated from the harmonic mean of the onsite Coulomb interaction on adjacent sites of the Hubbard model. Numerical calculations corroborate these findings also in interfaces and superlattices, where analytical calculations are more complicated.