Shape-Aware Matching of Implicit Surfaces Based on Thin Shell Energies

We present a variational model for the matching of surfaces implicitly represented as level sets. The approach is inspired by the mathematical theory of nonlinear elasticity of thin shells. The model consists in an energy functional, which is to be minimized among deformations of a computational domain in which two given surfaces are embedded. A minimizer of this functional is a deformation that closely maps one (reference) surface onto the other (template) surface. As the underlying model we consider the reference surface as a thin elastic shell, i.e., a layer of an elastic material embedded in a volume of another several orders of magnitude softer isotropic elastic material. Subject to matching forces the volume is deformed in such a way that the thin shell is mapped onto the template surface. The functional reflects desired phenomena like resistance to compression and expansion of the surface, resistance to bending, and rotational invariance, while solely involving the deformation and the Jacobian of the deformation. The model is formulated in terms of projected derivatives from the tangent space of the reference surface onto the expected tangent space of the template surface. Taking into account a suitable factorization of the natural pullback under a deformation of shape operators enables us to formulate a model with appropriate convexity properties. The actual surface matching constraint is handled through a penalty, allowing for efficient numerical computation.

Through arguments of compensated compactness, we are able to show weak lower semicontinuity of the energy and consequently existence of minimizing deformations. We present a numerical approach based on a multilinear finite element ansatz for the deformation implemented on adaptive octree grids. The resulting discrete energy is minimized in a multiscale fashion applying a regularized gradient descent.

In the conference article [32], a preliminary version of this approach was presented. For the functional in that paper, lower semicontinuity could not be ensured for either the membrane or bending energies. This lack of lower semicontinuity manifests itself in applications, where compression of the surface is expected, and leads to undesired oscillations in almost-minimizing deformations, which we explore in the present work through explicit examples and computations. Additionally, to increase the efficiency the computational meshes are in the present paper adapted to the surfaces. Consequently, the number of degrees of freedom scales asymptotically almost like that of a surface problem.

Related work Linear elasticity has been extensively used in computer vision and in graphics. Prominent applications are image registration [33, 34, 38, 48, 55], optical flow extraction [35], and shape modeling [29]. Recently, theories of nonlinear elasticity have been applied in many computer vision and graphics problems such as mesh deformation [13], shape averaging [56], registration of medical images [12]. The advantage of nonlinear models is that they allow for intuitive deformations when the displacements are large.

In this paper, we present a model for nonlinear elastic matching of thin shells. A finite element method for the discretization of bending energies of biological membranes has been introduced in [5]. Their approach uses quadratic isoparametric finite elements to approximate the interface on which the gradient flow of an elastic energy of Helfrich type is considered. The papers [9, 10] discuss accurate convex relaxation of higher-order variational problems on curves described as jump sets of functions of bounded variation. In particular, it enables the numerical treatment of elastic energies on such curves.

One challenge in polyhedral surface processing is to provide consistent notions of curvatures and second fundamental forms, i.e., notions that converge (in an appropriate topology or in a measure theoretic sense) to their smooth counterparts, given a smooth limit surface. One computationally popular model for discretizing the second fundamental form is Grinspun’s et al.  [30] discrete shells model. Another efficient and robust method for nonlinear surface deformation and shape matching is PriMo [6]. This approach is based on replacing the triangles of a polyhedral surfaces by thin prisms. During a deformation, these prisms are required to stay rigid, while nonlinear elastic forces are acting between neighboring prisms to account for bending, twisting, and stretching of the surface. We refer to Botsch and Sorkine [7] for a discussion of pros and cons for various such methods. In comparison with methods based on polyhedral surfaces, level set approaches like ours are not dependent on specific triangulations of the shapes.

The matching of surfaces with elastic energies has recently been studied in [61]. Their energy contains a membrane energy depending on the Cauchy–Green strain tensor and a bending-type energy comparing the mean curvatures on the surfaces. The matching problem is formulated in terms of a binary linear program in the product space of sets of surface patches. For computations, a relaxation approach is used.

A different direction is the use of parametric approaches to reduce shape matching problems to the matching of functions on a fixed domain. For example, the methods presented in [60, 62] are based on conformal maps from the unit disk. A more general variant using conformal maps on surfaces with arbitrary topology is presented in [42]. Within the family of parametric methods, a surface matching approach related to ours is presented in [44], where nonlinear elastic energies are used for matching parametrized surface patches. In comparison with all these methods, our level set approach is nonparametric and allows surfaces of any topology, which does not need to be fixed in advance.

In [59], face matching based on a matching of corresponding level set curves on the facial surfaces is investigated. To match pairs of curves an optimal deformation between them is computed using an elastic shape analysis of curves. Compared to our approach, this model does not take into account bending dissipation of the curves.

A different direction in shape recognition and matching is exploiting the intrinsic geometry of the surfaces only, thereby producing isometry-invariant methods based on the first fundamental form, like those in [11, 24]. In comparison, bending is penalized in our model and we use all curvatures of the surfaces and their directions to be able to better match regions of edges and creases correctly.

A method for matching and blending of curves represented by level sets has been presented in [49]. Thereby, a level set evolution generates an interpolating family of curves, where the associated propagation speed of the level sets depends on differences of level set curvatures. In this class of approaches, geometric evolution problems are formulated, whereas here we focus on variational models for matching deformations. Variational registration of implicit surfaces was also considered in [40], but only through volume elasticity, in contrast to our shell terms.

To summarize, the main novelty of our contribution is the combination of independence of mesh topologies arising from the use of level sets, penalization of tangential distortion in a rotationally invariant framework, and awareness both of curvatures and curvature directions of the surfaces in the matching. We are not aware of any other methods possessing all of these features simultaneously.

Our approach is inspired by the articles [21, 22] in which surface PDE models are derived in terms of the signed distance function. Shape warping based on the framework of [21] has been discussed from a geometric perspective in [14].

Outline The paper is organized as follows. In Sect. 2, we review the required preliminaries about distance functions and formulate the geometric nondistortion and matching conditions that inspire our model. In Sect. 3, we present the different contributions to our energy. Section 4 is devoted to proving the existence of minimizing deformations under suitable Dirichlet and Neumann boundary conditions. Furthermore, the strong convergence of solutions for vanishing matching penalty parameter is discussed and counterexamples showing the lack of lower semicontinuity of related simpler models are given. In Sect. 5, a numerical strategy for minimizing the energy on adaptive octree grids is presented. Finally, Sect. 6 contains a range of numerical examples demonstrating the behavior of solutions corresponding to our design criteria and presents several potential applications.

\(T_x\mathcal {M}_i^{\mathbf{{d}}_i(x)}\) denotes the tangent space to \(\mathcal {M}_i^{\mathbf{{d}}_i(x)}\) at x. The outwards normal to \(\mathcal {M}_i^{\mathbf{{d}}_i(x)}\) is given by \(\mathbf{{n}}_i(x)\), and the set of points where \(\mathbf{{d}}_i\) is not differentiable is denoted by \({\text {sing}}\,\mathbf{{d}}_i\).

We are given two compact, connected embedded hypersurfaces \(\mathcal {M}_1,\mathcal {M}_2\) of class \(C^{2,1}\), which are diffeomorphic to each other, and both of which are contained in a bounded Lipschitz domain \(\Omega \subset \mathbb {R}^{n}\). In this section, we deal with the tangential distortion and the change of the shape operator under a deformation \(\phi : \Omega \rightarrow \mathbb {R}^n\).

For any \(c \in \mathbb {R}\), we denote the c-offsets to the hypersurface \(\mathcal {M}_i\) by \(\mathcal {M}_i^c := \{ x \in \Omega \,|\, \mathbf{{d}}_i(x) = c \}\,\). Furthermore, we define the singularity set \({\text {sing}}\,\mathbf{{d}}_i\) as the set of points where \(\mathbf{{d}}_i\) is not differentiable. With the regularity of \(\mathcal {M}_i\) that we have assumed, it is well known (e.g., Theorem 1.1, Corollary 1.3 and Remark 1.4 of [43]) that \({\text {sing}}\,\mathbf{{d}}_i\) has Lebesgue measure zero and \({\text {dist}}(\mathcal {M}_i, {\text {sing}}\,\mathbf{{d}}_i) > 0\). Furthermore, combining [21, Theorem 5.6] and [45, Proposition 4.6, 7.] we see that \(\mathbf{{d}}_i \in C^2(\overline{\Omega } \setminus \overline{{\text {sing}}\,\mathbf{{d}}_i})\).

2.1 Tangential Derivative and Area and Length Distortion

First, let us assume that \(\phi \) exactly maps \(\mathcal {M}_1^c\) onto \(\mathcal {M}_2^c\), for all \(c>0\). Then, \(T_x \mathcal {M}_1^{\mathbf{{d}}_1(x)}={\text {im}}\text {P}_1(x)\) and \(T_{\phi (x)}\phi (\mathcal {M}_1^{\mathbf{{d}}_1(x)})=T_{\phi (x)}\mathcal {M}_2^{\mathbf{{d}}_2(\phi (x))}={\text {im}}\text {P}_2 ( \phi (x) )\) and we define the tangential derivative induced by the deformation \(\phi \) as

$$\begin{aligned} \mathcal {D}_{\text {tg}}\phi (x):=\text {P}_2( \phi (x) ) \, \mathcal {D}\phi (x) \, \text {P}_1 (x)\,, \end{aligned}$$

(2.1)

capturing the tangential variation of \(\phi (x)\) on \(\mathcal {M}_2\) along tangential directions on \(\mathcal {M}_1\). In the variational model, we consider below an energy term depending on \(\mathcal {D}_{\text {tg}}\phi (x)\) will reflect the tangential distortion of the deformation in the context of a matching of the two hypersurfaces \(\mathcal {M}_1\) and \(\mathcal {M}_2\) even though \(\phi (\mathcal {M}_1)\) does not necessarily equal \(\mathcal {M}_2\). Indeed, in the case \(\mathcal {M}_2 \ne \phi (\mathcal {M}_1)\) the variation along a tangent direction on \(\mathcal {M}_1\) is still projected via \(\mathcal {D}_{\text {tg}}\phi (x)\) onto the tangent space \(T_{\phi (x)}\mathcal {M}_2^{\mathbf{{d}}_2(\phi (x))}\) and not onto the tangent space of the deformed hypersurface \(\phi (\mathcal {M}_1)\) (cf. Fig. 1). Therefore, there may exist tangential directions \(v \in T_x \mathcal {M}_1^{\mathbf{{d}}_1(x)}\), such that \(\mathcal {D}_{\text {tg}}\phi (x) v =0\) even though \(\mathcal {D}\phi v \ne 0\). Thus \(\mathcal {D}_{\text {tg}}\phi (x)\) can only be considered a measure of tangential distortion if \(\phi (\mathcal {M}_1)\) is sufficiently close to \(\mathcal {M}_2\) in the sense of closeness of tangent bundles.

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For a general deformation \(\psi :\mathbb {R}^n\rightarrow \mathbb {R}^n\) the Cauchy–Green strain tensor \(\mathcal {D}\psi ^T \mathcal {D}\psi \) describes (up to first order) the deformation in a frame invariant (with respect to rigid body motions) way. Since we are interested in the effect of such a deformation between two hypersurfaces, for a suitably extended tangential gradient \(\mathcal {D}_{\text {tg}}\phi + \mathbf{{n}}_2\circ \phi \otimes \mathbf{{n}}_1\) we define the extended tangential part of the Cauchy–Green strain tensor, measuring only tangential distortion:

$$\begin{aligned} \begin{aligned}&\left( \mathcal {D}_{\text {tg}}\phi + (\mathbf{{n}}_2\circ \phi ) \otimes \mathbf{{n}}_1\right) ^T \left( \mathcal {D}_{\text {tg}}\phi + (\mathbf{{n}}_2 \circ \phi ) \otimes \mathbf{{n}}_1\right) \\&\quad =\mathcal {D}_{\text {tg}}\phi ^T \mathcal {D}_{\text {tg}}\phi + \mathbf{{n}}_1 \otimes \mathbf{{n}}_1\,. \end{aligned} \end{aligned}$$

(2.2)

The term \(\mathbf{{n}}_2(\phi (x)) \otimes \mathbf{{n}}_1(x)\) is used to complement directions that are removed by the projections in the definition of the tangential distortion \(\mathcal {D}_{\text {tg}}\phi \) and can be seen to realize a nonlinear Kirchhoff–Love assumption [18, Page 336], which postulates that lines normal to the middle surface of a shell remain normal after the deformation, without stretching.

Next, we investigate the area and length distortion due to the tangential derivative \(\mathcal {D}_{\text {tg}}\phi \). For a given vector \(e \in \mathbb {R}^n\) we denote by \(\text {Q}(e)\) any proper rotation such that \(\text {Q}(e) e_n = e\), where \(e_n\) denotes the n-th element of the canonical basis of \(\mathbb {R}^n\). Note that this condition does not specify a unique \(\text {Q}(e)\). Then, for every \(B \in \mathbb {R}^{n \times n}\) satisfying \(w \in \ker B\) and \({\text {im}}B \subseteq v^\perp \) for some unit vectors \(v,w \in \mathbb {S}^{n-1}\), we have

$$\begin{aligned} \begin{gathered} \text {Q}(v)^T ( B + v \otimes w ) \text {Q}(w) = \text {Q}(v)^T B \text {Q}(w) + e_n \otimes e_n=\left( \begin{array}{c|c} \tilde{B} &{} 0 \\ \hline 0 &{} 1 \end{array} \right) , \end{gathered} \end{aligned}$$

(2.3)

where \(\tilde{B}\) is the upper left \((n-1)\times (n-1)\) submatrix of \(\text {Q}(v)^T B \text {Q}(w)\). Obviously, (2.3) implies

$$\begin{aligned} \det (B + v \otimes w)= & {} \det (\tilde{B})\,,\\ |B+ v\otimes w|^2= & {} {\mathrm {tr}}\big ( (B + v\otimes w)^T(B+ v\otimes w) \big )= 1+ | \tilde{B} |^2\,. \end{aligned}$$

Hence, for \(\phi (\mathcal {M}_1)=\mathcal {M}_2\) and \(v=n_2(\phi (x))\), \(w=n_1(x)\) the area distortion under the hypersurface matching deformation \(\phi \) at some position x is described by \(\det (\mathcal {D}_{\text {tg}}\phi (x) + \mathbf{{n}}_2(\phi (x)) \otimes \mathbf{{n}}_1(x))\), which equals the positive square root of the determinant of the above Cauchy–Green strain tensor \(\mathcal {D}_{\text {tg}}\phi ^T \mathcal {D}_{\text {tg}}\phi + \mathbf{{n}}_1 \otimes \mathbf{{n}}_1\). The squared tangential length distortion (in the sense of summing all squared distortions with respect to an orthogonal basis) is described by \(|\mathcal {D}_{\text {tg}}\phi (x) + \mathbf{{n}}_2(\phi (x)) \otimes \mathbf{{n}}_1(x))|^2\) and equals the trace of the Cauchy–Green strain tensor.

In what follows we introduce the four energy contributions separately.

3.1 Tangential Distortion Energy

Picking up the insight gained in Sect. 2.1, we formulate the membrane energy in terms of the length and area change associated with the tangential distortion \(\mathcal {D}_{\text {tg}}\phi \):

$$\begin{aligned} E_{\text {mem}}[\phi ]= \delta \int _\Omega \eta _\sigma (\mathbf{{d}}_1(x)) W\big ( \mathcal {D}_{\text {tg}}\phi (x) + \mathbf{{n}}_2(\phi (x)) \otimes \mathbf{{n}}_1(x) \big ) \,{{\text {d}}}x, \end{aligned}$$

(3.1)

where \(W\) is a nonnegative polyconvex energy density vanishing at \(\text {SO}(n)\). The weight \(\delta \) reflects the proper scaling of the tangential distortion energy in case of a thin shell model with shell thickness \(\delta \).

The energy (3.1) vanishes only on deformations \(\phi \) whose Jacobian matrix \(\mathcal {D}\phi (x)\) maps \(T\mathcal {M}_1^{\mathbf{{d}}_1(x)}\) isometrically onto \(T\mathcal {M}_2^{\mathbf{{d}}_2(\phi (x))}\) for every point \(x \in {\text {supp}}\,{\eta _\sigma \circ \mathbf{{d}}_1}\). In consequence, both tangential expansion and compression are penalized.

Let us remark, that the extension \(\mathcal {D}_{\text {tg}}\phi + \mathbf{{n}}_2(\phi (x)) \otimes \mathbf{{n}}_1(x)\) of the tangential derivative \(\mathcal {D}_{\text {tg}}\phi \) defined in (2.1) with rank \(n-1\) can degenerate or be orientation-reversing depending on the local configuration of \(\mathcal {M}_1\) and \(\mathcal {M}_2\) at x (cf. Fig. 2 for examples).

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Furthermore, the energy density W should not satisfy \(W(B) \rightarrow \infty \) for \(\det B \rightarrow 0\). A straightforward modification of the arguments of Ciarlet and Geymonat ([16, 17] Theorem 4.10-2) leads to a smooth integrand \(W\) which has isometries as local minimizers, with the correct invariance properties, and with a Hessian for \(B={\mathbb {1}}\) which matches the quadratic energy integrand of the Lamé-Navier model of linearized elasticity. With given Lamé coefficients \(\lambda , \mu > 0\), we select the energy

$$\begin{aligned} W(A)=\frac{\mu }{2}| A |^2 + \frac{\lambda }{4} (\det A)^2 + \left( \mu + \frac{\lambda }{2}\right) e^{- ( \det A - 1 ) } - \frac{(n+2) \mu }{2} - \frac{3\lambda }{4}\,. \end{aligned}$$

This density fits into the notation of (1.1), if we choose \(p=q=2\) and \(\Gamma (t)=c t^2 + d e^{- (t-1) }\;.\)

First, we prove the following weak continuity lemma, which is a generalization of the classical result given in [50, Theorem 4.1]. Here, the coefficients may depend on the deformed configuration.

Lemma 4.1

Let \(\phi ^k \rightharpoonup \phi \in W^{1,p}(\Omega ; \mathbb {R}^n)\) and \(p>n\). Moreover, let \(V_i \in C^0(\overline{\Omega } \times \mathbb {R}^{n}; \mathbb {S}^{n-1})\), \(i=1,2\) and we denote

$$\begin{aligned} \mathbf{{v}}_i^k(\cdot ):=V_i\big (\cdot , \phi ^k(\cdot )\big ) \text { and } \mathbf{{v}}_i:=V_i\big (\cdot , \phi (\cdot )\big )\,,\quad i=1,2\;. \end{aligned}$$

Then

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(4.1)

Moreover, for every symmetric positive definite \(M_i\), \(i=1,2\) with \(M_1^{-\frac{1}{2}} \in C^0(\overline{\Omega } \times \mathbb {R}^{n}; \mathbb {R}^{n \times n})\) and \(M_2^{\frac{1}{2}} \in C^0(\overline{\Omega } \times \mathbb {R}^{n}; \mathbb {R}^{n \times n})\) and the corresponding compositions

$$\begin{aligned} M_i^k(\cdot ):=M_i\big (\cdot , \phi ^k(\cdot )\big ) \text { and } \overline{M}_i:=M_i\big (\cdot , \phi (\cdot )\big ) \end{aligned}$$

we have

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(4.2)

We are now in a position to prove existence of a minimizing deformation for the hypersurface matching energy E in a suitable set of admissible deformations. Of particular difficulty is that derivatives of \(\mathbf{{d}}_2\) are not defined in the whole of \(\Omega \) and that in the functional these derivatives are evaluated at deformed positions. We handle this by ensuring that the involved deformations are such that terms involving these derivatives are not evaluated near the singularities. We obtain the following theorem:

Proof

We proceed in several steps.

Step 1 Coercivity First, we point out the coercivity enjoyed by our functional. Using the Poincaré and Morrey inequalities ([41], Theorem 12.30 and 11.34), and the Dirichlet boundary conditions we have

$$\begin{aligned} \Vert \phi \Vert _{C^{0,\alpha }(\Omega )} \le C \Vert \phi \Vert _{W^{1,p}(\Omega )} \le C ( 1 + \Vert \mathcal {D}\phi \Vert _{L^p(\Omega )}) \le C (1 + E_\nu [\phi ]^{\frac{1}{p}}), \end{aligned}$$

(4.5)

for any \(\phi \in W^{1,p}_0(\Omega )+{\text {id}}\) and \(\alpha =1-n/p\).

Step 2 Lower semicontinuity along sequences of constrained deformations For the remainder of the proof, a deformation \(\phi \in W^{1,p}_0(\Omega ; \mathbb {R}^n) + {\text {id}}\), \(p>n\) is termed \(\rho \)-admissible for \(\rho > 0 \), if

• \(E_{\text {vol}}[\phi ]< +\infty \),

• \(\det \mathcal {D}\phi (x) > 0\) for a.e. \(x \in \Omega \), and

• for all \(x \in {\text {supp}}\,\big ( \eta _\sigma \circ \mathbf{{d}}_1 \big )\) and every \(y \in {\text {sing}}\,(\mathbf{{d}}_2)\), we have \(\left| \phi (x) - y \right| \ge \rho \).

Notice that since \(p>n\), \(\phi \) has a unique continuous representative, so the third property is well defined.

First, notice that with the assumption (4.4) we have

$$\begin{aligned} {\text {supp}}\,(\eta _\sigma \circ \mathbf{{d}}_i) = \{|\mathbf{{d}}_i|\le \sigma \} \subset \Omega \setminus \overline{{\text {sing}}\,(\mathbf{{d}}_i)}\,,\quad i=1,2. \end{aligned}$$

(4.6)

Let \(\phi ^k\) be a sequence of \(\rho \)-admissible deformations with \(E_{\text {vol}}[\phi ^k]\le C\). By (4.5) and using the Banach–Alaoglu and Rellich–Kondrakov theorems, a subsequence (again denoted by \((\phi ^k)\)) converges to a deformation \(\phi \), both in the \(W^{1,p}\)-weak and uniform topologies.

Now, we have ([20, Theorem 8.20])

$$\begin{aligned} (\det \mathcal {D}\phi ^k, {\text {Cof}}\,\mathcal {D}\phi ^k) \rightharpoonup (\det \mathcal {D}\phi , {\text {Cof}}\,\mathcal {D}\phi ) \text { in } L^{\frac{p}{n}}(\Omega ) \times \left( L^{\frac{p}{n-1}}(\Omega )\right) ^{n^2}. \end{aligned}$$

(4.7)

Additionally, since (4.7) holds and because \(E_\nu [\phi _k]\) is bounded, \(\int _\Omega (\det \mathcal {D}\phi _k)^{-s} d x\) is bounded by the definition of \(\hat{W}\) and \(\det \mathcal {D}\phi _k \ge 0\) a.e. Together with (4.7), we have

$$\begin{aligned} \det \mathcal {D}\phi (x) > 0 \text { a.e.,} \end{aligned}$$

(4.8)

so that \(\phi \) is again \(\rho \)-admissible.

Notice also that by a.e. positivity of the determinants, (4.7) and a standard lower semicontinuity result for convex integrands (see, e.g., [20, Theorem 3.23]) implies

$$\begin{aligned} E_{\text {vol}}[\phi ] \le \liminf _{k \rightarrow \infty } E_{\text {vol}}[\phi ^k], \end{aligned}$$

and uniform convergence of \(\phi ^k\) immediately leads to

$$\begin{aligned} E_{\text {match}}[\phi ] = \lim _{k \rightarrow \infty } E_{\text {match}}[\phi ^k]. \end{aligned}$$

We claim that under the assumptions of this theorem, we also have that

$$\begin{aligned} E_{\text {mem}}\left[ \,\phi \,\right] \le \liminf _{k \rightarrow \infty } E_{\text {mem}}[\phi ^k] \end{aligned}$$

(4.9)

and

$$\begin{aligned} E_{\text {bend}}\left[ \,\phi \,\right] \le \liminf _{k \rightarrow \infty } E_{\text {bend}}[\phi ^k]. \end{aligned}$$

(4.10)

To see this, notice that \(\phi ^k, \phi \) being \(\rho \)-admissible ensures that the normal vectors satisfy

$$\begin{aligned} \mathbf{{n}}_1,\, \mathbf{{n}}_2 \circ \phi ^k,\, \mathbf{{n}}_2 \circ \phi \in C^{0}( \{|\mathbf{{d}}_1|\le \sigma \}; \mathbb {R}^{n}). \end{aligned}$$

Consequently, the first part of Lemma 4.1 (with \(V_i=\mathbf{{n}}_i\)) implies

$$\begin{aligned} \begin{aligned}&\chi _{\{|\mathbf{{d}}_1|\le \sigma \}}\Big ( \mathcal {D}_{\text {tg}}\phi ^k, \det \big (\mathcal {D}_{\text {tg}}\phi ^k + (\mathbf{{n}}_2\circ \phi ^k) \otimes \mathbf{{n}}_1\big )\Big ) \\&\rightharpoonup \chi _{\{|\mathbf{{d}}_1|\le \sigma \}}\Big ( \mathcal {D}_{\text {tg}}\phi , \det \big (\mathcal {D}_{\text {tg}}\phi + (\mathbf{{n}}_2\circ \phi ) \otimes \mathbf{{n}}_1\big )\Big )\text { in } \left( L^p(\Omega ) \right) ^{n^2} \times L^{\frac{p}{n}}(\Omega ), \end{aligned} \end{aligned}$$

(4.11)

with \(\chi _{\{|\mathbf{{d}}_1|\le \sigma \}}\) denoting the indicator function. Combining (4.11) with the polyconvexity of W, defining the function \(E_{\text {mem}}\), both introduced in (3.1) we find the assertion (4.9).

Furthermore, by our assumptions on \(\mathcal {M}_i\) (see Sect. 2), we have that

$$\begin{aligned} \chi _{\{|\mathbf{{d}}_i|\le \sigma \}}{\mathcal {S}}_i = \chi _{\{|\mathbf{{d}}_i|\le \sigma \}}\mathcal {D}^2 \mathbf{{d}}_i \in C^0( \overline{\Omega }; \mathbb {R}^{n \times n}). \end{aligned}$$

Since \({\mathcal {C}}\) produces uniformly positive matrices, we have \(\chi _{\{|\mathbf{{d}}_1|\le \sigma \}}({\mathcal {C}}({\mathcal {S}}^{ext}_1))^{-1}\in C^0(\overline{\Omega }; \mathbb {R}^{n \times n})\). We can then use a continuity result for square roots of nonnegative definite matrix-valued functions defined on \(\Omega \) [15, Theorem 1.1] to see that

$$\begin{aligned} \chi _{\{|\mathbf{{d}}_1|\le \sigma \}}({\mathcal {C}}({\mathcal {S}}^{ext}_1))^{-\frac{1}{2}}\in & {} C^0(\overline{\Omega }; \mathbb {R}^{n \times n}),\\ \chi _{\{|\mathbf{{d}}_1|\le \sigma \}}( {\mathcal {C}}({\mathcal {S}}^{ext}_2) \circ \phi ^k )^{\frac{1}{2}}\in & {} C^0(\overline{\Omega }; \mathbb {R}^{n \times n}),\\ \chi _{\{|\mathbf{{d}}_1|\le \sigma \}}({\mathcal {C}}({\mathcal {S}}^{ext}_2) \circ \phi )^{\frac{1}{2}}\in & {} C^0(\overline{\Omega }; \mathbb {R}^{n \times n}). \end{aligned}$$

The second part of Lemma 4.1 implies the weak convergence

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from which (4.10) follows by using the polyconvexity of W.

Step 3 Existence of minimizers restricted to admissible deformations Since we have already seen that the set of \(\rho \)-admissible deformations is weakly closed and weakly compact, and that every term of E is weakly lower semicontinuous on this set, we just need to check that for all fixed \(\nu > 0\), the set of \(\rho \)-admissible deformations, with adequate \(\rho \), is not empty.

For some given \(\sigma \) satisfying \({\text {dist}}( \mathcal {M}_2, {\text {sing}}\,\mathbf{{d}}_2 ) - \sigma > 0\) let \(\rho \) satisfy

$$\begin{aligned} 0< \rho < {\text {dist}}( \mathcal {M}_2, {\text {sing}}\,\mathbf{{d}}_2 ) - \sigma \;. \end{aligned}$$

(4.12)

We construct a deformation \(\hat{\varphi }\), which is \(\rho \)-admissible and satisfies \(E_\nu [\hat{\varphi }] < \infty \). By assumption, there exists a diffeomorphism \(\varphi :\mathcal {M}_1\rightarrow \mathcal {M}_2\). Thus, we construct an extension of this diffeomorphism to \(\{|\mathbf{{d}}_1|\le \sigma \}\) along the normal directions using

$$\begin{aligned} \hat{\varphi }( x+s\mathbf{{n}}_1(x) ):=\varphi (x)+s \mathbf{{n}}_2(\varphi (x)), \text { for }x \in \mathcal {M}_1, -\sigma \le s \le \sigma . \end{aligned}$$

(4.13)

We can then extend \(\hat{\varphi }\) to the inside and outside components \(\Omega _i, \Omega _o\) of \(\Omega \setminus \{|\mathbf{{d}}_1|\le \sigma \}\) by solving the minimization problems for \(E_{{\text {vol}}}\) with Dirichlet boundary conditions given by (4.13) on \(\partial \Omega _i\) and \(\partial \Omega _o \setminus \partial \Omega \), and by \(\hat{\varphi }(x)=x\) on \(\partial \Omega \). For the resulting \(\hat{\varphi }\), we have

$$\begin{aligned} E_{\text {match}}[\hat{\varphi }]=0,\, E_{\text {vol}}[\hat{\varphi }]<\infty ,\, E_{\text {mem}}[\hat{\varphi }]<\infty ,\, E_{\text {bend}}[\hat{\varphi }]<\infty , \end{aligned}$$

where the first two statements follow by construction, and the last two by virtue of \(\varphi \) being a diffeomorphism and the choice of \(\sigma \). Moreover, we note that since \(\hat{\varphi }\) has finite energy and the growth conditions assumed for \(\hat{W}\) [see (3.7)], the condition \(\det \mathcal {D}\hat{\varphi }(x) > 0\) for a.e. x is also satisfied [4].

Step 4 A priori estimate to remove the constraint Next, we show that for any \(\rho \) satisfying (4.12) there exists a parameter \(\nu _0 > 0\) such that for all \(0< \nu < \nu _0\) the constrained minimizers of \(E_\nu \) subject to (4.4) solves the unconstrained optimization problem, consisting in minimizing \(E_\nu \) on \(W_0^{1,p}+{\text {id}}\).

To this end, we verify that every \(\phi \) that satisfies

$$\begin{aligned} E_\nu [ \phi ] \le E_\nu [ \hat{\varphi }] \end{aligned}$$

(4.14)

is \(\rho \)-admissible. It is immediate from (4.14) that \(E_{\text {vol}}(\phi ) < + \infty \), and from the definition of \(\hat{W}\) in (3.7) it follows with the same arguments as in (4.8) that \(\det \phi > 0\) a.e.

We prove now that for all deformations \(\phi \) satisfying (4.14) also satisfy

$$\begin{aligned} \Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})} \le {\text {dist}}(\mathcal {M}_2,{\text {sing}}\,\mathbf{{d}}_2) - \rho \;. \end{aligned}$$

(4.15)

This is sufficient because from (4.15) it follows for all x satisfying \(|\mathbf{{d}}_1(x)| \le \sigma \) by the triangle inequality that

$$\begin{aligned} \begin{aligned} \rho&\le {\text {dist}}(\mathcal {M}_2,{\text {sing}}\,\mathbf{{d}}_2) - \Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})}\\&= {\text {dist}}(\mathcal {M}_2,{\text {sing}}\,\mathbf{{d}}_2) - {\text {dist}}(\phi (x),\mathcal {M}_2) \\&\le {\text {dist}}(\phi (x),{\text {sing}}\,\mathbf{{d}}_2), \end{aligned} \end{aligned}$$

which is the third property of a \(\rho \)-admissible deformation \(\phi \).

To prove (4.15), we use the triangle inequality and estimate

$$\begin{aligned} \Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})} \le \sigma + \Vert \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})}. \end{aligned}$$

(4.16)

By the monotonicity of \(\eta _\sigma \) and the fact that the signed distance functions \(\mathbf{{d}}_i\) are Lipschitz continuous with constant 1 we have, for each \(\hat{\sigma } \in (0, \sigma )\) that

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})} \\&\quad \le \left( 1 + \Vert \phi \Vert _{C^{0,\alpha }(\{\sigma - \hat{\sigma } \le |\mathbf{{d}}_1| \le \sigma \})}\right) \hat{\sigma }^\alpha \\&\qquad + \frac{ \Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^\infty ( \{ |\mathbf{{d}}_1| < \sigma -\hat{\sigma }\} )}}{\eta _\sigma (\sigma - \hat{\sigma } )}. \end{aligned} \end{aligned}$$

(4.17)

Estimates (4.5) and (4.14) imply in turn

$$\begin{aligned} \Vert \phi \Vert _{C^{0,\alpha }(\{\sigma - \hat{\sigma } \le |\mathbf{{d}}_1| \le \sigma \})} \le C \Vert \phi \Vert _{W^{1,p}(\Omega )} \le C (1+E_\nu [\hat{\varphi }]^{\frac{1}{p}}). \end{aligned}$$

(4.18)

Finally, combining (4.16), (4.17), and (4.18) we obtain

$$\begin{aligned} \begin{aligned}&\Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})} \\&\quad \le \sigma + \left( 1 + C ( 1 + E_\nu [\hat{\varphi }]^{\frac{1}{p}} ) \right) \hat{\sigma }^\alpha \\&\qquad + \frac{1}{\eta _\sigma (\sigma - \hat{\sigma } )} \Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^\infty ( \{ |\mathbf{{d}}_1| < \sigma -\hat{\sigma }\} )}. \end{aligned} \end{aligned}$$

(4.19)

Now we can apply Ehrling’s lemma [54, Theorem 7.30] for the embeddings \( W^{1,p}(\Omega ) \subset \subset L^\infty (\Omega ) \subset L^2(\Omega )\) to control the last term in (4.19). Taking into account, the Poincaré inequality and Dirichlet boundary conditions, we obtain for any \(\epsilon > 0\) a constant \(C(\epsilon ) > 0\) such that

$$\begin{aligned} \begin{aligned}&\Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^\infty ( \{ |\mathbf{{d}}_1| < \sigma -\hat{\sigma }\} )} \\&\quad \le \Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^\infty ( \Omega )} \\&\quad \le C(\epsilon )\Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^2( \Omega )}\\&\qquad + \epsilon \, C \left( \Vert \nabla ( \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) )\Vert _{L^p( \Omega )}+1\right) . \end{aligned} \end{aligned}$$

(4.20)

Now, for the first term in the right-hand side of (4.20) we can estimate

$$\begin{aligned} \Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^2( \Omega )} = \nu ^{\frac{1}{2}} E_{\text {match}}[\phi ]^{\frac{1}{2}} \le \nu ^{\frac{1}{2}} E_\nu [\hat{\varphi }]^{\frac{1}{2}}. \end{aligned}$$

(4.21)

For the second term, denoting \({\text {diam}}\,\Omega = \sup _{x,y \in \Omega }|x-y|\),

$$\begin{aligned} \begin{aligned}&\Vert \nabla ( \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) )\Vert _{L^p( \Omega )} \\&\quad \le \Vert \nabla ( \eta _\sigma \circ \mathbf{{d}}_1 ) ( \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 ) \Vert _{L^p(\Omega )} \\&\qquad + \Vert ( \eta _\sigma \circ \mathbf{{d}}_1 ) \nabla ( \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 ) \Vert _{L^p(\Omega )} + 1 \\&\quad \le C \nu ^{\frac{1}{p}} \left( \Vert \mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1 \Vert ^{\frac{p-2}{p}}_{L^\infty ( \Omega )} E_{\text {match}}[\phi ]^{\frac{1}{p}}\right) \\&\qquad + C \Big ( \Vert \mathcal {D}\phi \Vert _{L^p(\Omega )} + \Vert \nabla \mathbf{{d}}_1 \Vert _{L^{p}(\Omega )} + 1 \Big ) \\&\quad \le C \nu ^{\frac{1}{p}} \left( ( \Vert \phi \Vert _{C^{0,\alpha }(\Omega )} + 2 {\text {diam}}\,\Omega )^{\frac{p-2}{p}}E_\nu [\hat{\varphi }]^{\frac{1}{p}} \right) + C \left( E_\nu [\hat{\varphi }]^{\frac{1}{p}} + 1 \right) \\&\quad \le C \nu ^{\frac{1}{p}} \left( ( 1 + E_\nu [\hat{\varphi }]^{\frac{1}{p}} )^{\frac{p-2}{p}} E_\nu [\hat{\varphi }]^{\frac{1}{p}}\right) + C \left( E_\nu [\hat{\varphi }]^{\frac{1}{p}} + 1\right) , \end{aligned} \end{aligned}$$

(4.22)

where we have applied the product rule, the definition of \(E_{\text {match}}\), \(\eta _\sigma \in C^\infty _0\), \(\eta _\sigma \le C\), that \(| \nabla \mathbf{{d}}_i | = 1\) a.e., \(i=1,2\), the chain rule, and (4.14). The use of the chain rule is justified by [46, Theorem 2.2], since \(\mathbf{{d}}_2\) has Lipschitz constant 1.

Together, (4.20), (4.21), and (4.22) imply

$$\begin{aligned} \begin{aligned}&\Vert \eta _\sigma \circ \mathbf{{d}}_1 (\mathbf{{d}}_2 \circ \phi - \mathbf{{d}}_1) \Vert _{L^\infty ( \{ |\mathbf{{d}}_1| < \sigma -\hat{\sigma }\} )} \\&\quad \le \nu ^{\frac{1}{p}} \left( C(\epsilon ) \nu ^{\frac{1}{2}-\frac{1}{p}} E_\nu [\hat{\varphi }]^{\frac{1}{2}} + \epsilon \, C ( 1 + E_\nu [\hat{\varphi }]^{\frac{1}{p}} )^{\frac{p-2}{p}} E_\nu [\hat{\varphi }]^{\frac{1}{p}}\right) \\&\qquad + \epsilon \, C \left( E_\nu [\hat{\varphi }]^{\frac{1}{p}} + 1\right) . \end{aligned} \end{aligned}$$

(4.23)

In light of (4.19) and (4.23), and since \(E_\nu [\hat{\varphi }]\) is independent of \(\nu \), we can now choose first \(\hat{\sigma }\), then \(\epsilon \) and finally \(\nu \) small enough to obtain

$$\begin{aligned} \Vert \mathbf{{d}}_2 \circ \phi \Vert _{L^\infty (\{|\mathbf{{d}}_1| \le \sigma \})}&\le \sigma + ( {\text {dist}}( \mathcal {M}_2, {\text {sing}}\,\mathbf{{d}}_2 ) - \sigma - \rho ) \\&\le {\text {dist}}( \mathcal {M}_2, {\text {sing}}\,\mathbf{{d}}_2 ) - \rho \;. \end{aligned}$$

Step 5 Injectivity The injectivity and regularity of the inverse follow by the growth conditions satisfied by \(E_\text {vol}\) and classical results of Ball [4, Theorems 2 and 3]. Note that Theorem 3 in [4] is stated in the mechanical application context in dimension \(n=3\), but it holds also in \(\mathbb {R}^n\) following the same proof and using the condition \(s>(n-1)q/(q-n)\). \(\square \)

We have particularized the statement of Theorem 4.2 to the case of Dirichlet boundary conditions to ensure global invertibility. In fact, we also have existence of minimizing deformations for the case of Neumann boundary conditions.

We conclude this section with the following proposition, which explores the penalization limit in which the parameter \(\nu \) tends to zero.

4.1 Oscillations and Lack of Rank-One Convexity for the Naive Approach

To model the tangential distortion energy we have considered a frame indifferent energy density with the argument \(\mathcal {D}_{\text {tg}}\phi +(\mathbf{{n}}_2 \circ \phi ) \otimes \mathbf{{n}}_1\). Let us now consider the case \(n=2\) and a simpler version of the membrane energy (3.6), where we use as an argument of the energy density directly the tangential Cauchy–Green strain tensor [cf (2.2)] \(({{{\tilde{\mathcal {D}}}}}_{tg}\phi (x))^T ({{{\tilde{\mathcal {D}}}}}_{tg}\phi (x)) + \mathbf{{n}}_1(x) \otimes \mathbf{{n}}_1(x)\), and define the membrane energy

$$\begin{aligned} {\tilde{E}}_\text {mem}[\phi ]:=\int _{\Omega } \eta _\sigma (\mathbf{{d}}_1(x)) W \left( \big ({{{\tilde{\mathcal {D}}}}}_{tg}\phi (x)\big )^T {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x) + \mathbf{{n}}_1(x) \otimes \mathbf{{n}}_1(x) \right) {{\text {d}}}x, \end{aligned}$$

(4.28)

with \({{{\tilde{\mathcal {D}}}}}_{tg}\phi := \mathcal {D}\phi \text {P}_1\) defined as the tangential part of the derivative along \(T_x \mathcal {M}_1^{\mathbf{{d}}_1(x)}\), and \(W:\mathbb {R}^{2 \times 2}\rightarrow \mathbb {R}\) a frame indifferent energy density that has a strict minimum at \(\text {SO}(2)\). In fact, this energy is no longer lower semicontinuous and we will present counterexamples.

Example 4.7

(Oscillation patterns) We construct an explicit sequence for which lower semicontinuity of the membrane energy \({\tilde{E}}_\text {mem}\) fails. Fix \(0<R<1\) and \(\mathcal {M}_1 = \mathbb {S}^1\) with the parametrization \(\xi \rightarrow e^{i \xi }\). Consider a sequence of deformations \(\varphi _k: \mathbb {S}^1 \rightarrow \mathbb {R}^2\) defined in polar coordinates of \((r, \theta )\) by the condition

$$\begin{aligned} \begin{aligned} \partial _\xi \varphi _k(\xi ) =&\left( R \sin k\xi \right) e_r\big ( r(\varphi _k( \xi ) ), \theta (\varphi _k( \xi ) ) \big )\\&+ \left( 1-R^2 \sin ^2 k \xi \right) ^{\frac{1}{2}} e_\theta \big (r(\varphi _k( \xi )), \theta (\varphi _k( \xi ) ) \big ), \end{aligned} \end{aligned}$$

(4.29)

where \(e_r=(\cos \theta , \sin \theta )^T, e_\theta =(-\sin \theta , \cos \theta )^T\) for given \(\phi _k(0)\). Note that for any k and \(\theta \) that \(|\partial _\theta \varphi _k(\theta )|=1\), so that the transformations are tangentially isometric. We define \(\varphi _k(0)\) via two integration constants \(r_0\) and \(\theta _0\) for the initialization of r and \(\theta \) at \(\xi =0\). We set \(\theta _0=0\) and choose \(r_0\) such that the curve \(\varphi _k\) is closed and simple, which imposes \(r_0=r(\varphi _k(0))=r(\varphi _k(2\pi ))\) since the first term in (4.29) has zero average. From the second term, taking into account that \(e_\theta (r, \theta )\) is independent of r, we get the condition

$$\begin{aligned} 2 \pi r_0 = \int _0^{2\pi }\left( 1-R^2 \sin ^2 k \xi \right) ^{\frac{1}{2}} {{\text {d}}}\xi =\frac{1}{k} \int _0^{2\pi k}\left( 1-R^2 \sin ^2 \zeta \right) ^{\frac{1}{2}} {{\text {d}}}\zeta , \end{aligned}$$

where we have applied the change of variables \(\zeta = k \xi \). By periodicity the right-hand side (an incomplete elliptic integral of the second kind with modulus R) is independent of k and thus determines \(r_0\). The resulting \(\varphi _k\) for several values of k are depicted in Fig. 3.

We observe that \(\partial _\theta \varphi _k(\theta ) \rightharpoonup r_0 e_\theta \text { in }L^p,\) for any \(1 \le p < \infty \) (and also weak-* in \(L^\infty \)). Therefore, the weak \(W^{1,p}\)-limit \(\varphi \) of the \(\varphi _k\) is the function defined by \(\varphi (\theta )=r_0 e_r\) and obviously not an isometry. Assuming \(0< \sigma < 1\) and extending \(\varphi _k, \varphi \) along the radial direction \(e_r\) to the annulus \(\{1-\sigma \le r \le 1+\sigma \}\), we obtain corresponding deformations given by

$$\begin{aligned} \phi ^k(r,\theta )&=\varphi _k(\theta )+(r-1)Q_{\frac{\pi }{2}}\partial _\theta \varphi _k(\theta ) \text { and }\\ \phi (r,\theta )&=\varphi (\theta )+(r-1)r_0 e_r=r \, r_0 e_r, \end{aligned}$$

where \(Q_{\frac{\pi }{2}}\) stands for clockwise rotation by \(\pi /2\), so that \(Q_{\frac{\pi }{2}}\partial _\theta \varphi _k(\theta )\) is the unit outward normal to \(\varphi _k(\mathbb {S}^1)\). Clearly also \(\phi ^k \rightharpoonup \phi \) in \(W^{1,p}\) on the annulus. We observe that \({\tilde{E}}_{\text {mem}}[\phi ^k]=0\), but \({\tilde{E}}_{\text {mem}}[\phi ]>0\). Hence, \({\tilde{E}}_\text {mem}\) is not weakly lower semicontinuous.

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The celebrated Nash–Kuiper theorem [37, 51] states that it is possible to uniformly approximate any short \(C^\infty \) immersion by \(C^1\) isometric ones. Our explicit oscillations around \(r_0 \mathbb {S}^1\) is just one example of this phenomenon. Notice that a bending term of the type \(E_{\text {bend}}\) introduced in our model only compares the curvatures of \(\mathcal {M}_1^{\mathbf{{d}}_1(x)}\) and \(\mathcal {M}_2^{\mathbf{{d}}_2(\phi (x))}\). It therefore does not penalize oscillations, since it does not detect the curvature of \(\phi (\mathcal {M}_1)\) at all.

Example 4.8

(Lack of rank-one convexity) We present an additional example of a configuration for which the integrand of an energy of the type \({\tilde{E}}_{\text {mem}}\) is not rank-one convex. Rank-one convexity of the complete energy density, i.e., , convexity in \(t \in \mathbb {R}\) when composed with the function \(A+tB\) for any matrix A and any rank-one matrix B, is known to be a necessary condition for quasiconvexity ([20], Theorem 5.3). Quasiconvexity, in turn, is necessary for weak lower semicontinuity of integral functionals in Sobolev spaces ([20], Theorem 8.1 and Remark 8.2).

Let \(\Omega =(-2,2)^2\), and \(\mathcal {M}_1\) be a closed \(C^2\) curve such that \(\mathcal {M}_1 \cap (-1,1)\times (0,2)=(-1,1)\times \left\{ 1\right\} \). At any point \(x_0 \in (-1,1)\times \left\{ 1\right\} \), the tangential derivatives are just partial derivatives along the first coordinate, yielding

$$\begin{aligned} {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)= & {} \mathcal {D}\phi (x_0) \text {P}(e_2)=\left( \begin{array}{cc} \partial _1 \phi _1(x_0) &{} 0 \\ \partial _1 \phi _2(x_0) &{} 0 \end{array}\right) , \text { and }\\ {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)^T {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)= & {} \left( \begin{array}{cc} \left( \partial _1 \phi _1(x_0)\right) ^2 + \left( \partial _1 \phi _2(x_0)\right) ^2 &{} 0 \\ 0 &{} 0 \end{array}\right) . \end{aligned}$$

Hence, the tangential area distortion measure reduces to

$$\begin{aligned} \begin{aligned} {\mathrm {tr}}({{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)^T {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0))&=\det ({{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)^T {{{\tilde{\mathcal {D}}}}}_{tg}\phi (x_0)+e_2 \otimes e_2)\\&=\left( \partial _1 \phi _1(x_0)\right) ^2 + \left( \partial _1 \phi _2(x_0)\right) ^2, \end{aligned} \end{aligned}$$

(4.30)

where \(e_2=(0,1)^T\). Defining now the convex function

$$\begin{aligned} F(a,d)=\frac{1}{2} a + \frac{1}{2} d + d^{-1}-2, \end{aligned}$$

which has a unique minimum with value 0 for \(a=d=1\), we have that the energy density

$$\begin{aligned} W_F(B)=F\left( {\mathrm {tr}}(B^TB), \det (B^TB+e_2 \otimes e_2) \right) \end{aligned}$$

has a pointwise minimum, with value zero, whenever \(\mathcal {D}\phi \) is such that \(\left( \partial _1 \phi _1\right) ^2 + \left( \partial _1 \phi _2\right) ^2=1\).

Consider now, for \(0\le \lambda \le 1\), the family of matrices

$$\begin{aligned} B(\lambda )=\left( \begin{array}{cc} \lambda &{} 0 \\ (1-\lambda ) &{} 0 \end{array}\right) =\lambda \left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} 0 \end{array}\right) +(1-\lambda ) \left( \begin{array}{cc} 0 &{} 0 \\ 1 &{} 0 \end{array}\right) . \end{aligned}$$

(4.31)

Clearly \(B(\lambda )\) is rank one. But we have \(W_F(B(\lambda ))=\lambda ^2+(1-\lambda )^2+\frac{1}{\lambda ^2+(1-\lambda )^2}-2\) and therefore

$$\begin{aligned} W_F(B(0))=F(B(1))=0, \text { but }W_F( B(1/2))=\frac{1}{2}, \end{aligned}$$

which demonstrates that \(W_F\) is not rank-one convex.

We adopt a ‘discretize, then optimize’ approach and consider a finite element approximation and optimize for the coefficients of the solution. Since the energy \(E_\nu \) is highly nonlinear and nonconvex, we use a cascadic multilevel minimization scheme in which the solution for one grid level is used as the initial data for the minimization on the next finer grid. We use adaptive refinement of the underlying meshes around the surfaces \(\mathcal {M}_1, \mathcal {M}_2 \subset (0,1)^n\) for \(n=2,3\) (Algorithm 1).

One of the main characteristics of our functional is the pervasive presence of coefficients depending on the deformed position \(\phi (x)\). Indeed, this is how the functional takes into account the geometry of target surface, through the projection \(\text {P}_2\) and shape operator \({\mathcal {S}}_2\). From an implementation perspective, however, this means that frequently discrete functions have to evaluated at deformed positions. Therefore, the ability to efficiently search the index of an element containing a given position is of paramount importance, so a hierarchical data structure that allows for efficient searching is needed. The model only contains first derivatives of the unknown deformation. Hence, multilinear finite elements already allow a conforming discretization. For these reasons we use multilinear FEM on octree grids. The grids used are such that all of the elements are either squares or cubes of side length \(h=2^{-\ell }\), for an integer \(\ell \) to which we refer as grid level of the element. In what follows let us detail the different ingredients of the algorithm.

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Multilinear Finite Elements on Octrees We assume \(n=3\) for the presentation here. Using an adaptive octree grid based on cubic cells leads to hanging nodes (see Fig. 4), nodes which are on the facet of a cell without being one of its vertices. Enforcing continuity of the finite element functions leads to constraints for function values on hanging nodes and these hanging nodes are not degrees of freedom. Additionally, to minimize the complexity of the required interpolation rules, the subdivision is propagated in such a way that the grid level of neighboring elements sharing a cell facet differs at most by one.

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Octrees and the access to degrees of freedom via hashtables Even though the tree structure gives a natural hierarchical structure to the elements of the mesh, maintaining consistent linear indices for degrees of freedom, hanging nodes, and elements can be delicate. Consistent rules could be devised to maintain consistency with the element octree for a given mesh, but these would not be easy to update when the grid is refined. In order to keep track of vertex indices in a simple manner without sacrificing efficiency, hash maps ([19], Chapter 11) are maintained to keep track of the indices of degrees of freedom, hanging nodes, and cells. The keys used in the hashmap are a combination of a level value \(\ell \) and point coordinates as integer multiples of \(h=2^{-\ell }\). These keys uniquely identify nodes or elements, with the convention that an element is identified with its lower-left-back corner. Whenever a query for a node or cell is made, there are two possible outcomes. If it is already contained in the corresponding hash table, a linear index for it can be retrieved. Otherwise, a new entry of the hash table is created and the node or cell is given the next unused index. Since we do not require coarsening of the mesh, this scheme guarantees a consistent linear set of indices with a computational cost for insertions and queries that is, on average, independent of the mesh size.

Computing distance functions on octrees In our model, we have assumed that the distance functions to our surfaces are given. In practice, especially when using adaptive grids, we need to compute signed distance functions on such grids. This has been accomplished by a straightforward adaptation of the Fast Marching Method on cartesian grids [57] exploiting the fact that our grids still are subgrids of a regular cartesian grid. In the implemented variant hanging nodes are not taken into account for the propagation, their values being linearly interpolated to accommodate the constraints needed for conformality. The initialization for the distance computation has been performed starting from triangular meshes of the surfaces (for \(n=3\); for \(n=2\) two-bit segmentation of interior and exterior of the curves has been used). The signs of the distance functions have to be computed separately, by detecting which points of the grid are inside (resp. outside) the initial surface data. In our case, they have been computed with the provably correct algorithm given in [2].

Computation of the coefficients The discretization for the unknown deformation \(\phi \), as already mentioned, is done by multilinear finite elements. However, the coefficients of our model include first and second derivatives of the signed distance functions \(\mathbf{{d}}_i\), for the normal vectors \(\mathbf{{n}}_i\) and shape operators \({\mathcal {S}}_i\) (\(i=1,2\)), respectively. The approximations are required to be robust, since they appear in the highest order terms of the model. For the normal vectors \(\mathbf{{n}}_i\), we compute the \(L^2\) projection of the finite element derivative of \(\mathbf{{d}}_i\) to recover the nodal values of a piecewise multilinear function, followed by a orthogonal projection to the unit sphere to restore the constraint \(| \mathbf{{n}}_i | = 1\).

For the computation of matrix square roots and their inverses, we have used the method described in [26], taking appropriate care to truncate almost-singular matrices, since the resulting square roots also appear inverted.

Minimization strategy For the minimization at each level, we have opted for a Fletcher-Reeves nonlinear conjugate gradient method ([52], Section 5.2). The \(L^2\) gradient of \(E_\nu \), whose computation is involved but elementary, was implemented directly. The parameter \(\alpha \) is progressively reduced when a further feasible descent step is not found, according to an Armijo line search ([52], Section 3.1).

All of our results have been computed on the unit cube \(\Omega =[0,1]^3\) for the matching of surfaces in 3D, and the unit square \([0,1]^2\) for the matching of contour curves in 2D. In practice, we have used homogeneous Neumann boundary conditions, since this allows to have relatively large shapes \(\mathcal {M}_i\) in comparison with the size of the domain \(\Omega \) without creating excessive volume energies near the boundary (for the justification we refer to Corollary 4.3). However, if the boundary is not fixed, the deformed domain \(\phi (\Omega )\) is not necessarily contained in \(\Omega \), so evaluation of coefficients on deformed positions has to be appropriately handled numerically. We use a projection of outside position onto the boundary of \(\Omega \) for sufficient large \({\text {dist}}(\mathcal {M}_2, \partial \Omega )\).

In all examples, we have used the identity as the initial deformation. It should be noted that although the energy is geometric by design, we are using a first-order descent method for its minimization. In consequence, an adequate rigid pre-alignment can be beneficial for intricate shapes. Figure 8 shows results for the matching of two different dolphin shapes. Our variational approach is highly nonlinear and nonconvex. Thus, the numerical approximation of the globally optimal deformation depends on the initialization of the deformation. Figure 9 shows that the identity deformation as the initial deformation is advisable only if the expected optimal deformation is not too large. This is demonstrated by applying different rigid body motions to \(\mathcal {M}_1\).

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All figures have been produced by deforming the input data (polygonal curve or triangulated surface) via the resulting deformation \(\phi \). This is in contrast to deforming the grid and plotting the resulting extracted level sets (which effectively visualizes the inverse deformation), as commonly done in the registration literature, and also in [32].

Test case First, we present a simple test case to underline the qualitative properties of our model. Figure 5 shows a configuration in which a high amount of compression, combined with rotation, is required. Our model finds the intuitively correct deformation, but oscillations typical for the lack of lower semicontinuity of the underlying energy are induced when \(\text {P}_2\) is not used in the membrane and bending terms. The bending term assists in matching the curvatures even if the deformation is not rigid. Note, however, that for the optimal match the curvature energy \(E_{\text {bend}}\) is not expected to vanish, as can easily be seen from (2.6), (3.4) and the related discussion in Sect. 3.

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