Slide Number 1IntroductionCurve Evolution EquationLagrangian FormulationHamilton-Jacobi Skeleton FLowShock DetectionThresholdingHomotopy Preserving SkeletonsShock Detection Results (2D)Shock Detection Results (3D)Shock GrammarExamples of shock graphWorm ExampleShock grammar definitionHamilton-Jacobi Skeleton and Shock GraphsPeihong ZhuUniversity of UtahPapers: Hamilton-Jacobi Skeleton (Siddiqi et al.)Shock Grammar (Kimia, Siddiqi)Introduction■ Skeleton (medial axis) A thin representation of shape.■ good skeleton: Thin set Homotopic to the original shape Invariant under Euclidean transformations Given the radius, the object can be reconstructed exactlyCurve Evolution EquationEikonal Equation:--vector of curve coordinates-- inward normal-- speed of the frontShocks (skeletal points):Where the curves collapseFrom: PhD thesis Hui Sun, U-PennLagrangian FormulationAction function:--coordinates --velocitiesBy minimizing S, we got:In the special case ofHamilton-Jacobi Skeleton FLowLegendre transformation:Huygen's principle:Hamilton-Jacobi skeleton flow formalism:Shock DetectionAverage outward flux of : Via the divergence theorem:■ Conclusion:Non-medial points give values close to zero;while medial points(shocks) which corresponding to a strong singularities give large values.ThresholdingHigh threshold:Low threshold:Homotopy Preserving Skeletons■ 'simple' point:Its removal does not change the topology of the object.■ Goal:To move the simple points as many as possible and get a thin skeleton.Shock Detection Results (2D)Shock Detection Results (3D)Shock Grammar■ Four types of shocks:Examples of shock graphSize and rotation invariantWorm ExampleAllow deformation:straight bended spiralShock grammar definitionGrammar-- alphabet-- terminal symbols -- start symbols--
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