Ellipsoidal approximations in problems of control

2006 ◽  
pp. 361-384 ◽  
Author(s):  
István Vályi
Keyword(s):  
Ellipsoidal Approximations

scholarly journals Calculus rules for combinations of ellipsoids and applications

1993 ◽  
Vol 47 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alberto Seeger
Keyword(s):  
Convex Hull ◽  
Convex Polytope ◽  
Finite Family ◽  
Minkowski Sum ◽  
Calculus Rules ◽  
Ellipsoidal Approximations

We derive formulas for the Minkowski sum, the convex hull, the intersection, and the inverse sum of a finite family of ellipsoids. We show how these formulas can be used to obtain inner and outer ellipsoidal approximations of a convex polytope.

Neurosurgery planning based on automated image recognition and optimal path design

2021 ◽  
Vol 69 (8) ◽  
pp. 708-721
Author(s):  
Annika Hackenberg ◽  
Karl Worthmann ◽  
Torben Pätz ◽  
Dörthe Keiner ◽  
Joachim Oertel ◽  
...  
Keyword(s):  
Optimal Path ◽  
Planning Problem ◽  
Simulation And Optimization ◽  
Ellipsoidal Approximations ◽  
Set Up ◽  
The Brain ◽  
Path Planning Problem ◽  
Automated Image Recognition ◽  
Modelling Simulation

Abstract Stereotactic neurosurgery requires a careful planning of cannulae paths to spare eloquent areas of the brain that, if damaged, will result in loss of essential neurological function such as sensory processing, linguistic ability, vision, or motor function. We present an approach based on modelling, simulation, and optimization to set up a computational assistant tool. Thereby, we focus on the modeling of the brain topology, where we construct ellipsoidal approximations of voxel clouds based on processed MRI data. The outcome is integrated in a path-planning problem either via constraints or by penalization terms in the objective function. The surgical planning problem with obstacle avoidance is solved for different types of stereotactic cannulae using numerical simulations. We illustrate our method with a case study using real MRI data.

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