# Asymptotic Result for a Decoupled Nonlinear Elasticity-Based Multiscale Registration Model

N Debroux, C Le Guyader

► To cite this version:

N Debroux, C Le Guyader. Asymptotic Result for a Decoupled Nonlinear Elasticity-Based Multiscale Registration Model. Scale Space and Variational Methods in Computer Vision, May 2023, Santa Margherita di Pula, Italy. Springer International Publishing, cale Space and Variational Methods in Computer Vision 2023, Scale Space and Variational Methods in Computer Vision, Lecture Notes in Computer Science. hal-04873916

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# Asymptotic Result for a Decoupled Nonlinear Elasticity-Based Multiscale Registration Model

N. Debroux* and C. Le Guyader†

*Université Clermont Auvergne, CNRS, SIGMA Clermont, Institut Pascal, France, noemie.debroux@uca.fr

†INSA Rouen Normandie, Normandie Univ, LMI UR 3226, F-76000 Rouen, France, carole.le-guyader@insa-rouen.fr

## 1. Motivations

Motivated by Tadmor et al.'s work ([5]) dedicated to multiscale image representation using hierarchical ($BV, L^2$) decompositions, we propose transposing their approach to the case of registration, task which consists in determining a smooth deformation aligning the salient constituents visible in an image into their counterpart in another.

<figure><img src="image_2.png" /><figcaption>Illustration of the (BV, L²) hierarchical decomposition of an image representing a finger print. Taken from Tadmor et al.'s work [5].</figcaption></figure>

The underlying goal is to obtain a **hierarchical decomposition** of the deformation into the form of a **composition of intermediate deformations**: the **coarser** one, computed from versions of the two images capturing the essential features, **encodes the main structural/geometrical deformation**, while iterating the procedure and refining the versions of the two images yields **more accurate deformations** that map faithfully small-scale features.

<figure><img src="image_3.png" /><figcaption>Multiscale registration results on synthetic T-shape images with texture [3].</figcaption></figure>

This contribution focuses more specifically on a **theoretical asymptotic result**, establishing a link between the **original minimisation problem** comprising **high non linearity** and **non convexity**, and the one derived from **splitting techniques** by means of **auxiliary variables** and $L^p$-penalisations which exhibits increased numerical manageability. This latter problem appears to be a **good compromise** between **physical realism** and **practical feasibility**.

## 2. Original Mathematical Problem

* **Rationale underpinning the study:** build a **hierarchical** decomposition of the deformation matching two images in the form of a **composition** of intermediate deformations

coarser deformation

$$ \underbrace{\varphi_0}_{\text{finer deformation}} \circ \dots \circ \varphi_k \circ \dots \circ \varphi_n \Rightarrow \text{hierarchise} $$

the information

carried by the deformation.

Mathematically, the resulting iterative refinement scheme that takes into account the granulometry of the deformation is broken down as follows:

$$ (P_k) \quad \left\{ \begin{array}{l} \varphi_0 = \underset{\varphi \in W}{\arg\min} \{\mathcal{F}(\varphi, T_0, R_0)\}, \\ \varphi_k = \underset{\varphi \in \mathcal{X}_k}{\arg\min} \{\mathcal{F}(\varphi_0 \circ \varphi_1 \circ \dots \circ \varphi_{k-1} \circ \varphi, \sum_{j=0}^k T_j, \sum_{j=0}^k R_j)\}, \end{array} \right. $$

with:

* $\bar{R}_k := \sum_{j=0}^k R_j$ (resp. $\bar{T}_k := \sum_{j=0}^k T_j$), **hierarchical decomposition** of the **Reference** image $BV(\Omega) \ni R : \bar{\Omega} \to \mathbb{R}$ (resp. the **Template** image $BV(\Omega) \ni T : \bar{\Omega} \to \mathbb{R}$ with ess supp($T$) $\subset \Omega' \subset \subset \Omega$);

* $\mathcal{X}_k = \{ \psi \mid \Phi := \varphi_0 \circ \varphi_1 \circ \dots \circ \varphi_{k-1} \circ \psi \in \text{Id} + W_0^{1,\infty}(\Omega, \mathbb{R}^2), \det \nabla \Phi > 0 \text{ a.e.,} \\ \|\nabla \Phi\|_{L^\infty(\Omega, M_2(\mathbb{R}))} \le \alpha, \|(\nabla \Phi)^{-1}\|_{L^\infty(\Omega, M_2(\mathbb{R}))} \le \beta \}.$

* **Functional $\mathcal{F}$** $\Rightarrow$ viewed as a **parent functional** from which the **successive minimisation subproblems** are **derived**. Devised according to arguments from the **theory of mechanics** (hyperelasticity [2, Part A, Chapter 4 and Part B, Chapter 7]):

## 2. Original Mathematical Problem - follow-up

* $\mathcal{F}(\varphi, T, R) = \frac{\lambda}{2} \|T \circ \varphi - R\|_{L^2(\Omega)}^2 + \int_{\Omega} W_{Op}(\nabla \varphi, \det(\nabla \varphi)) dx + 1_{\{\|\cdot\|_{L^\infty(\Omega, M_2(\mathbb{R}))} \le \alpha\}} (\nabla \varphi) + 1_{\{\|\cdot\|_{L^\infty(\Omega, M_2(\mathbb{R}))} \le \beta\}} ((\nabla \varphi)^{-1}),$

where

$$ W_{Op}(F, \delta) = \begin{cases} a_1 \|F\|^4 + a_2(\delta - 1)^2 + \frac{a_3}{(\delta)^{10}} - 4a_1 - a_3 & \text{if } \delta > 0 \\ \infty & \text{otherwise} \end{cases} $$

is the stored energy of an **Ogden material** ([2]) —$Op$ standing for ‘Ogden particular’ and $\| \cdot \|$ being the Frobenius norm —.

## 3. Towards a Decoupled Problem

* Problem ($P_k$) exhibits fine theoretical properties $\Rightarrow$ (i) it admits at least **one minimiser** (bi-Lipschitz homeomorphism from $\bar{\Omega}$ to $\bar{\Omega}$), (ii) an **asymptotic result** can be established when $k$ tends to $\infty$, etc.). Nevertheless, part of the class of **non convex, non differentiable** problems;

* To **balance** the numerical complexity $\Rightarrow$ introduction of several **auxiliary variables**, yielding a **splitting** of the original problem ($P_k$) into subproblems that are more easily tractable. $\Psi \simeq \phi^{-1}$, $\phi \simeq \varphi_0 \circ \dots \circ \varphi_{k-1} \circ \varphi$, $V \simeq \nabla \phi$, $W \simeq \nabla \Psi$ and $\zeta_{k-1} = \varphi_0 \circ \varphi_1 \circ \dots \circ \varphi_{k-1}$.

* Derived **decoupled** problem ($DP_{k,i}$), $(\gamma_{k,i})_{i \in \mathbb{N}}$ being an increasing sequence of positive real numbers such that $\lim_{i \to \infty} \gamma_{k,i} = \infty$ for fixed $k$.

$$ \inf_{\varphi, \phi, \psi, V, W} \left\{ \begin{aligned} & \bar{F}_{k,i}(\varphi, \phi, \psi, V, W) = \frac{\lambda}{2} \int_{\Omega} (\bar{T}_k - \bar{R}_k \circ \psi)^2 \det \nabla \psi \, dx && (DP_{k,i}) \\ & + \int_{\Omega} W_{Op}(V, \det V) \, dx + 1_{\{\|\cdot\|_{L^\infty(\Omega, M_2(\mathbb{R}))} \le \alpha\}} (V) + 1_{\{\|\cdot\|_{L^\infty(\Omega, M_2(\mathbb{R}))} \le \beta\}} (W) \\ & + \frac{\gamma_{k,i}}{4} \|V - \nabla \phi\|_{L^4(\Omega, M_2(\mathbb{R}))}^4 + \frac{\gamma_{k,i}}{4} \|W - \nabla \psi\|_{L^4(\Omega, M_2(\mathbb{R}))}^4 \\ & + \frac{\gamma_{k,i}}{2} \|\zeta_{k-1}^{-1} \circ \phi - \varphi\|_{L^2(\Omega, \mathbb{R}^2)}^2 + \frac{\gamma_{k,i}}{2} \|\psi \circ \phi - \text{Id}\|_{L^2(\Omega, \mathbb{R}^2)}^2 \end{aligned} \right\}, $$

## 4. Main Result

**Theorem 1 (Asymptotic result)** Let $(\varphi_{i,n}, \phi_{i,n}, \Psi_{i,n}, V_{i,n}, W_{i,n})_n$ be a minimising sequence of $\bar{F}_{k,i}$. Then there exists a subsequence denoted by $(\varphi_{\tau(i),N(\tau(i))}, \phi_{\tau(i),N(\tau(i))}, \Psi_{\tau(i),N(\tau(i))}, V_{\tau(i),N(\tau(i))}, W_{\tau(i),N(\tau(i))})$ that we now write $(\varphi_i, \phi_i, \Psi_i, V_i, W_i)$ for the sake of simplicity, and a minimiser $\bar{\varphi}$ of $F_k$ such that

$$ \lim_{i \to \infty} \bar{F}_{k,i}(\varphi_i, \phi_i, \Psi_i, V_i, W_i) = F_k(\bar{\varphi}), $$

with $\phi_i \stackrel{i \to \infty}{\to} \bar{\phi}$ in $W^{1,4}(\Omega, \mathbb{R}^2)$, $\varphi_i \stackrel{i \to \infty}{\to} \bar{\varphi} = (\varphi_0 \circ \dots \circ \varphi_{k-1})^{-1} \circ \bar{\phi}$ in $L^2(\Omega, \mathbb{R}^2)$, $V_i \stackrel{i \to \infty}{\to} \nabla \bar{\phi}$ in $L^\infty(\Omega, M_2(\mathbb{R}))$, $\det V_i \stackrel{i \to \infty}{\to} \det \nabla \bar{\phi}$ in $L^2(\Omega)$, $W_i \stackrel{i \to \infty}{\to} \nabla \bar{\phi}^{-1}$ in $L^\infty(\Omega, M_2(\mathbb{R}))$ and $\Psi_i \stackrel{i \to \infty}{\to} \bar{\phi}^{-1}$ in $W^{1,4}(\Omega, \mathbb{R}^2)$.

## Bibliographical References

[1] J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, P. Roy. Soc. Edin. A, 88(3-4) (2011), pp. 315-328.

[2] P. G. Ciarlet, Three-Dimensional Elasticity, Mathematical Elasticity, Elsevier Science (1994).

[3] N. Debroux, C. Le Guyader and L. A. Vese, A Multiscale Deformation Representation, accepted for publication in SIAM J. Imaging Sci. (2023).

[4] P. V. Negrón Marrero, A numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity, Numer. Math., 58 (1990), pp. 135-144.

[5] E. Tadmor, S. Nezzar and L. A. Vese, A Multiscale Image Representation Using Hierarchical (BV, L²) Decompositions, Multiscale Model. Simul., 2(4) (2004), pp. 554-579.

## Acknowledgements

This project was co-financed by the European Union with the European regional development fund (ERDF, 18P03390/18E01750/18P02733), by the Haute-Normandie Régional Council via the M2SINUM project and by the French Research National Agency ANR via AAP CE23 MEDISEG ANR project.

INSTITUT PASCAL
sciences de l'ingénierie et des systèmes

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