Differential Geometric Aspects in Image Processing
Dr. Marcelo C´ ardenas
Classroom exercises: November 21, 2019
Problem C3.0
Describe a geometric construction of the level sets corresponding to the positive values of the distance function for a convex closed curve.
Problem C3.1
Show that a path connected regular oriented surface has umbilic points every- where, then it is either contained in a plane or a sphere.
Problem C3.2
Let f : Ω → R, Ω⊂ R2 be a greyscale image and consider the parametrised surfaceσ: Ω→R3
σ(u, v) = (f(u, v)
√2 − u
√2, v, u
√2 +f(u, v)
√2 ) i) Check thatσ(U) is a regular surface
ii) Compute a Gauss map
iii) Express the first and second fundamental form of σ in terms of deriva- tives of f.
iv) Show that the Gaussian curvatureK and the mean curvature H are given by
K= fuufvv−fvu2 (1 +fv2+fu2)2 H= 1
2(1 +fv2+fu2)−32((1 +fv2)fuu−2fufvfuv+ (1 +fu2)fvv))
1
