Geodesic Snakes Level-Set EvolutionJonathan BronsonCS7960Advanced Image ProcessingApril 8th, 2010ContentMotivationImplicit Contour FormulationHypersurface Embedding ExamplesConclusionsMotivationDrawbacks of previous Snake formulations:Explicit RepresentationParameterization / Reparameterization issuesApproximating Discrete DerivativesFixed TopologyExtention to 3D very complex (active meshes)MotivationNew Approach:Embed contour in higher order surfaceImplicit RepresentationInsensitive to TopologyEasily extends to 3DMathematical FrameworkdCdt=N: SpeedN:NormaldCdt=N:CurvatureMathematical FrameworkCombining terms simple: Still want:Ability to slow/stop on edges/lines/etcImage force term Where have we seen this before?dCdt=NdCdt=gINMathematical FrameworkAnisotropic Diffusion (Perona & Malik)Use gradient magnitude for diffusion speedgI=11∥∇I∥2gI=e−∥∇I∥2(Quadratic) (Exponential)Mathematical FormulationWhat if we overshoot?Want to pull toward edges∂C∂t=gIN−∇gI⋅NNAdvection TermEmbedding Contour C(s,t)into Surface u(x,t)EmbeddingEmbedding function:Contour:ux,tCs,tuC,t=0(Zero level-set)u∈ℜ3C∈ℜ2ux,tCs,tEmbeddingEmbedding FormulationHow does surface vary over time?uCt ,t=0ddtuCt,t=∂u∂t∂C∂x∂x∂t∂C∂y∂y∂t∂C∂z∂z∂tddtuCt ,t=∂ u∂ t∇u⋅dCdtChain RuledCdt=N=−∇u∣∇u∣ddtuCt,t=∂u∂t−∣∇u∣=0∇u∇u∣∇u∣=∣∇u∣∂u∂t=∣∇u∣Hamilton-Jacobi Equation for certain speeds InterpretationSummaryImplicit Solution Solvable using PDE's (stable)Parameterization FreeSeamlessly handles Topological ChangesExtends to 3D in Straightfoward MannerCommon ImplementationsFast Marching MethodFast Iterative
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