Zoltan Kato: PhD Course on Variational and Level Set Methods in Image processing Zoltan Kato: PhD Course on Kato: PhD Course on Variational Variational and Level Set Methods in Image processing and Level Set Methods in Image processing Zoltan Snakes: Snakes: Snakes: Snakes: Active Contours Active Contours Active Contours Active Contours Zoltan Kato Kato Zoltan http://www.cab.u- -szeged.hu/~kato/variational/ szeged.hu/~kato/variational/ http://www.cab.u
Zoltan Kato: PhD Course on Variational and Level Set Methods in Image processing Zoltan Kato: PhD Course on Kato: PhD Course on Variational Variational and Level Set Methods in Image processing and Level Set Methods in Image processing Zoltan 2 2 Introduction Introduction Introduction Introduction � Proposed by Proposed by � � Michael Michael Kass Kass � � Andrew Andrew Witkin Witkin � � http:// � http://www.ri.cmu.edu/people/witkin_andrew.html www.ri.cmu.edu/people/witkin_andrew.html � Demetri Demetri Terzopoulos Terzopoulos � � http:// � http://mrl.nyu.edu/~dt mrl.nyu.edu/~dt/ / Snakes: Active Contour Models . . Snakes: Active Contour Models International Journal of Computer Vision, International Journal of Computer Vision, Vol. 1, pp 321- -331, 1988. 331, 1988. Vol. 1, pp 321
Zoltan Kato: PhD Course on Variational and Level Set Methods in Image processing Zoltan Kato: PhD Course on Kato: PhD Course on Variational Variational and Level Set Methods in Image processing and Level Set Methods in Image processing Zoltan 3 3 What is a snake? What is a snake? What is a snake? What is a snake? � An energy minimizing An energy minimizing spline spline guided by external constraint guided by external constraint � forces and pulled by image forces toward features: forces and pulled by image forces toward features: � Edge detection Edge detection � � Subjective contours Subjective contours � � Motion tracking Motion tracking � � Stereo matching Stereo matching � � …. …. � Images taken from the GVF website: Images taken from the GVF website: http:// http://iacl.ece.jhu.edu/projects/gvf iacl.ece.jhu.edu/projects/gvf/ /
Zoltan Kato: PhD Course on Variational and Level Set Methods in Image processing Zoltan Kato: PhD Course on Kato: PhD Course on Variational Variational and Level Set Methods in Image processing and Level Set Methods in Image processing Zoltan 4 4 Snake behavior Snake behavior Snake behavior Snake behavior � A snake falls into the closest A snake falls into the closest local local energy energy � minimum. minimum. � The local minima of the snake energy comprise The local minima of the snake energy comprise � the set of alternative solutions the set of alternative solutions � A higher level knowledge is needed to choose A higher level knowledge is needed to choose � the „correct one” „correct one” from these solutions from these solutions the � High High- -level reasoning level reasoning � � User interaction User interaction � � These high These high- -level methods can level methods can interact interact with the with the � contour model by pushing it toward an contour model by pushing it toward an appropriate local minimum appropriate local minimum
Zoltan Kato: PhD Course on Variational and Level Set Methods in Image processing Zoltan Kato: PhD Course on Kato: PhD Course on Variational Variational and Level Set Methods in Image processing and Level Set Methods in Image processing Zoltan 5 5 Snake behavior Snake behavior Snake behavior Snake behavior � They rely on other mechanisms They rely on other mechanisms � to place them near near the desired the desired to place them contour. contour. � The existence of such an The existence of such an initializer initializer is 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/gvf iacl.ece.jhu.edu/projects/gvf/ /
Zoltan Kato: PhD Course on Variational and Level Set Methods in Image processing Zoltan Kato: PhD Course on Kato: PhD Course on Variational Variational and Level Set Methods in Image processing and Level Set Methods in Image processing Zoltan 6 6 Snake energy Snake energy Snake energy Snake energy � Parametric representation: Parametric representation: v(s v(s)=( )=(x(s),y(s x(s),y(s)) )) � 1 ∫ = + + E E ( v ( s )) E ( v ( s )) E ( v ( s )) ds snake int image con 0 � E E int = internal energy due to bending. Serves to impose � int = internal energy due to bending. Serves to impose piecewise smoothness constraint. piecewise smoothness constraint. � E E image = image forces pushing the snake toward image � image = image forces pushing the snake toward image features (edges, etc…). features (edges, etc…). � E E con = external constraints are responsible for putting � con = 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 interpretation Higher level interpretation � � User interaction, etc… User interaction, etc… �
Zoltan Kato: PhD Course on Variational and Level Set Methods in Image processing Zoltan Kato: PhD Course on Kato: PhD Course on Variational Variational and Level Set Methods in Image processing and Level Set Methods in Image processing Zoltan 7 7 Internal energy Internal energy Internal energy Internal energy � The snake is a The snake is a controlled continuity controlled continuity spline spline � � Regularizes the problem Regularizes the problem � = α + β 2 2 E ( ( s ) | v ( s ) | ( s ) | v ( s ) | ) / 2 int s ss � The first order derivative The first order derivative v v s (s) ) makes the makes the spline spline act like act like s (s � a membrane membrane („elasticity”). („elasticity”). a � The second order derivative The second order derivative v v ss (s) ) makes it act like a makes it act like a � ss (s thin- -plate plate („rigidity”). („rigidity”). thin and β β (s) � α α (s (s) ) and (s) controls the relative importance of controls the relative importance of � membrane and thin- -plate terms plate terms membrane and thin β (s)=0 Setting β � Setting (s)=0 for a point allows the snake to become second for a point allows the snake to become second- - � order discontinuous and develop a corner. order discontinuous and develop a corner.
Zoltan Kato: PhD Course on Variational and Level Set Methods in Image processing Zoltan Kato: PhD Course on Kato: PhD Course on Variational Variational and Level Set Methods in Image processing and Level Set Methods in Image processing Zoltan 8 8 Image forces Image forces Image forces Image forces � Attracts the snake to features (data term) Attracts the snake to features (data term) � = + + E w E w E w E image line line edge edge term term � lines lines : the simplest functional is the image : the simplest functional is the image � intensity: E E line =I(x,y I(x,y) ) intensity: line = � Depending on the sign of Depending on the sign of w w line , the snake will be � line , the snake will be attracted to the lightest or darkest nerby nerby contour contour attracted to the lightest or darkest ∇ I(x,y)| | ∇ � edges edges : one can simply set : one can simply set E E edge = - - | I(x,y)| 2 2 � edge = � attracts the snake to large intensity gradients. attracts the snake to large intensity gradients. � � terminations terminations : discussed later : discussed later �
Zoltan Kato: PhD Course on Variational and Level Set Methods in Image processing Zoltan Kato: PhD Course on Kato: PhD Course on Variational Variational and Level Set Methods in Image processing and Level Set Methods in Image processing Zoltan 9 9 Snake convergence Snake convergence Snake convergence Snake convergence � If part of a snake finds a low If part of a snake finds a low- -energy energy � feature � � the the spline spline term will pull term will pull feature neighboring parts toward a possible neighboring parts toward a possible continuation of the feature found. continuation of the feature found. � In fact, this places a large energy well In fact, this places a large energy well � around a good local minimum around a good local minimum Video taken from the website: Video taken from the website: http://www http://www- -2.cs.cmu.edu/afs/cs/user/aw/www/gallery.html 2.cs.cmu.edu/afs/cs/user/aw/www/gallery.html
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