THESIS
2020
xi, 66 pages : illustrations ; 30 cm
Abstract
In this thesis, we would summarize and discuss some concepts in Optimal Transport(OT) from theoretical and practical aspects. First, we are going to introduce
the original OT problem formulated by French geometer Gaspard Monge. Then
we will consider the Kantorovich Relaxation to Monge's problem, since the original
problem is so strict. After introducing the relaxation, Kantorovitch's dual
framework could be applied to OT, which will be transformed into a standard
linear programming problem. Hence solvers can be designed. Some well-known
results, like Brenier theory will also be discussed. Later on semi-discrete OT and
spherical OT will be included. Since the essence of OT is to find the most economic
way to transform one probability distribution to the other, it turned out
to...[
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In this thesis, we would summarize and discuss some concepts in Optimal Transport(OT) from theoretical and practical aspects. First, we are going to introduce
the original OT problem formulated by French geometer Gaspard Monge. Then
we will consider the Kantorovich Relaxation to Monge's problem, since the original
problem is so strict. After introducing the relaxation, Kantorovitch's dual
framework could be applied to OT, which will be transformed into a standard
linear programming problem. Hence solvers can be designed. Some well-known
results, like Brenier theory will also be discussed. Later on semi-discrete OT and
spherical OT will be included. Since the essence of OT is to find the most economic
way to transform one probability distribution to the other, it turned out
to be a very powerful tool in various fields, such as computer graphics, medical
imaging, computer vision, machine learning, etc. In our context, we will apply
OT to inverse problem as an application. To be more specific, the reflector design
problem will be elegantly solved using spherical OT approach.
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