Convex Sets and Convex Functions

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Convex Sets and Convex Functions
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The Barrier Method
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Diagram copied from Boyd and Vandenberghe
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Diagram copied from Boyd and Vandenberghe
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Diagram copied from Boyd and Vandenberghe
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Diagram copied from Boyd and Vandenberghe
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Diagram copied from Boyd and Vandenberghe
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Quasi-Convex Minimization
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Set maximumand minimum s
Is (Smax Smin) lt epsilon
Finish withminimum s
Yes
Smax s
Choose s between Smin and Smax
Smin s
Feasible for this s?
No
Yes
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x
y
z
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Specific Geometric Vision Problems
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X
x1
x2
x3
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X
x
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Direction (unit) vectors from cameras (blue) to
points (black) are given Find the positions of
the cameras and points.
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This does not happen in strictly quasi-convex
optimization problems.
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Conclusions

• -framework for geometric reconstruction
  problems
• Efficient computations with SOCP
• Globally optimal solutions, invariant to
  coordinate systems

Algebraic methods
methods
norm