[Paper Review] Multiple Shape Registration using Constrained Optimal Control

Sylvain Arguillère, Emmanuel Trélat+2 more|arXiv (Cornell University)|Mar 2, 2015

Medical Image Segmentation Techniques47 references3 citations

TL;DR

This paper proposes a constrained optimal control framework for simultaneous registration of multiple shapes using independent diffeomorphisms for each shape, coupled with a background deformation and contact constraints to preserve configuration consistency. The method enables accurate, artifact-free registration of complex anatomical structures by decoupling shape-specific deformations while enforcing geometric consistency, outperforming single-diffeomorphism LDDMM in preserving local Jacobian metrics.

ABSTRACT

Lagrangian particle formulations of the large deformation diffeomorphic metric mapping algorithm (LDDMM) only allow for the study of a single shape. In this paper, we introduce and discuss both a theoretical and practical setting for the simultaneous study of multiple shapes that are either stitched to one another or slide along a submanifold. The method is described within the optimal control formalism, and optimality conditions are given, together with the equations that are needed to implement augmented Lagrangian methods. Experimental results are provided for stitched and sliding surfaces.

Motivation & Objective

• To address the limitation of classical LDDMM, which applies a single global diffeomorphism to all shapes, causing unrealistic deformations when shapes are close.
• To enable independent, shape-specific deformation modeling while maintaining configuration consistency through contact constraints.
• To develop a general framework for multishape registration that supports identity and sliding constraints on surfaces.
• To provide a numerically stable method using augmented Lagrangian techniques for solving the constrained optimal control problem.
• To demonstrate that deformation markers (Jacobian determinants) are more meaningful and consistent when using the proposed multishape approach than in standard LDDMM.

Proposed method

• Formulates multishape registration as a constrained optimal control problem within the LDDMM framework, using separate diffeomorphisms for each shape and a background deformation.
• Implements contact constraints via a background manifold that deforms independently and enforces shape boundary consistency through equality constraints.
• Applies the augmented Lagrangian method to numerically solve the constrained optimization problem, enabling stable convergence.
• Derives optimality conditions using the Pontryagin maximum principle, leading to a two-point boundary value problem with adjoint variables.
• Uses a discrete formulation of the diffeomorphism flow via time-integrated vector fields in a reproducing kernel Hilbert space.
• Employs deformation markers based on normal and tangential Jacobian determinants to analyze local stretching and compression.

Experimental results

Research questions

• RQ1 Can multiple shapes be registered simultaneously with independent deformation fields while preserving geometric consistency?
• RQ2 How can contact constraints between shapes be mathematically modeled and enforced in a large deformation diffeomorphic framework?
• RQ3 Does the proposed method reduce spurious compression artifacts in regions where shapes are close, compared to single-diffeomorphism LDDMM?
• RQ4 To what extent do the normal and tangential Jacobian metrics reflect true shape deformation when using multishape constraints?
• RQ5 Can the augmented Lagrangian method effectively handle the high-dimensional, constrained optimization problem arising in multishape registration?

Key findings

• The proposed method successfully prevents the strong compression artifacts observed in single-diffeomorphism LDDMM, especially when shapes are in close proximity.
• With identity constraints, the normal Jacobian remains uniformly expanded in regions like Ball A, avoiding the extreme distortions seen in the single-diffeomorphism case.
• Sliding constraints preserve consistent deformation patterns across shapes, with minor differences in tangent Jacobians compared to identity constraints, indicating robustness.
• The deformation markers (Jacobian determinants) are more reliable and interpretable in the multishape framework, as they reflect true shape-specific changes rather than spurious interactions.
• The augmented Lagrangian method enables stable numerical implementation, though convergence is slow due to high-dimensional optimization.
• Experimental results on synthetic and real subcortical structures (hippocampus, amygdala, ERC) confirm that the method produces consistent, anatomically plausible deformations.