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 lowersemicontinuous with respect
to the
L1norm.]
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
αDv(Ω) + 1
2 ∥v − g∥2
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09

Click to edit the document details