Mathematics Colloquia and Seminars

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The problem of image decomposition arises from the application of removing noise from an image. We seek to decompose an image $f$ into a sum $f=u+v,$ where the component $u$ is piecewise smooth and representing the 'cartoon' part of the image. The residual $v$ is oscillatory and representing the fine details or 'texture' part of the image. We model the cartoon component by using the total variation, denoted $|\cdot|_{BV}$, a semi-norm on space of Bounded Variation. The decomposition is obtained from the energy minimization problem $\inf \{ \lambda |u|_{BV} + || f-u ||^p_{X} \},$ where $\lambda$ is a positive parameter, $p$ is some fixed positive constant, and $X$ is a normed space. We will explore several different choices for the space $X$ and compare how well they capture oscillatory or 'texture' patterns.