-
8/18/2019 06Analysis.pdf
1/15
Basic Mathematical Toolsfor
Imaging and Visualization
Dr. Tobias Lasser
November 17th, 2011
http://campar.in.tum.de/Chair/TeachingWs11BasicMathTools
Computer Aided Medical Procedures (I-16)
Technische Universität München
http://campar.in.tum.de/Chair/TeachingWs11BasicMathToolshttp://campar.in.tum.de/Chair/TeachingWs11BasicMathToolshttp://campar.in.tum.de/Chair/TeachingWs11BasicMathTools
-
8/18/2019 06Analysis.pdf
2/15
Basic Mathematical Tools for Imaging and Visualization - Tobias Lasser
So far...0. Introduction
1. Linear Algebra
2
-
8/18/2019 06Analysis.pdf
3/15
Basic Mathematical Tools for Imaging and Visualization - Tobias Lasser
Today0. Introduction
1. Linear Algebra
2. Analysis
2.1. Metric spaces
2.2. Topology in metric spaces
2.3. Convergence and compactness
3
-
8/18/2019 06Analysis.pdf
4/15
-
8/18/2019 06Analysis.pdf
5/15
: × → R+ ∀ , , ∈
( , ) = ⇐⇒ =
( , ) = ( , )
( , ) ≤ ( , ) + ( , )
( , )
-
8/18/2019 06Analysis.pdf
6/15
p = 1 p = 2
p = ∞ A =
1 0
0 2
=
-
8/18/2019 06Analysis.pdf
7/15
-
8/18/2019 06Analysis.pdf
8/15
M
x
B r (x)
U
Neighborhood
M
x
B ! (x)
y
B ! (y)
Hausdorff separation property
an open set boundary
-
8/18/2019 06Analysis.pdf
9/15
-
8/18/2019 06Analysis.pdf
10/15
a bounded set
( , ) ⊂
sup , ∈ ( , ) < ∞.
⇐⇒ ∃ε > , ∈ ⊂ ε( ).
-
8/18/2019 06Analysis.pdf
11/15
( , ) ( , ) ( )∈N ⊂
⊂ ( )∈N ⊂
( , ) ⊂
=⇒ ,
=⇒
(R, · ∞) ⊂ R
⇐⇒ R
-
8/18/2019 06Analysis.pdf
12/15
R
· R , >
∞ ≤ ≤ ∞ ∀ ∈ R
.
R
-
8/18/2019 06Analysis.pdf
13/15
-
8/18/2019 06Analysis.pdf
14/15
Basic Mathematical Tools for Imaging and Visualization - Tobias Lasser
Table of Contents0. Introduction
1. Linear Algebra
2. Analysis
2.1. Metric spaces
2.2. Topology in metric spaces
2.3. Convergence and compactness
! next: Continuity, Differentiability
14
-
8/18/2019 06Analysis.pdf
15/15
Thank you for your attention!