Jointsegmentation/registrationmodelbyshape ......1 - Introduction Background 5/7 Deformation model ⇒ geometric transformations derived from physical models: elastic models, viscous

Joint segmentation/registration model by shapealignment via weighted total variation

minimization and nonlinear elasticity

Solene Ozere1 and Carole Le Guyader1

1Laboratory of Mathematics, INSA Rouen, Saint-Etienne-du-Rouvray, France

Journee scientifique SMAI-SIGMA 2015, Institut Henri Poincare, Paris

November, 2nd 2015

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Solene Ozere and Carole Le Guyader A Joint Segmentation/Registration Model

Outline

1 Introduction

2 Mathematical Modelling

3 Theoretical Results

4 A Numerical Method of Resolution

5 Numerical Simulations

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1 - Introduction Background 1/7

Image segmentation and image registration are fundamentaltasks:

Image segmentation : aims to partition a given image intomeaningful components.

Fig. 1 - Example of contour segmentation of a slice of a brain. (Permission of ’Laboratory Of NeuroImaging,

UCLA’). On the left, the initial contour. On the right, the obtained result at the end of the process.

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Image registration :

Given two images called Template and Reference ⇒registration consists in determining an optimal diffeomorphictransformation ϕ such that the deformed Template image isaligned with the Reference.

⇒ interest in clinical studies, when comparing an image to adatabase or for volumetric purposes.

For images of the same modality: ⇒ correlate thegeometrical features and the intensity level distribution of theReference and those of the deformed Template.

For images acquired through different mecanisms: ⇒correlate both images while preserving the modality of theTemplate.

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From left to right, top to bottom: Reference R; Template T (mouse atlas and gene expression data); deformed

Template; distortion map drawing the vectors from the grid points from the Reference image to non grid points

after registration every 7 rows and columns; deformed grid.

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In [13, 2013], Sotiras et al. provide an overview of the differentexisting registration methods and according to them, an imageregistration algorithm consists of three main components:

1 a deformation model;

2 an objective function;

3 an optimization method.

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Deformationmodel

⇒ geometric transformations derived from physical models:elastic models, viscous fluid model, diffusion models,curvature-based models and flows of diffeomorphisms;

⇒ geometric transformations derived from interpolation theory:radial basis functions, elastic body splines,piecewise affine models, free-form deformations, etc.;

⇒ knowledge-based geometric transformations:statistically-constrained geometric transformations,biomechanical/biophysical models.

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Objectivefunction

⇒ geometric methods: landmark information;

⇒ iconic methods: intensity-based methods,attribute-based methods,information-theoretic approaches;

⇒ hybrid methods.

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Optimization

⇒ continuous methods: gradient descent, conjugate gradient,Quasi-Newton, stochastic gradient descent methods;

⇒ discrete methods: graph-based, belief propagation,linear programming methods;

⇒ greedy approaches and evolutionary algorithms.

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1 - Introduction Goal 1/3

Proposed Method: falls within the scope of

non-parametric methods: deformation not restricted to aparameterizable set;iconic methods: introduction of an intensity and shape-basedcriterion.

Scope:

devise a theoretically well-motivated registration model in avariational formulation, authorizing large and smoothdeformations, while keeping the deformation maptopology-preserving.

Physical arguments often motivate the way the regularizer isdesigned ⇒ classical regularizers such as linear elasticity notsuitable for problems involving large deformations sinceassuming small strains and the validity of Hooke’s law.

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Proposed Methodology:

⇒ introduction of a geometric dissimilarity measure based onshape comparisons and allowing for joint segmentation andregistration.

⇒ use of the weighted total variation in order to align theedges of the objects (Bresson et al. ([3, 2007])), Chan et al.([5, 2006]), Strang ([14, 1983]));

⇒ the algorithm produces both a smooth mapping betweenthe two shapes and the segmentation of the object containedin the Reference.

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Proposed Methodology: (continuation)

⇒ introduction of a nonlinear-elasticity-based smoother.

⇒ shapes to be matched viewed as isotropic, homogeneous,hyperelastic materials and more precisely as SaintVenant-Kirchhoff materials (see Ciarlet’s book [7, 1985]).⇒ Hyperelasticity is a suitable framework when dealing withlarge and nonlinear deformations.

Rubber, filled elastomers, biological tissues are often modeledwithin the hyperelastic framework.

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1 - Introduction Prior Related Works

Prior related works suggest to jointly treat segmentation andregistration. Among others:

Yezzi et al. ([16, 2001]): curve evolution approach in a levelset framework;

Vemuri et al. ([15, 2003]): coupled PDE model to performboth segmentation and registration by evolving the level setsof the source image;

Lord et al. ([11, 2007]): model based on metric structurecomparisons;

Le Guyader and Vese ([10, 2011]): combines a matchingcriterion founded on the active contour without edges model([6, 2001]) and a nonlinear elasticity-based regularizer(Ciarlet-Geymonat stored energy function).

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2 - Mathematical Modelling General Mathematical Background 1/2

There are forward and backward transformations:

the former is done in the Lagrangian framework where aforward transformation Ψ is sought and grid points withintensity values T (x) are moved and arrive at non-grid pointsy = Ψ(x) with intensity values T (x) = T (Ψ−1(y));

the latter is done in the Eulerian framework (considered here),where a backward transformation ϕ = Ψ−1 is searched andgrid points y in the deformed image originate from non-gridpoints x = ϕ(y) = Ψ−1(y) and are assigned intensity valuesT (ϕ(y)) = T (Ψ−1(y)) = T (x).

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2 - Mathematical Modelling General Mathematical Background 2/2

ϕ(y′)

ϕ(y) y

x

x′

y′

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2 - Mathematical Modelling General Notations

Ω ⊂ R2: connected bounded open subset of R2 with

boundary ∂Ω of class C1.

R : Ω → R the Reference image and T : Ω → R the Templateimage.

For theoretical purposes:

T compactly supported on Ω to ensure that T ϕ iswell-defined.

T is assumed to be Lipschitz continuous with Lipschitzconstant κ > 0 (⇒ T ∈ W 1,∞(R2,R)).

ϕ : Ω → R2: deformation (or transformation). A deformation

is a smooth mapping that is orientation-preserving andinjective, except possibly on ∂Ω (self-contact must be allowed(Ciarlet, [8, 1988])).

u: associated displacement s.t. ϕ = Id + u. The deformationgradient is defined by ∇ϕ = I2 +∇u, Ω → M2(R).

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2 - Mathematical Modelling Functional to be minimized 1/12

Goal: find a smooth deformation field ϕ s.t. the deformedTemplate matches the Reference.⇒ model phrased as a functional minimization problem withunknown ϕ, - inf

ϕI (ϕ) =Edist(ϕ) + Ereg (ϕ) -, combining a

distance measure criterion Edist(ϕ) and a regularizer on ϕ,Ereg (ϕ).

⇒ we start with the construction of the regularizer Ereg (ϕ) onthe deformation ϕ.

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Fundamental concepts/notations for the construction ofthe nonlinear-elasticity-based regularizer:

Right Cauchy-Green strain tensor:C = ∇ϕT∇ϕ = FTF ∈ S2 =

A ∈ M2(R), A = AT

.

⇒ quantifier of the square of local change in distances due todeformation.

Green-Saint Venant strain tensor:E = 1

2

(∇u +∇uT +∇uT∇u

).

⇒ measure of the deviation between ϕ and a rigid deformation.

Notations: A : B = trATB and ||A|| =√A : A (Frobenius

norm).

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Fundamental concepts/notations for the construction ofthe nonlinear-elasticity-based regularizer:

Stored energy function of an isotropic, homogeneous,hyperelastic material:

W(F) = W (E ) =λ

2(trE )

2+ µ trE 2 + o

(||E ||2

), (1)

FTF = I + 2E .

Stored energy function of a Saint Venant-Kirchhoffmaterial:

WSVK(F) = W (E ) =λ

2(trE )

2+ µ trE 2. (2)

Saint Venant-Kirchhoff material: ⇒ simplest one whosestored energy function agrees with expansion (1).

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Construction of the nonlinear-elasticity-based regularizer:

⇒ proposed regularizer on ϕ based on the coupling of thestored energy function WSVK of a Saint Venant-Kirchhoffmaterial and on a term controlling that the Jacobiandeterminant remains close to 1.⇒ thus the deformation map does not exhibit contractions orexpansions that are too large.To sum up, the regularization can be written as

Ereg (ϕ) =

∫

Ω

W (∇ϕ) dx ,

with

W (F ) = WSVK (F ) + µ (detF − 1)2.

(The weighting of the determinant component by parameter µis for technical purpose.)The regularizer on the deformation ϕ is complemented by adissimilarity measure Edist that we now address.

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Construction of the dissimilarity measure:

⇒ based on the unified active contour model developed byBresson et al., [3, 2007] - designed to overcome the limitationof local minima and to deal with global minimum.

Let g : R+ → R+ be an edge detector function satisfying

g(0) = 1, g strictly decreasing, and limr→+∞

g(r) = 0.

⇒ apply the edge detector function to the norm of theReference image gradient : g(|∇R |) with |∇R | : Ω → R

+.From now on, for the sake of conciseness, we set g := g(|∇R |)and for theoretical purposes, we assume that ∃c > 0 such that0 < c < g and that g is Lipschitz continuous.

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure: Left: Image R ; Right: g(|∇R |)

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⇒ Use of the generalization of the notion of function ofbounded variation to the setting of BV -spaces associated witha Muckenhoupt’s weight function (see Baldi [2, 2001]);

⇒ Definition of the weighted BV-space related to weight g ;

Hypotheses on general weight w : Ω0 being a neighborhoodof Ω, w ∈ L1loc(Ω0) assumed to belong to the globalMuckenhoupt’s A1 = A1(Ω) class of weight functions, i.e., wsatisfies the condition:

C w(x) ≥ 1

|B(x , r)|

∫

B(x,r)

w(y) dy , a.e.

in any ball B(x , r) ⊂ Ω0.

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Definition 1

[2, Definition 2] Let w be a weight function in the classA∗1 = v ∈ A1, v lsc. We denote by BV (Ω,w) the set of

functions u ∈ L1(Ω,w) (set of functions that are integrable withrespect to the measure w(x)dx) such that:

sup

∫

Ωu div(φ) dx : |φ| ≤ w everywhere, φ ∈ Lip0(Ω,R

2)

<∞,

(3)

with Lip0(Ω,R2) the space of Lipschitz functions with compact

support. We denote by varw u the quantity (3).

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Weighted Total Variation with weight g

varg u = sup

∫

Ωu div(φ) dx : |φ| ≤ g everywhere, φ ∈ Lip0

,

with g = 11+|∇R|2

.

Interpretation of varg u

If u = 1ΩCwith closed set ΩC ⊂ Ω, C being the boundary of ΩC ,

varg (u = 1ΩC) =

∫

Cg ds.

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Distance measure criterion

Edist(ϕ) = varg T ϕ+ν

2

∫

Ω(T ϕ− R)2 dx .

In the end, denoting by W 1,4(Ω,R2) the Sobolev space offunctions ϕ ∈ L4(Ω,R2) with distributional derivatives up toorder 1 which also belong to L4(Ω), the consideredminimization problem is:

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Functional minimization problem

inf

I (ϕ) = varg T ϕ+

∫

Ωf (x , ϕ(x),∇ϕ(x)) dx (P)

= varg T ϕ+

∫

Ω

[ν2(T (ϕ) − R)2 +W (∇ϕ(x))

]dx

,

with ϕ ∈ Id +W 1,40 (Ω,R2) and f (x , ϕ, ξ) = ν

2 (T (ϕ) − R)2++W (ξ).

Remarks

(i) ϕ ∈ Id +W 1,40 (Ω,R2) means that ϕ = Id on ∂Ω and

ϕ ∈ W 1,4(Ω,R2).

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Remarks (continuation)

(ii) From generalized Holder’s inequality, if ϕ ∈ W 1,4(Ω,R2),then det ∇ϕ ∈ L2(Ω).

(iii) From [1, Corollary 3.2; chain rule],T ϕ ∈ W 1,4(Ω) := W 1,4(Ω,R)⊂ BV (Ω), since Ω isbounded.As g ≤ w∗ with w∗ = 1, BV (Ω) ⊂ BV (Ω, g) andT ϕ ∈ BV (Ω, g).

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3 - Theoretical Results Technical Difficulty

Theoretical issue: Function f in (P) fails to be quasiconvex(weaker definition of convexity) ⇒ raises a drawback oftheoretical nature since we cannot obtain the weak lowersemi-continuity of the functional.

Idea: replace problem (P) by an associated relaxed problem(QP) formulated in terms of the quasiconvex envelope Qf off .Note that the introduced problem (QP) is not the relaxedproblem associated to (P) in the sense of Dacorogna ([9,2008]). One has inf QP ≤ inf P

We start by establishing the explicit expression of thequasiconvex envelope of f and derive the associated relaxedproblem (QP).

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3 - Theoretical Results Introduction of an associated relaxed problem 1/3

Proposition 2

The quasiconvex envelope Qf of f is defined byQf (x , ϕ, ξ) = ν

2 (T (ϕ) − R)2 + QW (ξ), with

QW (ξ) =

W (ξ) if ‖ξ‖2 ≥ 2 λ+µ

λ+2µ ,

Ψ(det ξ) if ‖ξ‖2 < 2 λ+µλ+2µ

and Ψ, the convex

mapping such that Ψ : t 7→ −µ2 t

2 + µ (t − 1)2 + µ(λ+µ)2(λ+2µ) .

The relaxed problem (QP) is chosen to be:

inf

I (ϕ) = varg T ϕ+

∫

ΩQf (x , ϕ(x),∇ϕ(x))

, (QP)

with ϕ ∈ Id+W 1,40 (Ω,R2).

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Remarks

(i) The judicious rewriting of W (ξ) intoW (ξ) = β (‖ξ‖2 − α)2 +Ψ(det ξ) allows to see thatW 1,4(Ω,R2) is a suitable functional space.

(ii) We can prove a stronger result, namely, that the polyconvexenvelope of W , PW , coincides with the quasiconvex envelopeof W : PW = QW .

(iii) We understand better through the proof of Proposition 2. thechoice of the weighting parameter balancing the component(det ξ − 1)2; it has been chosen in order that the mapping Ψis convex.

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Remarks (continuation)

(iv) We emphasize that the extension of the model to the 3D caseis not straightforward. Indeed, in three dimensions, theexpression of WSVK (ξ) involves the cofactor matrix denotedby Cof ξ as follows:

WSVK (ξ) =λ

8

(‖ξ‖2 −

(3 +

2µ

λ

))2

+µ

4

(‖ξ‖4 − 2‖Cof ξ‖2

)

− µ

4λ(2µ + 3λ).

⇒ it is not clear that one can derive the explicit expression of thequasiconvex envelope QW of W .

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3 - Theoretical Results Theoretical Results: Existence of Minimizers

Theorem 3 (Existence of Minimizers)

Assume that there exists ϕ ∈ Id+W 1,40 (Ω,R2) such that

I (ϕ) < +∞. Then the infimum of (QP) is attained.

We now concentrate upon the relation betweeninf QP(= min QP) and inf P when additional hypotheses areassumed. Consider the auxiliary problem (AP):

inf

F(ϕ) =

∫

Ωf (x , ϕ(x),∇ϕ(x)) dx : ϕ ∈ Id +W 1,4

0 (Ω,R2)

,

(AP)

with f given in (P).

The following results can be established successively:

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3 - Theoretical Results Theoretical Results: Relaxation 1/2

Proposition 4

The relaxed problem in the sense of Dacorogna associated to (AP)is defined by:

inf

F(ϕ) =

∫

ΩQf (x , ϕ(x),∇ϕ(x)) dx : ϕ ∈ Id+W 1,4

0 (Ω,R2)

,

(QAP)

with Qf given in Proposition 2.

Proposition 5

The infimum of (QAP) is attained. Let then ϕ∗ ∈ W 1,4(Ω,R2) bea minimizer of the relaxed problem (QAP). Then there exists asequence ϕν∞ν=1 ⊂ ϕ∗ +W 1,4

0 (Ω,R2) such that ϕν → ϕ∗ inL4(Ω,R2) as ν → +∞ and F(ϕν) → F(ϕ∗) as ν → +∞, yieldingto min(QAP) = inf(AP).Moreover, the following holds: ϕν ϕ∗ in W 1,4(Ω,R2).

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3 - Theoretical Results Theoretical Results: Relaxation 2/2

Proposition 6 (Relaxation result)

Let us assume that T ∈ W 2,∞(R2,R), ∇T being Lipschitzcontinuous with Lipschitz constant κ′. Let ϕ ∈ Id+W 1,4

0 (Ω,R2)be a minimizer of the relaxed problem (QP). Then there exists asequence ϕν∞ν=1 ⊂ ϕ+W 1,4

0 (Ω,R2) such that ϕν ϕ inW 1,4(Ω,R2) as ν → ∞ and∫

Ωf (x , ϕν(x),∇ϕν(x)) dx →

∫

ΩQf (x , ϕ(x),∇ϕ(x)) dx. If

moreover ∇ϕν strongly converges to ∇ϕ in L1(Ω,M2(R)), thenone has I (ϕν) → I (ϕ) as ν → ∞ and thereforeinf(QP) = min(QP) = inf(P).

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4 - A Numerical Method of Resolution Description/Analysis of the proposed numerical method 1/14

Motivations and Prior related work:

In [12, 1990], Negron Marrero describes and analyzes anumerical method to detect singular minimizers and to avoidthe Lavrentiev phenomenon for three dimensional problems innonlinear elasticity.⇒ consists in decoupling function ϕ from its gradient and informulating a related decoupled problem under inequalityconstraint.

Idea:

⇒ introduction of an auxiliary variable V simulating theJacobian deformation field ∇ϕ. The underlying idea is toremove the nonlinearity in the derivatives of the deformation;⇒ introduction of an auxiliary variable T simulating thedeformed Template T ϕ;⇒ derivation of a functional minimization problem phrased interms of ϕ, V and T .

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Differences with [12]:

We do not formulate a minimization problem under constraintsbut incorporate Lp-type penalizations.

In [12], the author focuses on the decoupled discretizedproblem (discretized with the finite element method) for whichthe existence of minimizers is guaranteed⇒ we consider the continuous problem.

The author assumes that the finite element approximationssatisfy some convergence hypotheses.

In our case, less regularity is required for the formulation of theinequality constraint.

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Decoupled Problem

Iγ(ϕ,V , T ) =varg T +ν

2‖T (ϕ) − R‖2L2(Ω) +

∫

ΩQW (V ) dx

+γ

2‖V −∇ϕ‖2L2(Ω,M2)

+ γ ‖T − T ϕ‖L1(Ω). (4)

We set W = Id +W 1,20 (Ω,R2) and χ =

V ∈ L4(Ω,M2(R))

.

The decoupled problem consists in minimizing (4) on

W × χ× BV (Ω, g). Then the following theorem holds.

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Theorem 7 (Γ-convergence)

Let (γj) be an increasing sequence of positive real numbers such

that limj→+∞

γj = +∞. Let also(ϕk(γj ),Vk(γj ), Tk(γj )

)be a

minimizing sequence of the decoupled problem with γ = γj . Thenthere exist a subsequence denoted by(ϕN(γζ(j))(γζ(j)),VN(γζ(j))(γζ(j)), TN(γζ(j))(γζ(j))

)of

(ϕk(γj ),Vk(γj ), Tk(γj )

)and a minimizer ϕ of I

(ϕ ∈ Id+W 1,40 (Ω,R2)) such that:

limj→+∞

Iγζ(j)

(ϕN(γζ(j))(γζ(j)),VN(γζ(j))(γζ(j)), TN(γζ(j))(γζ(j))

)= I (ϕ).

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Alternating algorithm

1 We first solve

infT

E (T ) = varg T + γ ‖T − T ϕ‖L1(Ω) (P1)

for fixed ϕ and V .

Convex regularization, Dual formulation of the weighted totalvariation, Chambolle’s projection algorithm ([4, 2004]);Important remark: If T ϕ is a characteristic function, if Tis a minimizer of E , then for almost every µ0 ∈ [0, 1], one hasthat the characteristic function 1x | T (x)>µ0 is a global

minimizer of E .

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Alternating algorithm (Continuation)

2 We then solve (for fixed T ):

infϕ,V

Jǫ(ϕ,V ) +γ

2‖∇ϕ− V ‖2L2(Ω,M2(R))

+ γ ‖T − T ϕ‖L1(Ω),

with

Jǫ(ϕ,V ) =

∫

ΩW (V )Hǫ

(‖V ‖2 − α

)dx

+

∫

ΩΨ(det V )Hǫ

(α− ‖V ‖2

)dx

+ν

2

∫

Ω(T (ϕ)− R)2 dx . (P2)

Hǫ : z 7→ 12

(1 + 2

πArctan z

ǫ

): regularized one-dimensional Heaviside

function.

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Solving of (P1)

Following the same strategy as Bresson’s in [3, 2007], weintroduce an auxiliary variable f such that the problemamounts to minimizing:

infT ,f

varg T + γ‖f ‖L1(Ω) +

1

2θ‖T − T ϕ+ f ‖2L2(Ω)

,

problem decoupled into the two following subproblems:

infT

varg T +

1

2θ‖T − T ϕ+ f ‖2L2(Ω)

; (P ′

1)

inff

γ‖f ‖L1(Ω) +

1

2θ‖T − T ϕ+ f ‖2L2(Ω)

. (P

′′

1 )

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Solving of (P1) (Continuation)

Proposition 8 (Adaptation of Chambolle’s projection algorithm [4])

The solution of (P ′1) is given by

T = T ϕ− f − ΠθK (T ϕ− f ) .

where Π represents the orthogonal projection, and K is the closureofdiv ξ : ξ ∈ C 1

c (Ω,R2), |ξ(x)| ≤ g(x), ∀x ∈ Ω

, | · | being the

Euclidean norm in R2.

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Proposition 9

The solution of (P′′

1 ) is given by

f =

T ϕ− T − θγ if T ϕ− T ≥ θγ

T ϕ− T + θγ if T ϕ− T ≤ −θγ0 otherwise

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Reminder

For fixed ϕ and V ,

infT ,f

varg T + γ ‖f ‖L1(Ω) +

1

2θ‖T − T ϕ+ f ‖2L2(Ω)

;

Then for fixed T and f ,

infϕ,V

ν

2

∫

Ω(T (ϕ)− R)2 dx +

∫

ΩW (V )Hǫ

(‖V ‖2 − α

)dx

+

∫

Ωψ(det V )Hǫ

(α− ‖V ‖2

)dx

+γ

2‖∇ϕ− V ‖2L2(Ω,M2(R))

+1

2θ

∫

Ω(T − T ϕ+ f )2 dx

.

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Solving of (P2)

The Euler-Lagrange equation for ϕ is given by:

0 = ν(Tϕ−R)∇T (ϕ)−γ∆ϕ+γ(divV1

divV2

)−1

θ(f−Tϕ+T )∇T (ϕ),

and the system of equations for V is:

0 = 2βc0V11 (2Hε(c0) + c0δε(c0)) + µV22(detV − 2)

+ γ(V11 −∂ϕ1

∂x);


+ γ(V12 −∂ϕ1

∂y);

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+ γ(V21 −∂ϕ2

∂x);


+ γ(V22 −∂ϕ2

∂y),

with c0 = (‖V ‖2 − α).

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Orientation Preservation

Figure: Obtained global deformation field when topology preservation isnot enforced.

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Regridding Algorithm

Algorithm 1: Regridding.

Initialization: V 0 = I , ϕ0 = Id , regrid count=0.

For k = 0, 1, . . .

(ϕk+1,V k+1) = argminϕ,V

Jε,γ

If det∇ϕk+1 < tol

regrid count=regrid count+1T = T ϕk

save ϕk (regrid count), ϕk+1 = Id , V k+1 = 0

If regrid count>0ϕfinal = ϕ(1) · · · ϕ(regrid count)

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5 - Numerical Simulations Synthetic Images

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5 - Numerical Simulations Mouse Atlas 1/2

Template Reference Template Reference

Deformed Template Segmented Reference Deformed Template Segmented Reference

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5 - Numerical Simulations Mouse Atlas 2/2

V11 V12 V21 V22

∂ϕ1∂x1

∂ϕ1∂x2

∂ϕ2∂x1

∂ϕ2∂x2

Figure: ‖V11 −∂ϕ1

∂x‖∞ = 0.09, ‖V12 −

∂ϕ1

∂y‖∞ = 0.03, ‖V21 −

∂ϕ2

∂x‖∞ = 0.06,

‖V22 −∂ϕ2

∂y‖∞ = 0.07

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5 - Numerical Simulations Slice of a brain

Template Reference Deformation

Deformed Template Segmented Reference Vector field

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5 - Numerical Simulations MRI images of cardiac cycle

Template Reference Deformation

Deformed Template Segmented Reference Vector field

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5 - Numerical Simulations Values of involved parameters

ν θ γ1 γ2 µ λ c

Letter C 1 1 0.1 60 000 3500 10 1

Triangle 1.5 1 0.1 90 000 500 10 1

Atlas11 0.5 1 0.1 80 000 1000 10 10

Atlas12 0.5 1 0.1 80 000 2000 10 10

Brain69 1 1 0.1 80 000 1000 10 10

Heart ED-ES 1 0.1 0.1 90 000 1000 10 10

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6 - ReferencesBibliographical References I

[1] L. Ambrosio and G. Dal Maso, A general chain rule fordistributional derivatives, Proceedings of the AmericanMathematical Society, 108 (1990), pp. 691–702.

[2] A. Baldi, Weighted BV functions, Houston Journal ofMathematics, 27 (2001), pp. 683–705.

[3] X. Bresson, S. Esedoglu, P. Vandergheynst, J.-P.

Thiran, and S. Osher, Fast Global Minimization of the ActiveContour/Snake Model, J. Math. Imaging Vis., 68 (2007),pp. 151–167.

[4] A. Chambolle, An algorithm for total variation minimization andapplications, J. Math. Imaging Vis., 20 (2004), pp. 89–97.

[5] T. Chan, S. Esedoglu, and M. Nikolova, Algorithms forfinding global minimizers of image segmentation and denoisingmodels, SIAM J. Appl. Math., 66 (2006), pp. 1632–1648.

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6 - ReferencesBibliographical References II

[6] T. Chan and L. Vese, Active Contours Without Edges, IEEETrans. Image Process., 10 (2001), pp. 266–277.

[7] P. Ciarlet, Elasticite Tridimensionnelle, Masson, 1985.

[8] , Mathematical Elasticity, Volume I: Three-dimensionalelasticity, Amsterdam etc., North-Holland, 1988.

[9] B. Dacorogna, Direct Methods in the Calculus of Variations,Second Edition, Springer, 2008.

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[15] B. Vemuri, J. Ye, Y. Chen, and C. Leonard, ImageRegistration via level-set motion: Applications to atlas-basedsegmentation, Medical Image Analysis, 7 (2003), pp. 1–20.

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7 - Conclusion

Thank you for your attention.

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Documents

Jointsegmentation/registrationmodelbyshape ......1 - Introduction Background 5/7 Deformation model ⇒ geometric transformations derived from physical models: elastic models, viscous