A Non-local Topology-Preserving Segmentation-Guided Registration Model

Abstract

In this paper, we address the issue of designing a theoretically well-motivated segmentation-guided registration method capable of handling large and smooth deformations. The shapes to be matched are viewed as hyperelastic materials and more precisely as Saint Venant–Kirchhoff ones and are implicitly modeled by level set functions. These are driven in order to minimize a functional containing both a nonlinear-elasticity-based regularizer prescribing the nature of the deformation, and a criterion that forces the evolving shape to match intermediate topology-preserving segmentation results. Theoretical results encompassing existence of minimizers, existence of a weak viscosity solution of the related evolution problem and asymptotic results are given. The study is then complemented by the derivation of the discrete counterparts of the asymptotic results provided in the continuous domain. Both a pure quadratic penalization method and an augmented Lagrangian technique (involving a related dual problem) are investigated with convergence results.

Appendices

Appendix 1: Alternative Proof of Proposition 2

Proof

First, by definition (see [19, Chapitre 9, p. 432]), for almost every $x\in \varOmega$ and for every $(\varphi ,\xi )\in {\mathbb {R}}^2 \times {\mathbb {R}}^4$, the quasi-convex envelope of f with respect to the last variable, denoted by Qf, is defined by:

$$\begin{aligned} Qf(x,\varphi ,\xi )&=\displaystyle {\inf }\,\left\{ \dfrac{1}{{\text{ meas }}(D)}\,\displaystyle {\int _{D}}\,f(x,\varphi ,\xi +\nabla \varPhi (y))\,dy\,:\, \right. \\&\qquad \quad \left. \varPhi \in W_0^{1,\infty }(D,{\mathbb {R}}^2)\right\} , \end{aligned}$$

$D \subset {\mathbb {R}}^2$ being a bounded open set. Consequently, in our case, $Qf(x,\varphi ,\xi )=\frac{\nu }{2}\,(H_{\varepsilon }(\varPhi _0 \circ \varphi )-H_{\varepsilon }({\tilde{\varPhi }}(\cdot ,{\bar{T}})))^2+QW(\xi )$.

We now recall a few useful definitions.

Definition 2

(Dacorogna [19, Notation 5.30, Definition 13.1])

• The set of orthogonal matrices is denoted by O(n). It is the set of matrices $R\in M_n({\mathbb {R}})$ (set of real square matrices of order n) such that

  $$\begin{aligned} RR^T=I. \end{aligned}$$

• The set of special orthogonal matrices, denoted by SO(n), is the subset of O(n) such that the matrices satisfy

  $$\begin{aligned} \det R=1. \end{aligned}$$

• Let $\xi \in M_n({\mathbb {R}})$. The singular values of $\xi$, denoted by

  $$\begin{aligned} 0\le \lambda _1\le \dots \le \lambda _n, \end{aligned}$$

  are defined to be the square roots of the eigenvalues of the symmetric positive semi-definite matrix $\xi \xi ^T \in M_n({\mathbb {R}})$.

• The signed singular values of $\xi$, denoted by

  $$\begin{aligned} 0\le |\mu _1(\xi )|\le \mu _2(\xi )\le \dots \le \mu _n(\xi ), \end{aligned}$$

  are defined by

  $$\begin{aligned}&\mu _1(\xi )=\lambda _1(\xi ){\text{ sign }}(\det \xi ) \, \text { and } \, \mu _j(\xi )=\lambda _j(\xi ) ,\\&\forall j=2,\dots , n. \end{aligned}$$

• A function $f:M_n({\mathbb {R}}) \rightarrow [-\infty , +\infty ]$ is said to be $SO(n)\times SO(n)$ invariant if:

  $$\begin{aligned}&\forall \xi \in M_n({\mathbb {R}}), \forall Q,R \in SO(n)\times SO(n),\nonumber \\&\quad f(Q\xi R) =f(\xi ). \end{aligned}$$

According to the singular value theorem ([19, Theorem 13.3]), for all $\xi \in M_n({\mathbb {R}})$, we can find Q and R in SO(n) such that $\xi =Q\varLambda R$, where $\varLambda ={\text{ diag }}\,({\text{ sign }}(\det \xi )\lambda _1,$ $\lambda _2,\dots , \lambda _n)$. In terms of singular values, function W can be rewritten by

$$\begin{aligned} W(\xi )&= \beta (\lambda _1^2+\lambda ^2_2 -\alpha )^2 + \psi ({\text{ sign }}(\det \xi )\lambda _1\lambda _2),\\&=\beta (\lambda _1^2+\lambda ^2_2 -\alpha )^2+ \psi (\mu _1\mu _2). \end{aligned}$$

Let

$$\begin{aligned} g(\xi )= \left\{ \begin{aligned} W(\xi )&\text { if } \Vert {\xi }\Vert ^2\ge \alpha \\ \psi (\det \xi )&\text { if } \Vert {\xi }\Vert ^2< \alpha . \end{aligned} \right. \end{aligned}$$

In view of the general results, we have $PW\le QW\le RW \le W$, PW denoting the polyconvex envelope of W, QW, its quasi-convex envelope and RW, its rank-1 convex envelope (see [19, Chapter 6]). We are going to prove that

$$\begin{aligned} RW(\xi )=PW(\xi ) =g(\xi ), \, \forall \xi \in M_n({\mathbb {R}}). \end{aligned}$$

The stored energy function $W(\xi )$ is $SO(2)\times SO(2)$ invariant as a function of the trace and the determinant. Indeed,

$$\begin{aligned} \left\{ \begin{array}{ll} &{}\Vert {\xi }\Vert ^2 = {\text{ tr }}(\xi ^T\xi ),\\ &{}\Vert {Q\xi R}\Vert ^2 = {\text{ tr }}(R^T\xi ^TQ^TQ\xi R) = {\text{ tr }}(R^T\xi ^T\xi R),\\ &{}= {\text{ tr }}(\xi ^T\xi R R^T)=\Vert {\xi }\Vert ^2,\\ &{}\det (Q\xi R)= \det (\xi ). \end{array}\right. \end{aligned}$$

Hence, according to [19, Theorem 6.20], PW, QW and RW are $SO(2)\times SO(2)$ invariant. Therefore, we can restrict ourselves to the case of matrices of the form:

$$\begin{aligned} \xi = \begin{pmatrix} x &{} 0 \\ 0 &{} y \end{pmatrix} \text { where } |x| \le y. \end{aligned}$$

Then, we have $\det \xi = x y$ and $\Vert {\xi }\Vert ^2 =x^2+y^2$.

Before proceeding further, it is convenient to introduce the two following functions defined on ${\mathbb {R}}^2$ by

$$\begin{aligned} \left\{ \begin{array}{ll} &{}\chi (x,y)= \beta (x^2 +y^2 -\alpha )^2 + \psi (xy)\\ &{}\phi (x,y)= \beta [x^2 +y^2 -\alpha ]^2_+ + \psi (xy) \end{array}\right. , \end{aligned}$$

with $[z]^2_+ = \left\{ \begin{aligned}&z^2\quad \text { if } z\ge 0 \\&0 \quad \,\,\text { otherwise.} \end{aligned}\right.$

A simple calculation leads to $W\begin{pmatrix} x &{} 0 \\ 0 &{} y \end{pmatrix} =\chi (x,y)$ and $g\begin{pmatrix} x &{} 0 \\ 0 &{} y \end{pmatrix}=\phi (x,y)$. According to [19, Definition 5.42], $\phi$ is polyconvex and thus g is polyconvex ([19, Theorem 5.43]). Since $g(\xi )\le W(\xi )$ and $PW=\sup \{h\le W,\, h\,\, \text {polyconvex}\}$, we have

$$\begin{aligned} g(\xi )\le PW(\xi ). \end{aligned}$$

This result is not surprising since if one sets $\chi (\xi )=\beta (\Vert \xi \Vert ^2-\alpha )^2$, it is well-known that

$C\chi (\xi )=P\chi (\xi )=Q\chi (\xi )=\left\{ \begin{array}{ll} \beta (\Vert \xi \Vert ^2-\alpha )^2 &{} \text {if } \Vert \xi \Vert ^2\ge \alpha \\ 0 &{} \text {otherwise} \end{array}\right.$, C denoting the convex envelope, and according to the definition of the polyconvex envelope, it is clear that $g(\xi )=\psi (\det \xi )+C\chi (\xi ) \le PW(\xi )$.

Case 1: $\Vert \xi \Vert ^2 < \alpha$.

Let us set

$$\begin{aligned} \xi _1= \begin{pmatrix} x &{} 0 \\ &{}\\ -\sqrt{\alpha -(x^2+y^2)} &{} y \end{pmatrix} ,\, \xi _2= \begin{pmatrix} x &{} 0 \\ &{}\\ \sqrt{\alpha -(x^2+y^2)} &{} y \end{pmatrix} \end{aligned}$$

Then

$$\begin{aligned} \left\{ \begin{array}{ll} &{}\xi = \frac{1}{2}\xi _1 + \frac{1}{2}\xi _2,\\ &{}\Vert \xi _1\Vert ^2=\Vert {\xi _2}\Vert ^2=\alpha ,\\ &{}\text {and }\,{\text{ rank }}(\xi _1-\xi _2)=1. \end{array}\right. \end{aligned}$$

This implies that

$$\begin{aligned} g(\xi )&\le PW(\xi )\le RW(\xi )\le \frac{1}{2}RW(\xi _1) +\frac{1}{2}RW(\xi _2)\\&\le \frac{1}{2}W(\xi _1) +\frac{1}{2}W(\xi _2) = g(\xi ). \end{aligned}$$

Case 2: $\Vert {\xi }\Vert ^2\ge \alpha$.

$$\begin{aligned} W(\xi )=g(\xi )\le PW(\xi )\le QW(\xi )\le RW(\xi )\le W(\xi ), \end{aligned}$$

which concludes the proof. $\square$

Remark 9

In fact, we have proved a stronger result, namely, that the polyconvex envelope of W, PW, equals both the quasi-convex envelope of W, QW and the rank-1 convex envelope of W, RW.

Remark 10

We understand better through this proof the choice of the weighting parameter balancing the component $\left( \det {\xi }-1\right) ^2$; it has been chosen in order that the mapping $\varPsi$ is convex.

Appendix 2: Discrete Generalized Poincaré Inequality

We first recall the continuous generalized Poincaré inequality.

Theorem 12

(extracted from [22, p.106]) Let $\varOmega$ be a bounded Lipschitz domain in ${\mathbb {R}}^N$. Let $p \in [1;+\infty [$ and let ${\mathcal {N}}$ be a continuous seminorm on $W^{1,p}(\varOmega )$, that is a norm on the constant functions. Suppose that $u \in W^{1,p}(\varOmega )$, then there exists a constant $C>0$ depending only on $N, p, \varOmega$ such that:

$$\begin{aligned} \left\| u \right\| _{W^{1,p}(\varOmega )} \le C \bigg ( \big (\int _{\varOmega } \mid \nabla u \mid ^p dx \big )^{\frac{1}{p}} + \mathcal {N}(u) \bigg ). \end{aligned}$$

We apply this result with $\mathcal {N}(u)=\int _{\varGamma _0}\mid u(x) \mid dx$ when $\varOmega$ is a $C^1$ open set and $\varGamma _0$ is a subset of $\partial \varOmega$ with positive (N-1)-dimensional Lebesgue measure.

Now, we will provide a similar discrete inequality. The following result is an adaption of the one from [57] given for real-valued functions which vanish on the boundary, whereas the results presented here stand for ${\mathbb {R}}^2$-valued functions which are equal to the identity on the boundary.

Theorem 13

(adapted from [57, Corollary 3.10]) (Discrete generalized Poincaré inequality). For any $f \in \mathcal {F}_1(\tilde{\varOmega })$, and any real number $q \ge 2$, we have:

$$\begin{aligned} \Vert f \Vert _{l^q(\tilde{\varOmega },\mathbb {R}^2)}^q \le B^{q}2^{\frac{7q+2}{2}} \Vert \nabla f \Vert _{l^q(\tilde{\varOmega },M_2(\mathbb {R}))}^q + c_4 \end{aligned}$$

with $c_4=B^{q} 2^{\frac{7q+2}{2}} \left[ 8NM+ \frac{4}{N} \frac{M(M+1)(2M+1)}{6} \right.$ $\left. + \frac{4}{M} \frac{N(N+1)(2N+1)}{6} \right] ^{\frac{q}{2}}$ with $B=\max \{M,N\}$.