Active Contours Snakes and Level Set Method

Transcription

1 Active Contours Snakes and Level Set Method Niels Chr. Overgaard N. Chr. Overgaard Lecture / 26

2 Outline What is an mage? What is an Active Contour? Deforming Force: The Edge Map The Snake Model The Level Set Representation The Chan-Vese Segmentation Model N. Chr. Overgaard Lecture / 26

3 Outline What is an mage? What is an Active Contour? Deforming Force: The Edge Map The Snake Model The Level Set Representation The Chan-Vese Segmentation Model N. Chr. Overgaard Lecture / 26

4 Outline What is an mage? What is an Active Contour? Deforming Force: The Edge Map The Snake Model The Level Set Representation The Chan-Vese Segmentation Model N. Chr. Overgaard Lecture / 26

5 Outline What is an mage? What is an Active Contour? Deforming Force: The Edge Map The Snake Model The Level Set Representation The Chan-Vese Segmentation Model N. Chr. Overgaard Lecture / 26

6 Outline What is an mage? What is an Active Contour? Deforming Force: The Edge Map The Snake Model The Level Set Representation The Chan-Vese Segmentation Model N. Chr. Overgaard Lecture / 26

7 Outline What is an mage? What is an Active Contour? Deforming Force: The Edge Map The Snake Model The Level Set Representation The Chan-Vese Segmentation Model N. Chr. Overgaard Lecture / 26

8 What is an mage? Consider a rectangular subset of the plane: D = [0, a] [0, b]. D R2 is called the image domain. A gray scale image is a function : D [0, 1] x = (x, y ) D is called a pixel or a point. (x ) is the gray level of at x. N. Chr. Overgaard Lecture 7 A digital image is a regular sampling of / 26

9 What is an mage? Consider a rectangular subset of the plane: D = [0, a] [0, b]. D R2 is called the image domain. A gray scale image is a function : D [0, 1] x = (x, y ) D is called a pixel or a point. (x ) is the gray level of at x. N. Chr. Overgaard Lecture 7 A digital image is a regular sampling of / 26

10 What is an mage? Consider a rectangular subset of the plane: D = [0, a] [0, b]. D R2 is called the image domain. A gray scale image is a function : D [0, 1] x = (x, y ) D is called a pixel or a point. (x ) is the gray level of at x. N. Chr. Overgaard Lecture 7 A digital image is a regular sampling of / 26

11 What is an mage? Consider a rectangular subset of the plane: D = [0, a] [0, b]. D R2 is called the image domain. A gray scale image is a function : D [0, 1] x = (x, y ) D is called a pixel or a point. (x ) is the gray level of at x. N. Chr. Overgaard Lecture 7 A digital image is a regular sampling of / 26

12 What is an Active Contour? mage segmentation with an active contour: xyz N. Chr. Overgaard Lecture / 26

13 Deforming Force The Edge Map Compute the image gradient: ( (x) (x) ) (x) =, x y An edge map is a function based on the image gradient, e.g. V (x) = (x) or V (x) = (x). The deforming force is (minus) the gradient of the edgemap: F = V. N. Chr. Overgaard Lecture / 26

14 The Snake Energy The contour is a parametrized curve: γ : [0, 2π] D R2. Thus γ(s ) is a pixel in D for every parameter s, 0 s 2π. The snake energy is a function (or functional) whose argument is an entire contour γ : E [γ] = α N. Chr. Overgaard Z 0 2π 1 0 γ (s ) 2 ds +β 2 Z 2π 0 Lecture 7 V (γ(s )) ds / 26

15 The Snake Energy The contour is a parametrized curve: γ : [0, 2π] D R2. Thus γ(s ) is a pixel in D for every parameter s, 0 s 2π. The snake energy is a function (or functional) whose argument is an entire contour γ : E [γ] = α N. Chr. Overgaard Z 0 2π 1 0 γ (s ) 2 ds +β 2 Z 2π 0 Lecture 7 V (γ(s )) ds / 26

16 Minimization of the Snake Energy The mathematical problem: Segmentation = The optimal contour = arg min γ E[γ] Structure: E[γ] = E reg [γ] + E data [γ]. Ereg Regularizing term. Will shorten the contour and make it more smooth. Edata External energy, delity term or data term. s small at image features e.g. edges. α, β Parameters used to weight the relative importance of Eint and E ext. N. Chr. Overgaard Lecture / 26

17 Minimization of the Snake Energy The mathematical problem: Segmentation = The optimal contour = arg min γ E[γ] Structure: E[γ] = E reg [γ] + E data [γ]. Ereg Regularizing term. Will shorten the contour and make it more smooth. Edata External energy, delity term or data term. s small at image features e.g. edges. α, β Parameters used to weight the relative importance of Eint and E ext. N. Chr. Overgaard Lecture / 26

18 Minimization of the Snake Energy The mathematical problem: Segmentation = The optimal contour = arg min γ E[γ] Structure: E[γ] = E reg [γ] + E data [γ]. Ereg Regularizing term. Will shorten the contour and make it more smooth. Edata External energy, delity term or data term. s small at image features e.g. edges. α, β Parameters used to weight the relative importance of Eint and E ext. N. Chr. Overgaard Lecture / 26

19 Minimization of the Snake Energy The mathematical problem: Segmentation = The optimal contour = arg min γ E[γ] Structure: E[γ] = E reg [γ] + E data [γ]. Ereg Regularizing term. Will shorten the contour and make it more smooth. Edata External energy, delity term or data term. s small at image features e.g. edges. α, β Parameters used to weight the relative importance of Eint and E ext. N. Chr. Overgaard Lecture / 26

20 Minimization of the Snake Energy The mathematical problem: Segmentation = The optimal contour = arg min γ E[γ] Structure: E[γ] = E reg [γ] + E data [γ]. Ereg Regularizing term. Will shorten the contour and make it more smooth. Edata External energy, delity term or data term. s small at image features e.g. edges. α, β Parameters used to weight the relative importance of Eint and E ext. N. Chr. Overgaard Lecture / 26

21 Steepest Descent Consider time-dependent contours: γ = γ(s; t) = γ t (s), t 0. The snake energy is minimized using gradient descent: γ t = E[γ t ], ( = d dt ) Here E[γ] is the variational gradient/derivative computed at the contour γ. For the snake energy the variational gradient is E[γ](s) = αγ (s) + β V (γ(s)), ( = d ds ) Derivation requires the theory of calculus of variations. N. Chr. Overgaard Lecture / 26

22 Steepest Descent Consider time-dependent contours: γ = γ(s; t) = γ t (s), t 0. The snake energy is minimized using gradient descent: γ t = E[γ t ], ( = d dt ) Here E[γ] is the variational gradient/derivative computed at the contour γ. For the snake energy the variational gradient is E[γ](s) = αγ (s) + β V (γ(s)), ( = d ds ) Derivation requires the theory of calculus of variations. N. Chr. Overgaard Lecture / 26

23 Steepest Descent Consider time-dependent contours: γ = γ(s; t) = γ t (s), t 0. The snake energy is minimized using gradient descent: γ t = E[γ t ], ( = d dt ) Here E[γ] is the variational gradient/derivative computed at the contour γ. For the snake energy the variational gradient is E[γ](s) = αγ (s) + β V (γ(s)), ( = d ds ) Derivation requires the theory of calculus of variations. N. Chr. Overgaard Lecture / 26

24 Discretizing the Snake N = number of discretization points in [0.2π] σ = 2π/N corresponding step length. si = i σ, i = 1,..., N discretization points. τ = time step. t k = k τ, k = 0, 1, 2,... time points. uki = γ(si, t k ) = γ(i σ, k τ ) uk = (uk1,..., unk )T = = i 'th node. discretized contour. N. Chr. Overgaard Lecture / 26

25 Discretizing the Snake N = number of discretization points in [0.2π] σ = 2π/N corresponding step length. si = i σ, i = 1,..., N discretization points. τ = time step. t k = k τ, k = 0, 1, 2,... time points. uki = γ(si, t k ) = γ(i σ, k τ ) uk = (uk1,..., unk )T = = i 'th node. discretized contour. N. Chr. Overgaard Lecture / 26

26 Discretizing the Snake N = number of discretization points in [0.2π] σ = 2π/N corresponding step length. si = i σ, i = 1,..., N discretization points. τ = time step. t k = k τ, k = 0, 1, 2,... time points. uki = γ(si, t k ) = γ(i σ, k τ ) uk = (uk1,..., unk )T = = i 'th node. discretized contour. N. Chr. Overgaard Lecture / 26

27 Discretizing the Snake N = number of discretization points in [0.2π] σ = 2π/N corresponding step length. si = i σ, i = 1,..., N discretization points. τ = time step. t k = k τ, k = 0, 1, 2,... time points. uki = γ(si, t k ) = γ(i σ, k τ ) uk = (uk1,..., unk )T = = i 'th node. discretized contour. N. Chr. Overgaard Lecture / 26

28 Discretizing the Snake N = number of discretization points in [0.2π] σ = 2π/N corresponding step length. si = i σ, i = 1,..., N discretization points. τ = time step. t k = k τ, k = 0, 1, 2,... time points. uki = γ(si, t k ) = γ(i σ, k τ ) uk = (uk1,..., unk )T = = i 'th node. discretized contour. N. Chr. Overgaard Lecture / 26

29 Discretizing the Snake N = number of discretization points in [0.2π] σ = 2π/N corresponding step length. si = i σ, i = 1,..., N discretization points. τ = time step. t k = k τ, k = 0, 1, 2,... time points. uki = γ(si, t k ) = γ(i σ, k τ ) uk = (uk1,..., unk )T = = i 'th node. discretized contour. N. Chr. Overgaard Lecture / 26

30 Discretizing the Snake Energy f we use the nite dierences for the velocity: γ (iσ) = u i+1 u i σ where D = forward dierence operator. (Obs! Cyclic indexing; u N+1 is interpreted as u 1 ) Discretized snake energy: E[u] = α N i=1 1 2 u i+1 u i σ = (Du) i 2 N σ + β i=1 V (u i )σ N. Chr. Overgaard Lecture / 26

31 Discretizing the Snake Energy f we use the nite dierences for the velocity: γ (iσ) = u i+1 u i σ where D = forward dierence operator. (Obs! Cyclic indexing; u N+1 is interpreted as u 1 ) Discretized snake energy: E[u] = α N i=1 1 2 u i+1 u i σ = (Du) i 2 N σ + β i=1 V (u i )σ N. Chr. Overgaard Lecture / 26

32 Gradient of the Snake Energy We throw away the common factor σ and redene: E[u] = α N i=1 1 2 u i+1 u i σ Compute the partial derivative wrt. the i'th node: 2 + β N i=1 V (u i ) E[u] = α u i+1 2u i u i 1 + u i σ 2 β V (u i ). The i'th component of the gradient of E may be written u i E[u] = α(d 2 u) i + β V (u i ). N. Chr. Overgaard Lecture / 26

33 Gradient of the Snake Energy We throw away the common factor σ and redene: E[u] = α N i=1 1 2 u i+1 u i σ Compute the partial derivative wrt. the i'th node: 2 + β N i=1 V (u i ) E[u] = α u i+1 2u i u i 1 + u i σ 2 β V (u i ). The i'th component of the gradient of E may be written u i E[u] = α(d 2 u) i + β V (u i ). N. Chr. Overgaard Lecture / 26

34 Gradient of the Snake Energy We throw away the common factor σ and redene: E[u] = α N i=1 1 2 u i+1 u i σ Compute the partial derivative wrt. the i'th node: 2 + β N i=1 V (u i ) E[u] = α u i+1 2u i u i 1 + u i σ 2 β V (u i ). The i'th component of the gradient of E may be written u i E[u] = α(d 2 u) i + β V (u i ). N. Chr. Overgaard Lecture / 26

35 Numerics : Two Descent Strategies We use forward dierences for the time derivative: Explicit Euler Scheme: γ(iσ, kτ) = uk+1 i τ u k i Semi-implicit Scheme: u k+1 i u k i τ = α(d 2 u k ) i + β V (u k i ) u k+1 i u k i τ Notice that the external force is non-linear. = α(d 2 u k+1 ) i + β V (u k i ) N. Chr. Overgaard Lecture / 26

36 Numerics : Two Descent Strategies We use forward dierences for the time derivative: Explicit Euler Scheme: γ(iσ, kτ) = uk+1 i τ u k i Semi-implicit Scheme: u k+1 i u k i τ = α(d 2 u k ) i + β V (u k i ) u k+1 i u k i τ Notice that the external force is non-linear. = α(d 2 u k+1 ) i + β V (u k i ) N. Chr. Overgaard Lecture / 26

37 Numerics : Two Descent Strategies We use forward dierences for the time derivative: Explicit Euler Scheme: γ(iσ, kτ) = uk+1 i τ u k i Semi-implicit Scheme: u k+1 i u k i τ = α(d 2 u k ) i + β V (u k i ) u k+1 i u k i τ Notice that the external force is non-linear. = α(d 2 u k+1 ) i + β V (u k i ) N. Chr. Overgaard Lecture / 26

38 Numerics : Evolving the Contour The discrete second derivative is computed as D 2 u = Au, A = Dene F (u) = (F 1 (u),... F N (u)) T where F i (u) = V (u i ), i = 1,..., N. Rearrange the explicit Euler vector equation as: ( ατ ) σ 2 A u k+1 = u k + βτ F (u k ). The contour is evolved u k u k+1 by solving this equation. N. Chr. Overgaard Lecture / 26

39 Numerics : Evolving the Contour The discrete second derivative is computed as D 2 u = Au, A = Dene F (u) = (F 1 (u),... F N (u)) T where F i (u) = V (u i ), i = 1,..., N. Rearrange the explicit Euler vector equation as: ( ατ ) σ 2 A u k+1 = u k + βτ F (u k ). The contour is evolved u k u k+1 by solving this equation. N. Chr. Overgaard Lecture / 26

40 Numerics : Evolving the Contour The discrete second derivative is computed as D 2 u = Au, A = Dene F (u) = (F 1 (u),... F N (u)) T where F i (u) = V (u i ), i = 1,..., N. Rearrange the explicit Euler vector equation as: ( ατ ) σ 2 A u k+1 = u k + βτ F (u k ). The contour is evolved u k u k+1 by solving this equation. N. Chr. Overgaard Lecture / 26

41 Stop Criterion and Parameter Settings...? N. Chr. Overgaard Lecture / 26

42 Examples: Success and Failure of the Snake Model xyz xyz N. Chr. Overgaard Lecture / 26

43 Local Optima: Trivial Equilibrium Solutions At equilibrium (steady state) u k+1 = u k. Thus equilibrium u satises: ( ατ ) σ 2 A u = u + βτ F (u). Suppose u = (c, c,..., c) T 1 = (1, 1,..., 1) T. for some c R 2, then u = 1c where Since A1 = 0 we nd ( ατ σ 2 A ) u = u. Suppose also that V (c) = 0, the F (u) = 0 and the right hand side of the steady state equation becomes u + βτ F (u) = u. Thus u = (c, c,..., c) T is a (trivial) equilibrium. N. Chr. Overgaard Lecture / 26

44 Local Optima: Trivial Equilibrium Solutions At equilibrium (steady state) u k+1 = u k. Thus equilibrium u satises: ( ατ ) σ 2 A u = u + βτ F (u). Suppose u = (c, c,..., c) T 1 = (1, 1,..., 1) T. for some c R 2, then u = 1c where Since A1 = 0 we nd ( ατ σ 2 A ) u = u. Suppose also that V (c) = 0, the F (u) = 0 and the right hand side of the steady state equation becomes u + βτ F (u) = u. Thus u = (c, c,..., c) T is a (trivial) equilibrium. N. Chr. Overgaard Lecture / 26

45 Local Optima: Trivial Equilibrium Solutions At equilibrium (steady state) u k+1 = u k. Thus equilibrium u satises: ( ατ ) σ 2 A u = u + βτ F (u). Suppose u = (c, c,..., c) T 1 = (1, 1,..., 1) T. for some c R 2, then u = 1c where Since A1 = 0 we nd ( ατ σ 2 A ) u = u. Suppose also that V (c) = 0, the F (u) = 0 and the right hand side of the steady state equation becomes u + βτ F (u) = u. Thus u = (c, c,..., c) T is a (trivial) equilibrium. N. Chr. Overgaard Lecture / 26

46 The Level Set Method Parametric representation of the unit circle (snake): γ(s) = (cos(s), sin(s)), 0 s 2π. mplicit representation of the unit circle (level set): Γ = {x = (x, y) R 2 x 2 + y 2 1 = 0}. Generally, a closed curve Γ in the plane is given by: Here φ is called the level set function. Γ = {x = (x, y) R 2 φ(x, y) = 0}. N. Chr. Overgaard Lecture / 26

47 The Level Set Method Parametric representation of the unit circle (snake): γ(s) = (cos(s), sin(s)), 0 s 2π. mplicit representation of the unit circle (level set): Γ = {x = (x, y) R 2 x 2 + y 2 1 = 0}. Generally, a closed curve Γ in the plane is given by: Here φ is called the level set function. Γ = {x = (x, y) R 2 φ(x, y) = 0}. N. Chr. Overgaard Lecture / 26

48 The Level Set Method Parametric representation of the unit circle (snake): γ(s) = (cos(s), sin(s)), 0 s 2π. mplicit representation of the unit circle (level set): Γ = {x = (x, y) R 2 x 2 + y 2 1 = 0}. Generally, a closed curve Γ in the plane is given by: Here φ is called the level set function. Γ = {x = (x, y) R 2 φ(x, y) = 0}. N. Chr. Overgaard Lecture / 26

49 Computing Geometric Entities from φ Outward unit normal: n(x ) = φ(x ), φ(x ) x Γ. Curvature of contour at x Γ: φ(x ) κ(x ) = div. φ(x ) Curve length element d σ : d σ = δ(φ(x )) φ(x ) dx where δ(z ) is the Dirac delta function on R. N. Chr. Overgaard Lecture / 26

50 Computing Geometric Entities from φ Outward unit normal: n(x ) = φ(x ), φ(x ) x Γ. Curvature of contour at x Γ: φ(x ). κ(x ) = div φ(x ) Curve length element d σ : d σ = δ(φ(x )) φ(x ) dx where δ(z ) is the Dirac delta function on R. N. Chr. Overgaard Lecture / 26

51 Computing Geometric Entities from φ Outward unit normal: n(x ) = φ(x ), φ(x ) x Γ. Curvature of contour at x Γ: φ(x ). κ(x ) = div φ(x ) Curve length element d σ : d σ = δ(φ(x )) φ(x ) dx where δ(z ) is the Dirac delta function on R. N. Chr. Overgaard Lecture / 26

52 Kinematics A moving contour is represented by a time-dependent level set function: Γ(t) = {x R 2 φ(x, t) = 0}. The normal velocity of Γ(t) is given by the formula: d dt Γ(t) = φ(x, t)/ t φ(x, t), x Γ(t). To see this, consider a particle t γ(t) moving along with the contour Γ(t). Then clearly φ(γ(t), t) = 0 for all t. Compute and interpret the derivative of this identity! N. Chr. Overgaard Lecture / 26

53 Kinematics A moving contour is represented by a time-dependent level set function: Γ(t) = {x R 2 φ(x, t) = 0}. The normal velocity of Γ(t) is given by the formula: d dt Γ(t) = φ(x, t)/ t φ(x, t), x Γ(t). To see this, consider a particle t γ(t) moving along with the contour Γ(t). Then clearly φ(γ(t), t) = 0 for all t. Compute and interpret the derivative of this identity! N. Chr. Overgaard Lecture / 26

54 Kinematics A moving contour is represented by a time-dependent level set function: Γ(t) = {x R 2 φ(x, t) = 0}. The normal velocity of Γ(t) is given by the formula: d dt Γ(t) = φ(x, t)/ t φ(x, t), x Γ(t). To see this, consider a particle t γ(t) moving along with the contour Γ(t). Then clearly φ(γ(t), t) = 0 for all t. Compute and interpret the derivative of this identity! N. Chr. Overgaard Lecture / 26

55 The Chan-Vese Model. Segment gray scale image : D [0, 1]. Model: Piecewise constant image φ(x ) < 0 model (x ) = cc ifif φ( x) > 0 + The constant areas separated by a contour Γ = {x φ(x ) = 0} Γ should be as short as possible. N. Chr. Overgaard Lecture / 26

56 The Chan-Vese Model. Segment gray scale image : D [0, 1]. Model: Piecewise constant image φ(x ) < 0 model (x ) = cc ifif φ( x) > 0 + The constant areas separated by a contour Γ = {x φ(x ) = 0} Γ should be as short as possible. N. Chr. Overgaard Lecture / 26

57 The Chan-Vese Model. Segment gray scale image : D [0, 1]. Model: Piecewise constant image φ(x ) < 0 model (x ) = cc ifif φ( x) > 0 + The constant areas separated by a contour Γ = {x φ(x ) = 0} Γ should be as short as possible. N. Chr. Overgaard Lecture / 26

58 The Chan-Vese Model. Given a gray scale image : D [0, 1] that we want to segment. The segmentation is described by the contour Γ and the gray levels c, c+ which minimize the following functional: E(Γ, c, c+) = α dσ + ( (x) c ) 2 dx + ( (x) c+) 2 dx Γ int(γ) ext(γ) where = (x) denotes the grayscale image. That is, we consider the variational problem: (Γ, c, c +) = argmin Γ,c,c + E CV (Γ, c, c+). Structure of E: Again E = E reg + E data. Here we have E reg = curve length the regularizing term. E data = squared distance model image/observed image the delity term. N. Chr. Overgaard Lecture / 26

59 The Chan-Vese Model. Given a gray scale image : D [0, 1] that we want to segment. The segmentation is described by the contour Γ and the gray levels c, c+ which minimize the following functional: E(Γ, c, c+) = α dσ + ( (x) c ) 2 dx + ( (x) c+) 2 dx Γ int(γ) ext(γ) where = (x) denotes the grayscale image. That is, we consider the variational problem: (Γ, c, c +) = argmin Γ,c,c + E CV (Γ, c, c+). Structure of E: Again E = E reg + E data. Here we have E reg = curve length the regularizing term. E data = squared distance model image/observed image the delity term. N. Chr. Overgaard Lecture / 26

60 The Chan-Vese Model. Given a gray scale image : D [0, 1] that we want to segment. The segmentation is described by the contour Γ and the gray levels c, c+ which minimize the following functional: E(Γ, c, c+) = α dσ + ( (x) c ) 2 dx + ( (x) c+) 2 dx Γ int(γ) ext(γ) where = (x) denotes the grayscale image. That is, we consider the variational problem: (Γ, c, c +) = argmin Γ,c,c + E CV (Γ, c, c+). Structure of E: Again E = E reg + E data. Here we have E reg = curve length the regularizing term. E data = squared distance model image/observed image the delity term. N. Chr. Overgaard Lecture / 26

61 The Chan-Vese Model. Given a gray scale image : D [0, 1] that we want to segment. The segmentation is described by the contour Γ and the gray levels c, c+ which minimize the following functional: E(Γ, c, c+) = α dσ + ( (x) c ) 2 dx + ( (x) c+) 2 dx Γ int(γ) ext(γ) where = (x) denotes the grayscale image. That is, we consider the variational problem: (Γ, c, c +) = argmin Γ,c,c + E CV (Γ, c, c+). Structure of E: Again E = E reg + E data. Here we have E reg = curve length the regularizing term. E data = squared distance model image/observed image the delity term. N. Chr. Overgaard Lecture / 26

62 The Chan-Vese Model. Given a gray scale image : D [0, 1] that we want to segment. The segmentation is described by the contour Γ and the gray levels c, c+ which minimize the following functional: E(Γ, c, c+) = α dσ + ( (x) c ) 2 dx + ( (x) c+) 2 dx Γ int(γ) ext(γ) where = (x) denotes the grayscale image. That is, we consider the variational problem: (Γ, c, c +) = argmin Γ,c,c + E CV (Γ, c, c+). Structure of E: Again E = E reg + E data. Here we have E reg = curve length the regularizing term. E data = squared distance model image/observed image the delity term. N. Chr. Overgaard Lecture / 26

63 The Chan-Vese Model. Given a gray scale image : D [0, 1] that we want to segment. The segmentation is described by the contour Γ and the gray levels c, c+ which minimize the following functional: E(Γ, c, c+) = α dσ + ( (x) c ) 2 dx + ( (x) c+) 2 dx Γ int(γ) ext(γ) where = (x) denotes the grayscale image. That is, we consider the variational problem: (Γ, c, c +) = argmin Γ,c,c + E CV (Γ, c, c+). Structure of E: Again E = E reg + E data. Here we have E reg = curve length the regularizing term. E data = squared distance model image/observed image the delity term. N. Chr. Overgaard Lecture / 26

64 Computing the Optimal Gray Levels f the contour Γ is xed, then the optimal gray level inside the contour must satisfy 0 = E(Γ, c, c+) = 2 ( (x) c ) dx c int(γ) Hence c = (x) dx int(γ) int(γ) dx similarly c+ = (x) dx ext(γ) ext(γ) dx That is, the mean gray levels inside resp. outside the contour Γ. N. Chr. Overgaard Lecture / 26

65 Computing the Optimal Gray Levels f the contour Γ is xed, then the optimal gray level inside the contour must satisfy 0 = E(Γ, c, c+) = 2 ( (x) c ) dx c int(γ) Hence c = (x) dx int(γ) int(γ) dx similarly c+ = (x) dx ext(γ) ext(γ) dx That is, the mean gray levels inside resp. outside the contour Γ. N. Chr. Overgaard Lecture / 26

66 Computing the Optimal Contour: Gradient Descent The gradient descent search for the optimal Γ is the gradient descent equation: d dt Γ(t) = E(Γ(t)), (0) = Γ 0 given. ( ) Here E(Γ) is the (L 2 -)shape gradient of E at Γ. Since the normal velocity is d Γ(t)/dt = ( φ/ t)/ φ Eq. ( ) becomes or φ/ t φ φ E(φ) φ = 0, φ(x, t This PDE is known as the level set equation. = E(φ) 0) = φ 0(x). N. Chr. Overgaard Lecture / 26

67 Computing the Optimal Contour: Gradient Descent The gradient descent search for the optimal Γ is the gradient descent equation: d dt Γ(t) = E(Γ(t)), (0) = Γ 0 given. ( ) Here E(Γ) is the (L 2 -)shape gradient of E at Γ. Since the normal velocity is d Γ(t)/dt = ( φ/ t)/ φ Eq. ( ) becomes or φ/ t φ φ E(φ) φ = 0, φ(x, t This PDE is known as the level set equation. = E(φ) 0) = φ 0(x). N. Chr. Overgaard Lecture / 26

68 The Shape Gradient and The Level Set Eq. The shape gradient of the Chan-Vese functional is E(Γ) = ακ + ( c ) 2 ( c+) 2 This result can be obtained in several ways. The level set equation for the Chan-Vese Model is: where φ(x, 0) = φ 0 (x) and (1 H(φ)) dx c = (1 H(φ)) dx φ [ ] = ακ + ( c ) 2 ( c+) 2 φ t H(z) denotes the Heaviside function of R. H(φ) dx and c+ = H(φ) dx N. Chr. Overgaard Lecture / 26

69 The Shape Gradient and The Level Set Eq. The shape gradient of the Chan-Vese functional is E(Γ) = ακ + ( c ) 2 ( c+) 2 This result can be obtained in several ways. The level set equation for the Chan-Vese Model is: where φ(x, 0) = φ 0 (x) and (1 H(φ)) dx c = (1 H(φ)) dx φ [ ] = ακ + ( c ) 2 ( c+) 2 φ t H(z) denotes the Heaviside function of R. H(φ) dx and c+ = H(φ) dx N. Chr. Overgaard Lecture / 26

70 Examples: Segmentation in on and two pieces xyz Movie: Segmentation of a banana. xyz Movie: Split and merge segmentation of two bananas. N. Chr. Overgaard Lecture / 26

71 Thank you for your attention! N. Chr. Overgaard Lecture / 26