A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2)

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where a SR-distance map is computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system derived via Pontryagin's Maximum Principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case $\mathcal{C}=1$. Regarding image analysis applications, tracking of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the sub-Riemannian geometry.Comment: Extended version of SSVM 2015 conference article "Data-driven Sub-Riemannian Geodesics in SE(2)

Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging

Left-invariant PDE-evolutions on the roto-translation group $SE(2)$ (and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) proposed by Citti & Sarti, and Petitot, and they include the direction process (for contour completion) proposed by Mumford. This paper presents a thorough study and comparison of the many numerical approaches, which, remarkably, is missing in the literature. Existing numerical approaches can be classified into 3 categories: Finite difference methods, Fourier based methods (equivalent to $SE(2)$-Fourier methods), and stochastic methods (Monte Carlo simulations). There are also 3 types of exact solutions to the PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in previous works by Duits and van Almsick in 2005. Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches. We compute relative errors of all numerical approaches to the exact solutions, and the Fourier based methods show us the best performance with smallest relative errors. We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions, crucial in implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying stochastic processes that overcomes the singular behavior in the origin of time-integrated kernels. Finally, we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.Comment: A final and corrected version of the manuscript is Published in Numerical Mathematics: Theory, Methods and Applications (NM-TMA), vol. (9), p.1-50, 201

The last Charrua Indian; (Uruguay): analysis of the remains of Chief Vaimaca Perú.

Uruguay is the only Latin American country that at present lacks Native populations and little is known about its prehistoric populations. In the construction of National identity, the unique reference to Natives is about Charra Indians, one of the most important ethnic groups that lived in the territory and exterminated in the 1830s. In 1833, four survivors were taken to be exhibited and studied in France, becoming martyrs and a symbol of their nation. The skeletal remains of Chief Perú were preserved and studied mainly by Rivet1; these are the only remains certainly identified as belonging to a Charrúa. In 2002, the French government returned the remains to Uruguay, where they were buried with honours at the National Pantheon. Before the burial, we performed morphological studies as well as extracted samples for DNA analysis. Peru's morphology is coherent with the one of a nomadic warrior: robust body with strong muscular insertions, wounds, and healthy diet based mainly on meat. Here we show that metric and morphological data as well as maternal inherited mitochondrial DNA (mtDNA) hypervariable region I (HVRI) and restriction fragment length polymorphisms (RFLPs), indicate a close relationship with Pampa-Patagonian Indians, and specially, with prehistoric Natives buried in mounds from eastern Uruguay. This last finding is particularly important to understand Uruguayan prehistory and history, raising the debate about who the mound builders were, and showing continuity between them, historic Charrúa Indians, and present populations

Laplacian-Regularized MAP-MRI: Improving Axonal Caliber Estimation

International audienceIn diffusion MRI, the accurate description of the entire diffusion signal from sparse measurements is essential to enable the recovery of microstructural information of the white matter. The recent Mean Apparent Propagator (MAP)-MRI basis is especially well suited for this task, but the basis fitting becomes unreliable in the presence of noise. As a solution we propose a fast and robust analytic Laplacian regularization for MAP-MRI. Using both synthetic diffusion data and human data from the Human Connectome Project we show that (1) MAP-MRI has more accurate microstructure recovery compared to classical techniques, (2) regularized MAP-MRI has lower signal fitting errors compared to the unregularized approach and a positivity constraint on the EAP and (3) that our regularization improves axon radius recovery on human data

La investigación en técnica vocal como herramienta de actualización pedagógica

En tanto equipo abocado al estudio y difusión de nuevas teorías vinculadas con el fenómeno vocal entendido holísticamente, GITeV (Grupo de Investigaciones en Técnica Vocal – FBA) propone una línea de investigación que abarca, además de cuestiones técnicas concernientes a la voz, elementos propios de la Historia de la Pedagogía Vocal. Se propone, así, comprender los avances actuales en relación a la producción vocal como progresos situados en paradigmas científicos bien determinados, que también deben ser analizados en pos de fomentar la ampliación y pertinencia de los conocimientos generados. Durante 2013, el debate por la Pedagogía Vocal Contemporánea se constituye como uno de los ejes de discusión más importantes a nivel mundial. En este sentido, el acceso a la bibliografía reciente (Aycan, Cimen, y Yerer, 2012; Chen, 2012; Moorcroft y Kenny, 2012; Nafisi, 2013; Walders, 2012) permitió profundizar en este tópico y plantear, tomando como referencias algunos antecedentes (Alessandroni, 2011, 2012a, 2013; Reyes, 2009), algunas preguntas técnicas y metodológicas a nivel local. Se plasman en este artículo algunas ideas vinculadas con el paradigma de la Pedagogía Vocal Contemporánea y sus vinculaciones con la Dirección Coral, la Corporalidad y otros campos de conocimiento.Eje temático 1 - La historia del arte revisada, la disciplina y las corrientes interdisciplinares, las instituciones de las artes y las instituciones académicas - Pedagogías en cruceFacultad de Bellas Arte

La investigación en técnica vocal como herramienta de actualización pedagógica

En tanto equipo abocado al estudio y difusión de nuevas teorías vinculadas con el fenómeno vocal entendido holísticamente, GITeV (Grupo de Investigaciones en Técnica Vocal – FBA) propone una línea de investigación que abarca, además de cuestiones técnicas concernientes a la voz, elementos propios de la Historia de la Pedagogía Vocal. Se propone, así, comprender los avances actuales en relación a la producción vocal como progresos situados en paradigmas científicos bien determinados, que también deben ser analizados en pos de fomentar la ampliación y pertinencia de los conocimientos generados. Durante 2013, el debate por la Pedagogía Vocal Contemporánea se constituye como uno de los ejes de discusión más importantes a nivel mundial. En este sentido, el acceso a la bibliografía reciente (Aycan, Cimen, y Yerer, 2012; Chen, 2012; Moorcroft y Kenny, 2012; Nafisi, 2013; Walders, 2012) permitió profundizar en este tópico y plantear, tomando como referencias algunos antecedentes (Alessandroni, 2011, 2012a, 2013; Reyes, 2009), algunas preguntas técnicas y metodológicas a nivel local. Se plasman en este artículo algunas ideas vinculadas con el paradigma de la Pedagogía Vocal Contemporánea y sus vinculaciones con la Dirección Coral, la Corporalidad y otros campos de conocimiento.Eje temático 1 - La historia del arte revisada, la disciplina y las corrientes interdisciplinares, las instituciones de las artes y las instituciones académicas - Pedagogías en cruceFacultad de Bellas Arte