CS计算机代考程序代写 Multiple View Geometry: Exercise Sheet 10

Multiple View Geometry: Exercise Sheet 10
Prof. Dr. Florian Bernard, Florian Hofherr, Tarun Yenamandra
Computer Vision Group, TU Munich
Link Zoom Room , Password: 307238

Exercise: July 17th, 2019

Part I: Theory

1. Variational Calculus and Euler-Lagrange

(a) Under the assumption that h vanishes at the boundary of Ω, prove that

dE(u)
du

=
∂L(u,∇u)

∂u
− div

(
∂L(u,∇u)
∂(∇u)

)
.

We can expand L(u+ �h,∇u+ �∇h) in terms of �:

L(u+ �h,∇u+ �∇h) = L(u,∇u) + �h
∂L
∂u

+ �∇h
∂L

∂(∇u)
+O(�2)

Inserting into δE(u)
δu

∣∣∣
h

gives

δE(u)

δu

∣∣∣∣
h

= lim
�→0

1


(
�h(x)

∂L
∂u

∣∣∣∣
u(x)

+ �∇h(x)
∂L

∂(∇u)

∣∣∣∣
∇u(x)

+O(�2)

)
dx

The � in the first two terms cancels, and in the O(�2) it will go to zero for �→ 0. Integra-
tion by parts of the second term yields∫


∇h(x)

∂L
∂(∇u)

dx =

∂Ω
h(x)

∂L
∂(∇u)

ds−


h(x)div

(
∂L

∂(∇u)

)
dx =

= −


h(x)div

(
∂L

∂(∇u)

)
dx .

Thus,
δE(u)

δu

∣∣∣∣
h

=



h(x)

(
∂L
∂u
− div

(
∂L

∂(∇u)

))
dx ⇒ claim .

(b) Which condition must hold true for a minimizer u0 of E(u) …

– … in general?

dE(u)
du

= 0 ⇒
∂L(u,∇u)

∂u
= div

(
∂L(u,∇u)
∂(∇u)

)
.

– … if L(u,∇u) = L(u)?
∂L(u)
∂u

= 0 .

1

https://tum-conf.zoom.us/s/62772800235?pwd=SUpZN2QrV0JpeXJyR2R1TWx5cHEwdz09

– … if L(u,∇u) = L(∇u)?

div
(
∂L(∇u)
∂(∇u)

)
= 0 .

2. Multiview Reconstruction as Shape Optimization

(a) Write down the Euler-Lagrange equation for the given energy E(u).
The E-L equations are

dE(u)
du

= −div
(
∂L(∇u)
∂(∇u)

)
= 0 with L(∇u) = ρ|∇u|

Taking the derivative w.r.t. ∇u gives

0 = −div
(
ρ(x)

∇u(x)
|∇u(x)|

)
.

It is also possible (but not neccessarily required) to expand this further using the product
rule for divergence:

div
(
ρ(x)

∇u(x)
|∇u(x)|

)
= ∇ρ(x)>

∇u(x)
|∇u(x)|

+ ρ(x)div
(
∇u(x)
|∇u(x)|

)

(b) Write down one gradient descent iteration for E(u).

u(k+1)(x) = u(k)(x)−τ
dE(u)

du
= u(k)(x)+τ

(
∇ρ(x)>

∇u(x)
|∇u(x)|

+ ρ(x)div
(
∇u(x)
|∇u(x)|

))

2