SE-Sync: a certifiably correct algorithm for synchronization over the special Euclidean group

Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown group elements (Formula presented.) given noisy measurements of a subset of their pairwise relative transforms (Formula presented.). Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a non-convex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation (MLE) whose minimizer provides an exact maximum-likelihood estimate so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so significantly faster than the Gauss–Newton-based approach that forms the basis of current state-of-the-art techniques

SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group

Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown poses given noisy measurements of a subset of their pairwise relative transforms. Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a nonconvex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides an exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so more than an order of magnitude faster than the Gauss-Newton-based approach that forms the basis of current state-of-the-art techniques

Microstructural and Mechanical Properties of Al2O3 and Al2O3/TiB2 Ceramics Consolidated by Plasma Pressure Compaction

Alumina oxide ceramics were produced by plasma pressure compaction (P2C) sintering process. Two types of pure α-alumina (Al2O3) and a mixture of alumina and titanium diboride (TiB2) powders were used as starting materials. Microstructure and mechanical properties, namely hardness, elastic modulus, and fracture toughness, were analyzed and correlated to the type of the sintered powders and the adopted manufacturing route. The microstructural development and the chemical composition variation induced by the sintering process were assessed by using scanning electron microscopy and x-ray diffraction. Nano-indentation and Chevron notch beam techniques were adopted to estimate the mechanical properties of the sintered ceramics. The conducted analyses show the capability of P2C technique to produce sound alumina ceramics. Pure alumina bulks exhibit a good level of compaction and mechanical properties close to those achievable with conventional sintering processes, such as hot isostatic pressing or spark plasma sintering. No significant alterations in the chemical composition of the ceramics were observed. The addition of the titanium diboride in the alumina powders caused a moderate increase in the grain size lowering the hardness and Young’s modulus of the sintered alumina and, at the same time, increased its fracture toughness to the occurrence of toughening mechanisms, like crack bridging and crack deflection

Perturbations of eigenvalues embedded at threshold: one, two and three dimensional solvable models

We examine perturbations of eigenvalues and resonances for a class of multi-channel quantum mechanical model-Hamiltonians describing a particle interacting with a localized spin in dimension $d=1,2,3$. We consider unperturbed Hamiltonians showing eigenvalues and resonances at the threshold of the continuous spectrum and we analyze the effect of various type of perturbations on the spectral singularities. We provide algorithms to obtain convergent series expansions for the coordinates of the singularities.Comment: 20 page

Asymmetric Organocatalysis and Continuous Chemistry for an Efficient and Cost-Competitive Process to Pregabalin

Herein, we present the scale up development of an innovative synthetic process to pregabalin. The process is underpinned by two enabling technologies critical to its success; continuous chemistry allowed a safe and clean production of nitroalkene, and asymmetric organocatalysis gave access to the chiral intermediate in an enantioenriched form. Crucial to the success of the process was the careful development of a continuous process to nitroalkene and optimization of the organocatalyst and of the reaction conditions to attain remarkably high turn-over frequency in the catalytic asymmetric reaction. Successful recycle of the organocatalysts was also developed in order to achieve a cost-competitive process

Turning renewable feedstocks into a valuable and efficient chiral phosphate salt catalyst

Solketal, the chiral acetonide of glycerol, has been employed as the starting material in the design of a novel punctually chiral phosphate sodium salt for catalytic applications in organic and asymmetric synthesis. The racemate and the two enantiomers of the substrate are economic and commercially available, straightforwardly prepared in high yields from naturally occurring feedstocks. Therefore, remarkably, both enantiomers of the final catalyst can be synthesized by simple procedures in high yield and in compliance with several principles of green chemistry. To further demonstrate the usefulness of the novel catalyst, its application in a solventless protocol for cyanohydrin synthesis from a series of aldehydes has been presented

Spin dependent point potentials in one and three dimensions

We consider a system realized with one spinless quantum particle and an array of $N$ spins 1/2 in dimension one and three. We characterize all the Hamiltonians obtained as point perturbations of an assigned free dynamics in terms of some ``generalized boundary conditions''. For every boundary condition we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin dependent point potentials that could be of interest as multi-component solvable models.Comment: 15 pages, some misprints corrected, one example added, some references modified or adde

Carboxyhemoglobin and oxygen affinity of human blood

We determined normal human blood p50 at various pH values (range 7.0 to 7.6) as a function of the proportion of carboxyhemoglobin (COHb) in total hemoglobin, from 0 to 23%. The d(log p50)/d[COHb] coefficient is 0.00848, independent of pH and 2,3-diphosphoglycerate. The derived equation allows the calculation of p50 as a function of COHb with an approximation of +/- 0.54 mmHg (about 72 Pa), and can be combined with other calculations (Clin Chem 27:1856-1861, 1981; Clin Chem 29:110-114, 1983) to predict p50 under any condition of pH within the range 7.0-7.6, ratio of [2,3-diphosphoglycerate] to [total hemoglobin] (range 0.3-2.5), pCO2 (range 20-90 mmHg), temperature (range 19-43 degrees C), and COHb (range 0-23%)

Dissociative electron attachment to the H2O molecule. I. Complex-valued potential-energy surfaces for the 2B1, 2A1, and 2B2 metastable states of the water anion

We present the results of calculations defining global, three-dimensional representations of the complex-valued potential-energy surfaces of the doublet B1, doublet A1, and doublet B2 metastable states of the water anion that underlie the physical process of dissociative electron attachment to water. The real part of the resonance energies is obtained from configuration-interaction calculations performed in a restricted Hilbert space, while the imaginary part of the energies (the widths) is derived from complex Kohn scattering calculations. A diabatization is performed on the 2A1 and 2B2 surfaces, due to the presence of a conical intersection between them. We discuss the implications that the shapes of the constructed potential-energy surfaces will have upon the nuclear dynamics of dissociative electron attachment to H2O. This work originally appeared as Phys Rev A 75, 012710 (2007). Typesetting errors in the published version have been corrected here.Comment: Corrected version of PRA 75, 012710 (2007