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Reliable estimations of a function and its derivatives & Theoretical and numerical study of a shape from shading problem

Abstract : The work presented in the first part of this thesis is the result of a collaboration between Alstom and the RATP. We present various models and algorithms that can be used to bound a real-valued function f defined on an interval I and its (d−1) first derivativesby knowingreliable boundson f in some discrete points and globalboundson its dth derivative. These results are applied to a situation inspired by the railway world. Finally, we present various extensions of our work, and we explain how the previous models can be easily generalized to vector-valuedapplications defined on an interval. The second part of this thesis is dedicated to the theoretical and numerical study of a shape from shading problem, which consists in a surface reconstitution from a black and white picture, by knowing only the shades of gray and the altitude of the surface at some points. We remind how the viscosity solutions framework allows us to obtain a well-posed formulation of this problem. Then we expose an explicit expression of an approximation scheme associated to this problem, and we propose a significant optimization of some algorithmsused to solve numerically such a problem. In the future, the works presented in the two parts of the thesis could be coupled to allow a real-time guidance of flying objects like drones overa given region.
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Jean-Francois Abadie. Reliable estimations of a function and its derivatives & Theoretical and numerical study of a shape from shading problem. Numerical Analysis [math.NA]. Sorbonne Université, 2019. English. ⟨NNT : 2019SORUS445⟩. ⟨tel-02931885⟩

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