Metrics of curves in shape optimization and analysis - Andrea Carlo ...
Metrics of curves in shape optimization and analysis - Andrea Carlo ...
Metrics of curves in shape optimization and analysis - Andrea Carlo ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
2.2.3 Examples <strong>of</strong> geometric energy functionals for segmentation 122.3 Geodesic active contour method . . . . . . . . . . . . . . . . . . . 132.3.1 The “geodesic active contour” paradigm . . . . . . . . . . 132.3.2 Example: geodesic active contour edge model . . . . . . . 132.3.3 Implicit assumption <strong>of</strong> H 0 <strong>in</strong>ner product . . . . . . . . . . 142.4 Problems & goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.1 Example: geometric heat flow . . . . . . . . . . . . . . . . 152.4.2 Short length bias . . . . . . . . . . . . . . . . . . . . . . . 152.4.3 Average weighted length . . . . . . . . . . . . . . . . . . . 162.4.4 Flow computations . . . . . . . . . . . . . . . . . . . . . . 162.4.5 Centroid energy . . . . . . . . . . . . . . . . . . . . . . . 172.4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Basic mathematical notions 203.1 Distance <strong>and</strong> metric space . . . . . . . . . . . . . . . . . . . . . . 203.1.1 Metric space . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Banach, Hilbert <strong>and</strong> Fréchet spaces . . . . . . . . . . . . . . . . . 213.2.1 Examples <strong>of</strong> spaces <strong>of</strong> functions . . . . . . . . . . . . . . . 223.2.2 Fréchet space . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 More examples <strong>of</strong> spaces <strong>of</strong> functions . . . . . . . . . . . . 233.2.4 Dual spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.5 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.6 Troubles <strong>in</strong> calculus . . . . . . . . . . . . . . . . . . . . . 253.3 Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Submanifold . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Tangent space <strong>and</strong> tangent bundle . . . . . . . . . . . . . . . . . 263.5 Fréchet Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.6 Riemann & F<strong>in</strong>sler geometry . . . . . . . . . . . . . . . . . . . . 273.6.1 Riemann metric, length . . . . . . . . . . . . . . . . . . . 283.6.2 F<strong>in</strong>sler metric, length . . . . . . . . . . . . . . . . . . . . 283.6.3 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.6.4 M<strong>in</strong>imal geodesics . . . . . . . . . . . . . . . . . . . . . . 293.6.5 Exponential map . . . . . . . . . . . . . . . . . . . . . . . 293.6.6 Hopf–R<strong>in</strong>ow theorem . . . . . . . . . . . . . . . . . . . . . 303.6.7 Drawbacks <strong>in</strong> <strong>in</strong>f<strong>in</strong>ite dimensions . . . . . . . . . . . . . . 303.7 Gradient <strong>in</strong> abstract differentiable manifold . . . . . . . . . . . . 313.8 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.8.1 Distances <strong>and</strong> groups . . . . . . . . . . . . . . . . . . . . 334 Spaces <strong>and</strong> metrics <strong>of</strong> <strong>curves</strong> 334.1 Classes <strong>of</strong> <strong>curves</strong> . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.2 Gâteaux differentials <strong>in</strong> the space <strong>of</strong> immersed <strong>curves</strong> . . . . . . . 354.3 Group actions on <strong>curves</strong> . . . . . . . . . . . . . . . . . . . . . . . 374.4 Two categories <strong>of</strong> <strong>shape</strong> spaces . . . . . . . . . . . . . . . . . . . 374.5 Geometric <strong>curves</strong> . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.5.1 Research path . . . . . . . . . . . . . . . . . . . . . . . . . 38102