On the normal exponential map in singular conformal metrics

Brake orbits and homoclinics of autonomous dynamical systems correspond, via Maupertuis principle, to geodesics in Riemannian manifolds endowed with a metric which is singular on the boundary (Jacobi metric). Motivated by the classical, yet still intriguing in many aspects, problem of establishing multiplicity results for brake orbits and homoclinics, as done in [6, 7, 10], and by the development of a Morse theory in [8] for geodesics in such kind of metric, in this paper we study the related normal exponential map from a global perspective.Comment: 10 page

Morse Theory for geodesics in singular conformal metrics

Motivated by the use of degenerate Jacobi metrics for the study of brake orbits and homoclinics, we develop a Morse theory for geodesics in conformal metrics having conformal factors vanishing on a regular hypersurface of a Riemannian manifold.Comment: 22 pages. To appear in Communications in Analysis and Geometr

Functions on the sphere with critical points in pairs and orthogonal geodesic chords

Using an estimate on the number of critical points for a Morse-even function on the sphere $\mathbb S^m$, $m\ge1$, we prove a multiplicity result for orthogonal geodesic chords in Riemannian manifolds with boundary that are diffeomorphic to Euclidean balls. This yields also a multiplicity result for brake orbits in a potential well.Comment: 12 pages, 3 figure

Natural risk warning: comparison of two methodologies

International audienceThe Italian network of "Centri Funzionali" is now reaching operational status both in hydro-meteorological risk forecasting and support to the decision making of administrations that issue natural risk warning. Each centre operates for its district of influence. In order to have a nationwide common standard the National Civil Protection Department proposed a quantitative warning methodology based on the definition of rainfall thresholds correlated to historical damages. In the first phase the thresholds have been defined using two studies that cover all Italy: the VAPI (statistics of extreme rainfall and discharges, see reference) and the AVI (database of historical flood and landslide events and reported damages, see reference). This work presents one year back analysis that compares the new methodology and the one that has been usied since 2000 by the Liguria Region Meteorological Centre with regard to flood warning, pinpointing the performance differences in terms of false and missed alerts

Hyperspectral imaging solutions for brain tissue metabolic and hemodynamic monitoring: past, current and future developments

Hyperspectral imaging (HSI) technologies have been used extensively in medical research, targeting various biological phenomena and multiple tissue types. Their high spectral resolution over a wide range of wavelengths enables acquisition of spatial information corresponding to different light-interacting biological compounds. This review focuses on the application of HSI to monitor brain tissue metabolism and hemodynamics in life sciences. Different approaches involving HSI have been investigated to assess and quantify cerebral activity, mainly focusing on: (1) mapping tissue oxygen delivery through measurement of changes in oxygenated (HbO₂) and deoxygenated (HHb) hemoglobin; and (2) the assessment of the cerebral metabolic rate of oxygen (CMRO₂) to estimate oxygen consumption by brain tissue. Finally, we introduce future perspectives of HSI of brain metabolism, including its potential use for imaging optical signals from molecules directly involved in cellular energy production. HSI solutions can provide remarkable insight in understanding cerebral tissue metabolism and oxygenation, aiding investigation on brain tissue physiological processes

Investigation of the quantification of hemoglobin and cytochrome-c-oxidase in the exposed cortex with near-infrared hyperspectral imaging: a simulation study

SIGNIFICANCE: We present a Monte Carlo (MC) computational framework that simulates near-infrared (NIR) hyperspectral imaging (HSI) aimed at assisting quantification of the in vivo hemodynamic and metabolic states of the exposed cerebral cortex in small animal experiments. This can be done by targeting the NIR spectral signatures of oxygenated (HbO2) and deoxygenated (HHb) hemoglobin for hemodynamics as well as the oxidative state of cytochrome-c-oxidase (oxCCO) for measuring tissue metabolism. AIM: The aim of this work is to investigate the performances of HSI for this specific application as well as to assess key factors for the future design and operation of a benchtop system. APPROACH: The MC framework, based on Mesh-based Monte Carlo (MMC), reproduces a section of the exposed cortex of a mouse from an in vivo image and replicates hyperspectral illumination and detection at multiple NIR wavelengths (up to 121). RESULTS: The results demonstrate: (1) the fitness of the MC framework to correctly simulate hyperspectral data acquisition; (2) the capability of HSI to reconstruct spatial changes in the concentrations of HbO2, HHb, and oxCCO during a simulated hypoxic condition; (3) that eight optimally selected wavelengths between 780 and 900 nm provide minimal differences in the accuracy of the hyperspectral results, compared to the "gold standard" of 121 wavelengths; and (4) the possibility to mitigate partial pathlength effects in the reconstructed data and to enhance quantification of the hemodynamic and metabolic responses. CONCLUSIONS: The MC framework is proved to be a flexible and useful tool for simulating HSI also for different applications and targets

A near-infrared hyperspectral imaging system for quantitative monitoring of hemodynamics and metabolism on the exposed cortex of mice

A near-infrared (NIR) hyperspectral imaging (HSI) system has been developed to measure the hemodynamic (changes in concentration of oxyhemoglobin and deoxyhemoglobin) and the metabolic (changes in concentration of oxidised cytochrome-c-oxidase) responses in the exposed cortex of small animals. Using the extended spectral information of multiple wavelengths in the NIR range between 780 and 900 nm optimal differentiation between the optical signatures of the chromophores (hemoglobin and cytochrome-c-oxidase) can be achieved. The system, called hNIR, is composed of: (1) a high-frame rate, large-format scientific CMOS (sCMOS) camera for image acquisition and (2) a broadband source coupled with a Pellin-Broca prism mounted on a rotating motor for sequential, fast-rate illumination of the target at different spectral bands. The system characterisation highlights the capability of the setup to achieve high spatial resolution over a ~1x1 mm field of view (FOV). Hyperspectral data analysis also includes simulations using a Monte Carlo optical model of HSI, to estimate the average photon pathlength and improve image reconstruction and quantification. The hNIR system described here is an improvement over a previously tested commercial snapshot HSI solution both in terms of spatial resolution and signal-to-noise ratio (SNR). This setup will be used to monitor brain hemodynamic and metabolic changes in the exposed cortex of mice during systemic oxygenation changes

The Use of Supercontinuum Laser Sources in Biomedical Diffuse Optics: Unlocking the Power of Multispectral Imaging

Optical techniques based on diffuse optics have been around for decades now and are making their way into the day-to-day medical applications. Even though the physics foundations of these techniques have been known for many years, practical implementation of these technique were hindered by technological limitations, mainly from the light sources and/or detection electronics. In the past 20 years, the developments of supercontinuum laser (SCL) enabled to unlock some of these limitations, enabling the development of system and methodologies relevant for medical use, notably in terms of spectral monitoring. In this review, we focus on the use of SCL in biomedical diffuse optics, from instrumentation and methods developments to their use for medical applications. A total of 95 publications were identified, from 1993 to 2021. We discuss the advantages of the SCL to cover a large spectral bandwidth with a high spectral power and fast switching against the disadvantages of cost, bulkiness, and long warm up times. Finally, we summarize the utility of using such light sources in the development and application of diffuse optics in biomedical sciences and clinical applications