ZoltanZoltanZoltanKato: PhD Course on Kato: PhD Course on Kato: PhD Course on VariationalVariationalVariationaland Level Set Methods in Image processingand Level Set Methods in Image processingand Level Set Methods in Image processingSnakes:Snakes:Active ContoursActive ContoursZoltanZoltanKatoKatohttp://www.cab.uhttp://www.cab.u--szeged.hu/~kato/variational/szeged.hu/~kato/variational/22ZoltanZoltanZoltanKato: PhD Course on Kato: PhD Course on Kato: PhD Course on VariationalVariationalVariationaland Level Set Methods in Image processingand Level Set Methods in Image processingand Level Set Methods in Image processingIntroductionIntroductionProposed by Proposed by Michael Michael KassKassAndrew Andrew WitkinWitkinhttp://http://www.ri.cmu.edu/people/witkin_andrew.htmlwww.ri.cmu.edu/people/witkin_andrew.htmlDemetriDemetriTerzopoulosTerzopouloshttp://http://mrl.nyu.edu/~dtmrl.nyu.edu/~dt//Snakes: Active Contour ModelsSnakes: Active Contour Models. . International Journal of Computer Vision, International Journal of Computer Vision, Vol. 1, pp 321Vol. 1, pp 321--331, 1988.331, 1988.33ZoltanZoltanZoltanKato: PhD Course on Kato: PhD Course on Kato: PhD Course on VariationalVariationalVariationaland Level Set Methods in Image processingand Level Set Methods in Image processingand Level Set Methods in Image processingWhat is a snake?What is a snake?An energy minimizing An energy minimizing splinesplineguided by external constraint guided by external constraint forces and pulled by image forces toward features:forces and pulled by image forces toward features:Edge detectionEdge detectionSubjective contoursSubjective contoursMotion trackingMotion trackingStereo matchingStereo matching….….Images taken from the GVF website: Images taken from the GVF website: http://http://iacl.ece.jhu.edu/projects/gvfiacl.ece.jhu.edu/projects/gvf//44ZoltanZoltanZoltanKato: PhD Course on Kato: PhD Course on Kato: PhD Course on VariationalVariationalVariationaland Level Set Methods in Image processingand Level Set Methods in Image processingand Level Set Methods in Image processingSnake behaviorSnake behaviorA snake falls into the closest A snake falls into the closest locallocalenergy energy minimum.minimum.The local minima of the snake energy comprise The local minima of the snake energy comprise the set of alternative solutionsthe set of alternative solutionsA higher level knowledge is needed to choose A higher level knowledge is needed to choose the the „correct one”„correct one”from these solutionsfrom these solutionsHighHigh--level reasoninglevel reasoningUser interactionUser interactionThese highThese high--level methods can level methods can interactinteractwith the with the contour model by pushing it toward an contour model by pushing it toward an appropriate local minimumappropriate local minimum55ZoltanZoltanZoltanKato: PhD Course on Kato: PhD Course on Kato: PhD Course on VariationalVariationalVariationaland Level Set Methods in Image processingand Level Set Methods in Image processingand Level Set Methods in Image processingSnake behaviorSnake behaviorThey rely on other mechanisms They rely on other mechanisms to place them to place them nearnearthe desired the desired contour.contour.The existence of such an The existence of such an initializerinitializeris is application dependent.application dependent.Even in the case of manual Even in the case of manual initialization, snakes are quite initialization, snakes are quite powerful in refining the user’s input.powerful in refining the user’s input.Basically, snakes are trying to Basically, snakes are trying to match a deformable model to an match a deformable model to an image by means of energy image by means of energy minimization.minimization.Image taken from the GVF website: Image taken from the GVF website: http://http://iacl.ece.jhu.edu/projects/gvfiacl.ece.jhu.edu/projects/gvf//66ZoltanZoltanZoltanKato: PhD Course on Kato: PhD Course on Kato: PhD Course on VariationalVariationalVariationaland Level Set Methods in Image processingand Level Set Methods in Image processingand Level Set Methods in Image processingSnake energySnake energyParametric representation: Parametric representation: v(sv(s)=()=(x(s),y(sx(s),y(s))))EEintint= internal energy due to bending. Serves to impose = internal energy due to bending. Serves to impose piecewise smoothness constraint.piecewise smoothness constraint.EEimageimage= image forces pushing the snake toward image = image forces pushing the snake toward image features (edges, etc…).features (edges, etc…).EEconcon= external constraints are responsible for putting = external constraints are responsible for putting the snake near the desired local minimum. It may come the snake near the desired local minimum. It may come from:from:Higher level interpretationHigher level interpretationUser interaction, etc…User interaction, etc…∫++=10int))(())(())(( dssvEsvEsvEEconimagesnake77ZoltanZoltanZoltanKato: PhD Course on Kato: PhD Course on Kato: PhD Course on VariationalVariationalVariationaland Level Set Methods in Image processingand Level Set Methods in Image processingand Level Set Methods in Image processingInternal energyInternal energyThe snake is a The snake is a controlled continuitycontrolled continuitysplinesplineRegularizes the problemRegularizes the problemThe first order derivative The first order derivative vvss(s(s))makes the makes the splinesplineact like act like a a membrane membrane („elasticity”).(„elasticity”).The second order derivative The second order derivative vvssss(s(s))makes it act like a makes it act like a thinthin--plate plate („rigidity”).(„rigidity”).αα(s(s))andandββ(s)(s)controls the relative importance of controls the relative importance of membrane and thinmembrane and thin--plate termsplate termsSetting Setting ββ(s)=0(s)=0for a point allows the snake to become secondfor a point allows the snake to become second--order discontinuous and develop a corner.order discontinuous and develop a corner.2/)|)(|)(|)(|)((22intsvssvsEsssβα+=88ZoltanZoltanZoltanKato: PhD Course on Kato: PhD Course on Kato: PhD Course on VariationalVariationalVariationaland Level Set Methods in Image processingand Level Set Methods in Image processingand Level
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