PaperIII_64_2 - MATHEMATICAL TRIPOS Part III Monday 10 June...

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MATHEMATICAL TRIPOS Part III Monday, 10 June, 2013 1:30 pm to 3:30 pm PAPER 64 IMAGE PROCESSING — VARIATIONAL AND PDE METHODS Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 Let Ω = (a, b)2 a square image domain and u Lloc(Ω) an image function. State the definition of the total variation of u and define the space BV (Ω) with corresponding norm. Prove that BV (Ω) is a Banach space. [Hint: You may assume here that BV (Ω) is a normed space and that the total variation is lower-semicontinuous with respect to the L1-norm.] Now, let Ω = R2 and g(x), x Ω be the characteristic function of a ball with centre in the origin and radius R, 0 < R < ∞. Derive an explicit formula for the ROF- minimiser u, that is for { } u = argminv α|Dv|(Ω) + 1 2 ∥v − g∥2
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