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images are constrained. As another example, muscles
such as the heart can shrink and expand, but within a
certain limit (Haber et al., 2010). In this case, it is the
relative change in volume that must be constrained.
Furthermore, and as it is the case with many fluid
registration techniques, optimal transport can result in
extreme deformations leading to topological changes
or unreasonable expansions or contractions of the dif-
ferent image regions. Figure 1 shows the interpolation
between two synthetic images that differ by a simple
translation using optimal transport. Figure 2, on the
other hand, shows the interpolation between the same
two images using a divergence free flow. The two re-
sults seem only slightly different. However, a closer
look at the divergence of the optimal transport flow
in Figure 3 shows that the image is not translated as
in the case of the divergence-free flow, but instead,
the region of high intensity is expanded, whereas the
region of low intensity is shrunk. This is the result
of minimizing the kinetic energy: only the difference
in mass is transported, not the whole mass in the re-
gion. Here again, limiting the relative change in vol-
ume would avoid such unreasonable transformations.
In this paper, we use the method of constrained
optimal transport (Kerrache and Nakauchi, 2011) to
interpolate between images. Constraints on the in-
terpolation: intermediate images as well as relative
change in volume can be imposed. Mathematically,
the relative change in volume is simply the Jacobian
of the transformation. Imposing a bound on the Jaco-
bian can be achieved by bounding the divergence of
the velocity field as it shown in Section 3. Aside from
the interpolation itself, an additional motivation for
this work is constrained image registration. Indeed,
once the interpolation is computed it becomes possi-
ble to compute the map between the two images by
integrating the velocity field of the flow. This point
will be the subject of a further study. The remain-
der of this paper is organized as follows. Section 2
presents the notion of constrained optimal transport.
Section 3 presents the proposed method. Section 4
contains the experimental results of the proposed al-
gorithms. Finally, Section 5 concludes the paper and
gives some future research directions.
2 CONSTRAINED OPTIMAL
TRANSPORT
Optimal transport admits two distinct formulations.
In the time-independent formulation, the problem
data consist in two spaces, each supporting a proba-
bility measure. The goal is to find a mass- preserving
map between the two spaces that minimizes a certain
transport cost. The cost can be the distance traveled
by each particle of mass, a function of the distance
or otherwise. In the time-dependent formulation, the
supporting space is the same, and the task consists
in transporting an initial mass distribution to a final
configuration. The cost in the time-dependent case is
also related to the distance between source and desti-
nation, but it has a continuous description. Each curve
joining any two points is given a cost, and therefore,
the transport cost is dependent on the path followed
and not only the start and ending points. Under ap-
propriate assumptions (Villani, 2009), the transport
between two mass distributions can be interpreted as
traversing a path in the set of probability measures
over the space under consideration. Finding an op-
timal transport consists then in finding a minimizing
curve in the space of probabilities. This formulation is
used by (Kerrache and Nakauchi, 2011) to introduce
the concept of constrained optimal transport, where
certain curves in the space of probability measures are
declared infeasible and can not be used as transport
paths. This allows to formulate a number of interest-
ing problems. For instance, it is possible to eliminate
certain mass distributions, or eliminate transport plans
that cause certain changes to the initial density. This
is the case in this paper, where transport plans that
cause undesired intermediate images or deformations
are eliminated.
In (Benamou and Brenier, 2000), the problem of
optimal mass transport in a closed convex subset D
of R
d
with the squared Euclidean distance as cost
is transformed to an optimal control problem of a
potential flow (Cohen and Kundu, 2004). The ap-
proach consists in computing a flow, in the sens of
fluid dynamics, having the minimum kinetic energy
that moves the initial density to the final one. More
precisely, the problem is formulated as
inf
ρ,m
Z
1
0
Z
D
|
m(t,x)
|
2
2ρ(t,x)
dxdt, (1)
s.t. ∂
t
ρ + ∇ · m = 0, (2)
ρ(0,·) = ρ
0
, ρ(1,·) = ρ
1
, (3)
where ρ (t,x) is the density, m (t,x) is the momentum
of the flow, ρ
0
(x) and ρ
1
(x) are two bounded positive
density functions defined on D, such as
Z
D
ρ
0
(x)dx =
Z
D
ρ
1
(x)dx = 1
This problem is then transformed to the following
saddle point problem
inf
φ,q
sup
µ
L (φ, q, µ) = F (q) + G (φ) +
h
µ,∇φ − q
i
, (4)
where µ = (ρ,m), G(φ) =
R
D
φ(0,x)ρ
0
(x) −
φ(1,x)ρ
1
(x)dx, F is defined by
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