[Paper Review] Geometric dynamics of optimization

François Gay–Balmaz, Darryl D. Holm+1 more|arXiv (Cornell University)|Dec 16, 2009

Mathematical Dynamics and Fractals62 references6 citations

TL;DR

This paper introduces optimization dynamics as a geometric framework for image registration using metamorphosis, where shape transformations are modeled via penalized, non-diffeomorphic deformations. By applying Lagrange-Poincaré and Hamilton-Poincaré reduction to optimal control problems, the authors derive Hamiltonian and Lagrangian equations of motion that generalize EPDiff and yield new integrable systems such as the two-component Camassa-Holm equations.

ABSTRACT

This paper investigates a family of dynamical systems arising from an evolutionary re-interpretation of certain optimal control and optimization problems. We focus particularly on the application in image registration of the theory of \emph{metamorphosis}. Metamorphosis is a means of tracking the optimal changes of shape that are necessary for registration of images with various types of data structures, without requiring that the transformations of shape be diffeomorphisms. This is a rich field whose possibilities are just beginning to be developed. In particular, metamorphosis and its related variants in the geometric approach to control and optimization can be expected to produce many exciting opportunities for new applications and analysis in geometric dynamics.

Motivation & Objective

• To develop a geometric mechanics framework for optimization problems in image registration, particularly using metamorphosis to relax the diffeomorphism constraint.
• To reinterpret image registration as an evolutionary dynamical system governed by variational principles and momentum maps.
• To generalize the EPDiff equation to non-diffeomorphic, penalized shape transformations through optimal control and reduction theory.
• To establish a unified geometric formulation—via Lagrange-Poincaré and Hamilton-Poincaré reduction—for optimization dynamics in computational anatomy.
• To demonstrate the framework on diverse systems, including fluid dynamics, rigid bodies, and integrable PDEs, revealing new dynamical structures.

Proposed method

• Adapt Lagrange-Poincaré reduction to optimal control problems with penalty terms, modeling inexact or soft constraints in shape deformation.
• Use momentum maps and Lie-Poisson brackets to derive Hamiltonian formulations from variational principles on semidirect product Lie algebras.
• Construct the Legendre transform to convert the Lagrangian system into a Hamiltonian system with variables (μ, n, β), representing momentum, density, and dual variables.
• Derive the Poisson bracket structure using the skew-symmetric matrix 𝒪, encoding the dynamics of the reduced system.
• Apply the framework to specific examples: heavy top, rigid body, fluid equations, and Camassa-Holm, showing how optimal reduction yields new integrable systems.
• Use the back-to-labels map and left action of diffeomorphisms to model fluid and shape evolution in a geometric, coordinate-free manner.

Experimental results

Research questions

• RQ1 How can optimal control problems in image registration be reinterpreted as dynamical systems using geometric mechanics?
• RQ2 What is the role of momentum maps and Lie-Poisson structures in connecting data structures (e.g., landmarks, contours) to evolutionary PDE solutions?
• RQ3 How does introducing a quadratic penalty for non-diffeomorphic transformations lead to new dynamical systems beyond EPDiff?
• RQ4 Can Lagrange-Poincaré and Hamilton-Poincaré reduction be systematically applied to derive equations of motion for optimization dynamics?
• RQ5 What new integrable systems emerge from optimal reduction of classical mechanical and fluid models, such as the Camassa-Holm equation?

Key findings

• The paper derives the two-component Camassa-Holm equations as an integrable Hamiltonian extension of the one-dimensional Camassa-Holm equation through optimal reduction.
• The metamorphosis equations on SE(2) are derived as a specific instance of the general framework, showing how shape and intensity changes are coupled via a penalty term.
• The Legendre-transformed variables (μ, n, β) satisfy a Lie-Poisson bracket structure defined by a skew-symmetric matrix 𝒪, which encodes the full dynamics of the system.
• The Poisson bracket (8.61) is shown to be linear in μ and quadratic in n and β, confirming the Lie-Poisson nature of the system on the dual of a semidirect product Lie algebra.
• The framework successfully generalizes the EPDiff equation to non-diffeomorphic, penalized deformations, enabling broader applications in computational anatomy.
• The momentum map provides a canonical isomorphism between image data (e.g., landmarks, contours) and the canonical variables of the evolutionary PDE, enabling optimal path computation.