Browsing by Author "Cueva, Evelyn"

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    Algebraic Reconstruction of Source and Attenuation in SPECT Using First Scattering Measurements
    (Springer, 2018) Cueva, Evelyn; Osses Alvarado, Axel Esteban; Quintana Fresno, Juan Carlos; Tejos Núñez, Cristián Andrés; Courdurier Bettancourt Matias Alejandro; Irarrazaval Mena, Pablo
    Here we present an Algebraic Reconstruction Technique (ART) for solving the identification problem in Single Photon Emission Computed Tomography (SPECT). Traditional reconstruction for SPECT is done by finding the radiation source, nevertheless the attenuation of the surrounding tissue affects the data. In this context, ballistic and first scattering information are used to recover source and attenuation simultaneously. Both measurements are related with the Attenuated Radon Transform and a Klein-Nishina angular type dependency is considered for the scattering. The proposed ART algorithm allow us to obtain good reconstructions of both objects in a few number of iterations.
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    Lipschitz stability for backward heat equation with application to fluorescence microscopy
    (SIAM PUBLICATIONS, 2021) Arratia, Pablo; Courdurier Bettancourt, Matías Alejandro; Cueva, Evelyn; Osses, Axel; Palacios Farias, Benjamin Pablo
    In this work we study a Lipschitz stability result in the reconstruction of a compactly supported initial temperature for the heat equation in R-n, from measurements along a positive time interval and over an open set containing its support. We employ a nonconstructive method which ensures the existence of the stability constant, but it is not explicit in terms of the parameters of the problem. The main ingredients in our method are the compactness of support of the initial condition and the explicit dependency of solutions to the heat equation with respect to it. By means of Carleman estimates we obtain an analogous result for the case when the observation is made along an exterior region omega x (t, T), such that the unobserved part R-n\omega is bounded. In the latter setting, the method of Carleman estimates gives a general conditional logarithmic stability result when initial temperatures belong to a certain admissible set, without the assumption of compactness of support and allowing an explicit stability constant. Furthermore, we apply these results to deduce similar stability inequalities for the heat equation in R and with measurements available on a curve contained in Rx[0,infinity), leading to the derivation of stability estimates for an inverse problem arising in 2D fluorescence microscopy. In order to further understand this Lipschitz stability, in particular, the magnitude of its stability constant with respect to the parameters of the problem, a numerical reconstruction is presented based on the construction of a linear system for the inverse problem in fluorescence microscopy. We investigate the stability constant by analyzing the condition number of the corresponding matrix.