Differential Geometric Aspects in Image Processing

Differential Geometric Aspects in Image Processing

Dr. Marcelo C´ ardenas

Classroom exercises: December 12, 2019

Problem C5.1

Let σ : U \l → V \L be a system of geodesic polar coordinates (ρ, θ) with p∈U ⊂TpS.Show that

i)E= 1 ii) Γ²₁₁= 0

iii) Γ¹₁₁=Eρ= 0 and Fρ= 0 iv)F = 0

Problem C5.2

Let

α_t=N×−→

t^α, α(·,0) =α₀(·)

be the equal distance contour propagation around somep∈S,withα₀a geodesic circle. Moreover, letube s.t. {(x, y) : u(x, y) =t}corresponds to the 2D pro- jection ofα(·, t).

i) Show that Π(−→

t^α) =c(−uy, ux) for somec∈R,where Π(x, y, z) = (x, y).

ii) Show that−→

t^α= √^{(−uy,ux,qux−puy)}

u²_x+u²_y+(qux−puy)²,with (p, q) = (^∂z_∂x,^∂z_∂y)

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