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  • 1/60

    Volume reconstruction from slices using phase field method

    Elie Bretin (with F. Dayrens and S. Masnou)

    Contrôle, Problème inverse et Applications Clermont-Ferrand

    Sep. 2017

  • 2/60

    Motivation : Magnetic resonance Imaging

  • 3/60

    Surface reconstruction from slices

    Find the set Ω∗ as a minimizer of

    JΩ1,Ω2(Ω) =

    J(Ω) if Ω1 ⊂ Ω ⊂ Ωc2+∞ otherwise , where J is a surface geometric energy as the perimeter or the Willmore energy.

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    Other application

    Regularization of discrete contours

    Ω1

    Ω2

    Find the set Ω∗ such as

    Ω∗ = arg min Ω1⊂Ω⊂Ωc2

    W(Ω), withW(Ω) = ∫ ∂Ω

    H2dHn−1

    where Ω1 and Ω2 are two given set such as Ω1 ⊂ Ωc2

  • 5/60

    Perimeter and mean curvature flow

    Perimeter P(Ω) =

    ∫ ∂Ω

    1dHn−1

    Shape derivative

    P′(Ω)(θ) = ∫ ∂Ω

    H θ.n dHn−1,

    where n and H denote the normal and the mean curvature. L2 gradient flow of P ⇒ the normal velocity Vn satisfies

    Vn = −H.

    Γ0

    Γt V(x)n����

    �� �� �� ��

  • 6/60

    Some properties of mean curvature flow t → Ω(t)

    Local existence for convex initial set. The set Ω(t) stay convex, converges to a point and becomes asymptotically spherical [Huisken 1984]

    In dimension 2, local existence for smooth closed curves. The set Ω(t) becomes convex in finite time, converges to a point and becomes asymptotically spherical [Gage and Hamilton 1986], [Grayson 1987]

    In dimension n > 2 : singularities in finite time [Grayson 1989]

    Inclusion principle [Ecker 2002]:

    Ω1(0) ⊂ Ω2(0) then Ω1(t) ⊂ Ω2(t), ∀t ∈ [0,T ]

  • 7/60

    Example of mean curvature flow in dimension two

    t = 3.0518e−05

    ε = 0.0039062, δ t = 1.5259e−05, N=256

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    t = 0.0020142

    ε = 0.0039062, δ t = 1.5259e−05, N=256

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    t = 0.01001

    ε = 0.0039062, δ t = 1.5259e−05, N=256

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    t = 0.027008

    ε = 0.0039062, δ t = 1.5259e−05, N=256

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    t = 3.0518e−05

    ε = 0.0039062, δ t = 1.5259e−05, N=256

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    t = 0.0070038

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    t = 0.020004

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    t = 0.030014

    ε = 0.0039062, δ t = 1.5259e−05, N=256

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  • 8/60

    Example of mean curvature flow in dimension three

  • 9/60

    Willmore Flow

    Willmore Energy

    W(Ω) = 1 2

    ∫ ∂Ω

    H2dHn−1,

    L2 gradient flow

    Vn = ∆SH + |A |2H − 1 2

    H3,

    where |A |2 = ∑ κ2i . In dimension 2 :

    Vn = ∆SH + 1 2

    H3,

    In dimension 3 :

    Vn = ∆SH + 1 2

    H(H2 − 4G),

  • 10/60

    Existence and regularity of Willmore Flow,

    Long time existence : single curve – [Dziuk Kuwert Schatzle-2002],

    Long time existence : higher dimension (small energy) [Kuwert Schatzle-2001]

    But in general, singularities in finite time !

  • 11/60

    Example of Willmore flow in dimension three

    fig: Two smooth evolutions by Willmore flow ; a Clifford’s torus and a Lawson-Kusner surface

  • 12/60

    Mean curvature flow Perimeter

    P(Ω) = ∫ ∂Ω

    1dHn−1

    L2 gradient flow of P ⇒ the normal velocity V satifies V = H

    where H denotes the mean curvature. Example of mean curvature flow

  • 13/60

    A Parametric approach ( [Deckelnick,Dziuk,Elliott])

    A parametric representation

    Γ = {X(s) = (x(s), y(s))}s∈[0,2π]

    Normal vector n(s) at X(s) :

    n(X(s)) = Xs(s)⊥

    |Xs(s)| =

    (y′(s),−x′(s, t))√ x′(s)2 + y′(s)2

    Mean curvature at X(s)

    κ(X(s)) = 1

    |Xs(s)|

    ( Xs(s) |Xs(s)|

    ) s · n(X(s)) = x

    ′(s)y′′(s) − y′(s)x′′(s) (x′(s)2 + y′(s)2)3/2

    Mean curvature flow

    Xt (s) = κ(X(s))n(X(s)) or equivalently Xt = 1 |Xs |

    ( Xs |Xs |

    ) s .

  • 14/60

    The level set method ( [Osher,Sethian])

    An implicit representation of the interface

    Γ = { x ; ϕ(x, t) = 0

    } Normal vector n and curvature κ :

    n(ϕ) = ∇ϕ |∇ϕ| and κ(ϕ) = div

    ( ∇ϕ |∇ϕ|

    )

    Mean curvature flow

    ∂tϕ = κ(ϕ)|∇ϕ| = div ( ∇ϕ |∇ϕ|

    ) |∇ϕ|

  • 15/60

    The level set method

    A Hamilton-Jacobi equation

    ∂tϕ = div ( ∇ϕ |∇ϕ|

    ) |∇ϕ| = ∆ϕ − 〈∇

    2ϕ∇ϕ,∇ϕ〉 |∇φ|2

    Weak solution in sense of viscosity [Evan,Spruck][Chen,Giga,Goto])

    Numerical approach : fast marching method (for transport equation) where the velocity κ(ϕ) is estimated explicitly

    Stability problems as for the explicit parametric approach

  • 16/60

    Phase field method and Cahn Hilliard energy

    Cahn Hilliard energy

    P�(u) = ∫ Rd

    ( � |∇u|2

    2 +

    1 �

    W(u) )

    dx,

    where W(s) = 12s 2(1 − s)2 is a double well potential,

    and � is a small parameter. Approximation in sense of Γ-convergence

    (1) ∀u� → u, lim inf �→0

    P�(u�) ≥ P(u)

    (2) ∀u ∃u� → u such that lim sup �→0

    P�(u�) ≤ P(u)

    Γ-convergence in L1(Rd) [Modica-Mortola77]

    Γ − lim �→0

    P� = cwP

  • 17/60

    Cahn Hilliard energy:

    P�(u) = ∫ (

    � |∇u|2

    2 +

    1 �

    W(u) )

    dx

    Allen Cahn equation as L2 gradient flow of P�

    ∂tu� = ∆u� − 1 �2

    W ′(u�)

    the solution u� is of the form

    u�(x, t) = q ( dist(x,Ω�(t))

    ) +O(�2)

    where q is a profile function.

    The normal velocity of Ω�(t) satisfies [Mottoni-Schatzmann89] [Chen92] [Bellettini-Paolini95]

    V� = H + O(�2)

  • 18/60

    Resolution of Allen Cahn equation

    Resolution of the Allen Cahn equation in Q = [0, 1]n with periodic boundary condition :ut (x, t) = ∆u(x, t) − 1�2 W ′(u(x, t)), for (x, t) ∈ Q × [0,T ],u(x, 0) = u0 ∈ [0, 1]

    Classical semi-implicit scheme :

    un+1 − un = δt ( ∆un+1 − 1

    �2 W ′(un)

    ) Stability constraint : δt ≤ M−1�2 where M = sups∈[0,1]{|W ′′(s)|}

  • 19/60

    A convex-concave decomposition

    Cahn Hilliard energy

    P�(u) = JC(u) + JN(u)

    Semi-implicit scheme is equivalent to minimize implicitly

    P�,un (u) = JC(u) + JN(un) + J′N(u n)(u − un)

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    J(x) = J

    c (x) + J

    N (x)

    J x

    0

    (x)= J c (x) + J

    N (x

    0 ) +J’

    N (x

    0 )*(x − x

    0 )

  • 20/60

    An unconditionally stable scheme

    Semi-implicit minimization

    P�(u) = ∫ Rd

    [ � |∇u|2

    2 + α

    u2

    2

    ] dx +

    ∫ Rd

    [ 1 �

    (W(u) − α 2

    u2) ]

    dx

    An P�-unconditionally stable scheme

    un+1 − un = δt ( ∆un+1 − α

    �2 un+1 − 1

    �2 (W ′(un) − αun)