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Basic Mathematical Toolsfor

Imaging and Visualization

Dr. Tobias Lasser

November 3rd, 2011

http://campar.in.tum.de/Chair/TeachingWs11BasicMathTools

Computer Aided Medical Procedures (I-16)

Technische Universitt Mnchen

http://campar.in.tum.de/Chair/TeachingWs11BasicMathToolshttp://campar.in.tum.de/Chair/TeachingWs11BasicMathToolshttp://campar.in.tum.de/Chair/TeachingWs11BasicMathTools

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Basic Mathematical Tools for Imaging and Visualization - Tobias Lasser

Admin Mid-term exam

- Tuesday, December 13th, 2011, start:16:15- duration: 60 or 90 minutes

- location: MW 0350

Final exam- Thursday, February 9th, 2012, start: 17:15

- duration: 90 minutes- location: tba

allowed material: one handwritten DIN A4 page (or US letter)

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Basic Mathematical Tools for Imaging and Visualization - Tobias Lasser

So far...0. Introduction

1. Linear Algebra

1.1. Three-dimensional space

1.2. Linear spaces

1.3. Bases

1.4. Linear mappings and matrices

1.5. Linear systems and matrices

1.6. Application: Series Expansion Method

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Original

Measurements

Reconstruction

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Basic Mathematical Tools for Imaging and Visualization - Tobias Lasser

Today0. Introduction

1. Linear Algebra

1.1. ...

1.4. Linear mappings and matrices

1.5. Linear systems and matrices

1.6. Application: Series Expansion Method

1.7. Orthogonality

1.8. Change of basis1.9. Solving linear systems

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R N)

,

:R

R

R

,

,

:=

=

= (),= () R

: R R, :=,.

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= . . .

() =

=, . . . ,

(+) = ()

=

a3x=y

3

a2x=y

2

a1x=y

1

x0=0

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R

= cos sin sin cos

,

=

cos sin sin cos

cos sin sin cos

=

= ,

(cos ) + (sin ) = det() =

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R , > R = R

=

.

= = = =

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R ||

=|| = , . . . ,

= ,

R R

= = = =

R

= ,

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R R

R

= .

= =

R > R R

= =

R

=

=

R

R()

,

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Thank you for your attention!