Moser Transport Map in Imaging and PDEs

• Moser Transport Map is a constructive method that generates a smooth diffeomorphic flow between strictly positive densities via an elliptic PDE and ODE system.
• It enforces mass conservation by combining a continuity equation with a linear density interpolation and minimizes energy through a variational formulation.
• The method is applied in imaging to create continuous motion between segmented images using a convergent finite-element least-squares algorithm.

Searching arXiv for relevant papers on the Moser transport map and the cited 2012 imaging paper. The Moser transport map, often presented in the Dacorogna–Moser form, is a constructive method for generating a diffeomorphic transport between two strictly positive densities on a bounded domain by coupling a prescribed density interpolation with an elliptic PDE and an ODE flow. In the presentation adopted by Besson et al., the method is formulated for imaging applications, notably the generation of an apparent continuous motion between segmented initial and final images $\rho_0$ and $\rho_1$, while retaining a rigorous PDE and variational structure (Besson et al., 2012).

1. Problem setting and admissible data

Let $\Omega \subset \mathbb{R}^d$ with $d=1,2,3$ be a bounded $C^{2,\alpha}$-domain satisfying the exterior-sphere condition. The prescribed endpoint densities satisfy

$$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$

The objective is to construct a one-parameter family $\rho(t,x)>0$ on $Q:=(0,1)\times \Omega$ and a velocity field $v(t,x)$, vanishing on $\partial\Omega$, such that

$\rho_1$0

and mass is conserved along the flow generated by $\rho_1$1 (Besson et al., 2012).

This formulation is the continuum core of the transport map. The endpoint densities are assumed strictly positive and uniformly bounded away from zero and infinity, and the boundary compatibility $\rho_1$2 is part of the standing hypothesis. These assumptions ensure that the interpolation used in the construction remains positive and that the subsequent elliptic step is well posed under standard elliptic regularity theory.

2. Continuity equation and the Moser divergence constraint

If $\rho_1$3 is the flow associated with $\rho_1$4,

$\rho_1$5

then the pull-back density satisfies

$\rho_1$6

This is equivalent to the continuity equation

$\rho_1$7

Moser’s construction begins by prescribing an explicit Ansatz for the density path, namely the linear interpolation

$\rho_1$8

Consequently,

$\rho_1$9

and the continuity equation is enforced by requiring, at each $\Omega \subset \mathbb{R}^d$0,

$\Omega \subset \mathbb{R}^d$1

Since $\Omega \subset \mathbb{R}^d$2, one may write $\Omega \subset \mathbb{R}^d$3 and impose

$\Omega \subset \mathbb{R}^d$4

instead (Besson et al., 2012).

The essential structural point is that the map is not obtained by directly solving a Monge-type optimization problem. Instead, the density path is fixed in advance and the velocity is recovered from a divergence equation that enforces exact mass conservation. This gives the Moser construction its characteristic elliptic-flow form.

3. Variational characterizations

The Moser velocity admits a global least-squares or minimum-energy interpretation. One formulation minimizes the kinetic-energy functional

$\Omega \subset \mathbb{R}^d$5

over pairs $\Omega \subset \mathbb{R}^d$6 satisfying

$\Omega \subset \mathbb{R}^d$7

Within this framework, the Moser choice

$\Omega \subset \mathbb{R}^d$8

is pinned down by the boundary conditions, and the Euler–Lagrange equations in $\Omega \subset \mathbb{R}^d$9 produce the elliptic problem used in the explicit construction (Besson et al., 2012).

An equivalent formulation uses the weighted $d=1,2,3$0-seminorm

$d=1,2,3$1

The pair $d=1,2,3$2 with

$d=1,2,3$3

minimizes

$d=1,2,3$4

under

$d=1,2,3$5

In particular, the optimal velocity satisfies $d=1,2,3$6 and has minimal divergence in the dual $d=1,2,3$7-norm. In the terminology of the 2012 study, existence results are given for a transport problem with a minimum divergence for a dual norm or a weighted $d=1,2,3$8-semi norm, for the velocity (Besson et al., 2012).

These variational statements clarify that the map is simultaneously a transport construction and an energy-minimizing procedure. A plausible implication is that the Moser map is best understood as an elliptically regularized mass-preserving flow rather than solely as an explicit interpolation device.

4. Explicit construction of the transport map

The constructive algorithm proceeds in four steps. First, define

$d=1,2,3$9

Second, for each $C^{2,\alpha}$0, solve the elliptic PDE

$C^{2,\alpha}$1

together with the stated boundary condition

$C^{2,\alpha}$2

and the normalization $C^{2,\alpha}$3. Third, define the velocity by

$C^{2,\alpha}$4

Fourth, solve the flow ODE

$C^{2,\alpha}$5

and set

$C^{2,\alpha}$6

The resulting map satisfies

$C^{2,\alpha}$7

(Besson et al., 2012).

The same exposition states that, because $C^{2,\alpha}$8 and $C^{2,\alpha}$9, one obtains $$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$0 by standard elliptic theory. Since $$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$1, the flow yields a unique $$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$2-diffeomorphism $$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$3 for each $$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$4. The transport map $$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$5 is therefore obtained as the terminal configuration of a time-dependent diffeomorphic flow rather than as a static pointwise rearrangement.

A recurrent source of confusion concerns boundary conditions in the potential formulation. The presentation includes the flux condition $$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$6 on $$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$7, a Neumann-type statement $$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$8, and a normalization written as $$\rho_0,\rho_1 \in C^{1,\alpha}(\overline{\Omega}), \qquad 0<\beta_1\le \rho_i(x)\le \beta_2<\infty, \qquad \rho_0|_{\partial\Omega}=\rho_1|_{\partial\Omega}.$$9 or $\rho(t,x)>0$0. This suggests that the central invariant is the boundary behavior required for the transport flux and the elliptic problem’s well-posedness, while the normalization fixes the additive indeterminacy of the potential.

5. Existence, uniqueness, and regularity

Under the hypotheses labeled H1–H2, the linear interpolation $\rho(t,x)>0$1 never vanishes and $\rho(t,x)>0$2 is Hölder continuous. The elliptic problem for $\rho(t,x)>0$3 has a unique solution in $\rho(t,x)>0$4 up to an additive constant, fixed by the normalization. Hence the velocity field is uniquely defined and belongs to $\rho(t,x)>0$5. The ODE flow of a Lipschitz field $\rho(t,x)>0$6 exists for all $\rho(t,x)>0$7 and defines a unique $\rho(t,x)>0$8-diffeomorphism $\rho(t,x)>0$9 (Besson et al., 2012).

These regularity statements are not merely ancillary. They establish that the Moser map is a genuinely diffeomorphic transport under the stated assumptions, with the positivity of the interpolated density playing a decisive role. The construction therefore links elliptic PDE regularity, continuity-equation constraints, and flow-map theory in a single existence-and-uniqueness framework.

6. Computational realization and imaging use

In the 2D implementation described by Besson et al., the construction is embedded in a fixed point formulation and a space-time least-squares finite-element method. The discretization proceeds simultaneously in time and space. A space-time FE space $Q:=(0,1)\times \Omega$0 of piecewise polynomials on $Q:=(0,1)\times \Omega$1 is chosen, vanishing at $Q:=(0,1)\times \Omega$2. At iteration $Q:=(0,1)\times \Omega$3, given $Q:=(0,1)\times \Omega$4, one solves, for each $Q:=(0,1)\times \Omega$5, the discrete elliptic problem in $Q:=(0,1)\times \Omega$6 to obtain $Q:=(0,1)\times \Omega$7, computes

$Q:=(0,1)\times \Omega$8

and updates $Q:=(0,1)\times \Omega$9 by minimizing

$v(t,x)$0

in $v(t,x)$1 via an $v(t,x)$2 least-squares variational form under the endpoint constraints $v(t,x)$3 and $v(t,x)$4. Positivity is enforced by projection in the convex set

$v(t,x)$5

and the iteration is continued to convergence in $v(t,x)$6 (Besson et al., 2012).

The reported implementation uses a first-order (brick) Lagrange FE in space-time together with preconditioned conjugate-gradients, with incomplete-Cholesky or Gram–Schmidt least-squares preconditioners cited as examples. The stated application domain is cardiac image tracking, where the goal is to generate an apparent continuous motion observable through intensity variation from one starting image to an ending one, both supposed segmented. In this context, the transport introduced in the paper is compared with the transport introduced by Dacorogna–Moser, and the numerical results are presented as showing the efficiency of the proposed strategy.

This computational perspective situates the Moser transport map at the intersection of PDE-constrained imaging, optical-flow-type transport, and diffeomorphic registration. The theoretical construction and the finite-element realization are presented together: theoretically as the unique solution of a divergence-minimizing elliptic-flow problem, and practically via a convergent space-time finite-element least-squares algorithm (Besson et al., 2012).