Optimal transport, level-set: applications to image
Ensimag 3A, Grenoble INP and Master MSIAM, IM2AG, UJF
Emmanuel Maitre∗
December 1, 2014
"C’est en traitant une de ces applications que Monge a donné sa théorie de la courbure des surfaces ; on la trouve exposée pour la
première fois dans le Mémoire sur les déblais et remblais publié dans les mémoires de l’académie des sciences de Paris, pour l’année
1781. Monge y fait voir que les routes suivies pour aller du déblai au remblai étant supposées rectilignes, elles sont les normales
d’une surface unique ; il part de là pour décomposer le faisceau de ces normales en groupes de surfaces développables qui ont pour
arêtes de rebroussement, les lignes, lieux des centres de courbure de la surface indiquée, et qui tracent autant de lignes de courbure
sur cette surface, limite la plus avantageuse du déblai ou du remblai supposés indéfinis d’un côté seulement. "
Charles Dupin, Essai historique sur les services et les travaux scientifiques de Gaspard Monge. Paris, Bachelier 1819, p. 230.
Image et texte récupérés sur le site http://alta.mathematica.pagesperso-orange.fr
∗
[email protected], http://ljk.imag.fr/membres/Emmanuel.Maitre/
1
1
Introduction
An image can be considered as an application of a domain of R2 (or R3 in the case of three-dimensional
images) with values in a subset of R (or R3 for color images). Many algorithms of edge detection, shape
recognition, denoising, are based partial differential models.
In a first part of this lecture we consider the Eulerian representation of region of interest in images.
We begin by giving the main principles of the Eulerian description of interfaces and what we can
derive from it as geometrical information. Then we develop few algorithms of active contours that
were developed up to last years. We will explicit numerical tools involved in the implementation of
these algorithms.
The optimal transport is an old research topic since it was first studied in 1781 by Gaspard Monge,
about transport optimization of building materials. It has applications in economy and has been set
in a fluid mechanics formulation by Brenier, as a minimization problem on kinetics energy under as
mass conservation equation.
The use of this tool in image analysis was introduced by Benamou - Brenier then by Haker et al.
Their work has helped to develop a method for interpolating between two images (or more) taking into
account the motion of an image on the other. Jusqu’à très récemment les énergies utilisées ne faisaient
intervenir que l’énergie cinétique, sans tenir compte de contraintes supplémentaires pouvant exister.
This lecture aims at bringing together these two topics, namely the study of optimal transport for
more general energies, in the context of image processing, by means of level-set formulations.
2
Segmentation methods using level-set in image analysis
2.1
Introduction
To represent an interface defining two moving regions of interest, several representations are possible.
In particular, we may consider a parametrization of the interface (for simplicity assume that this is
a curve Γt ). We call it a Lagrangian setting: let γ : [0, M ] × [0, T ] → γ(r, t) a parametrization of
Γt advected by the continuous medium velocity γ(r, t) = X(t; γ0 (r)) or equivalently the solution of
differential system:
(
∂t γ(r, t) = u(γ(r, t), t), r ∈ [0, M ], t ∈]0, T ]
(1)
γ(r, 0) = γ0 (r),
r ∈ [0, M ].
In practice, to capture a region, we will therefore consider a discretization of [0, M ] given by (ri )i=0...N
and then move γ(ri , t) to enclose the target region. The fact that the spacing between the points may
vary as a result of the curve deformation could cause over or under-sampling. Moreover, with this
representation, handling topology changes of the contour can be cumbersome.
2.2
Level set framework
We now change our representation of the interface to circumvent the pitfalls of the Lagrangian representation. We represent Γt , that we from now on consider as closed1 by introducing an auxiliary
function φ : Ω × [0, T ] → R such that
Γt = {x ∈ Ω,
φ(x, t) = 0}.
The bounded domain delimited by Γt , that we refer as the region interior, is Ω−
φ(x, t) <
t = {x ∈ Ω,
+
0} and its exterior as Ωt = {x ∈ Ω, φ(x, t) > 0}. As φ(γ(r, t), t) = 0 on [0, M ] × [0, T ], and
∂t γ = u(γ, t), we get differentiating with respect to t
∂t φ(γ(r, t), t) + u(γ(r, t), t) · ∇φ(γ(r, t), t) = 0
1
It is possible to remove this assumption with the addition of another auxiliary function.
2
The Level Set method [66] corresponds to consider a initial function φ0 representing Γ0 and to look
for a function φ which verifies the transport equation on the whole domain:
(
∂t φ + u · ∇φ = 0 on Ω×]0, T [
(2)
φ = φ0
on Ω × {0}.
Usually φ0 is chosen as a signed distance to interface:
(
− dist(x, Γ0 ) if x belongs to the interior of Γ0
φ0 (x) =
dist(x, Γ0 )
if x belongs to the exterior of Γ0
Remark 1
1. It is remarkable that a smooth function of time and space can be a very singular
evolution curve: melting of curves, or splitting of a curve. Is that we have added a dimension to
the problem, and in this new space, these situations are not unique! The price to pay is to work
with this additional dimension: instead of a 1D discretization, we will consider a discretization
of the whole space. Nevertheless, we will see that there are methods to restrict the calculations to
an area near the contour (narrow-band) .
2. The level-set method gives a direct way of knowing if a point in space is inside or outside of the
contour, by inspecting the sign of φ at this point. This is generally not trivial with the other
representation. On the contrary, access the coordinates of points of the contour is not at all
natural, and we’ll try to get around this need as possible.
With the choice of sign of φ0 above fact, that we will now assume systematically, the outer normal
to the area enclosed by Γt and its curvature are expressed at each of its points by
n(x) =
∇φ
|∇φ|
κ(x) = div
∇φ
|∇φ|
The general proof can be found in the book [50], pages 354 to 357. We justify the formula of curvature
in two dimensions. Consider a regular parametrization s → γ(s) of Γt such that, by following the curve
along the direction of ∂s γ, we have (φ < 0) on our left. The curvature is expressed in terms of the
parameterization as
[∂s γ, ∂ss γ]
κ=
|∂s γ|3
∇φ
By differentiating the identity φ(γ(s)) = 0 we have ∂s γ · ∇φ = 0. Differentiating ∂s γ · |∇φ|
= 0 we have
∇φ
∇φ
∂ss γ ·
+ ∂s γ · D
∂s γ = 0
|∇φ|
|∇φ|
As ∂s γ ⊥
∇φ
|∇φ|
∇φ
we have due to the chosen curve orientation, [∂s γ, |∇φ|
] < 0, therefore
∇φ
[∂s γ, ∂ss γ] = |∂s γ|∂s γ · D
∂s γ
|∇φ|
In dimension two, if we denote as ∇×φ the curl of phi φ, obtained by a + π2 rotation to the vector ∇φ,
the corresponding tangent vector is
∇×φ
τ (x) =
|∇φ|
Therefore
∇φ ∇×φ
∇×φ
κ=
D
|∇φ|
|∇φ| |∇φ|
The prof is achieved by the following lemma that the reader could prove (S1 is the unit circle).
Lemma 1 Let u, v : Ω ⊂ R2 → S1 , of class C 1 , such that u · v = 0 on Ω. Then [Du]v · v = div u.
3
2.3
Set operations expressed with level set functions
Let two bounded open sets Ω1 and Ω2 . If we already have two level set functions to capture those
sets, say φ1 and φ2 , such that φi < 0 in Ωi , then min(φ1 , φ2 ) is a level set function for Ω1 ∪ Ω2 and
max(φ1 , φ2 ) for Ω1 ∩ Ω2 . As φi is a level set function for Ωi , −φi plays the same role for Ωci . Therefore,
max(−φ1 , φ2 ) is a level set function for Ω2 \ Ω1 . At last min(max(−φ1 , φ2 ), max(φ1 , −φ2 )) is a level
set function for the symmetric difference Ω1 ∆Ω2 .
2.4
General principles of image segmentation and active contours
Consider an image (grayscale to start), that is a function defined on the domain Ω, I : Ω → R. To
identify areas in this image, a first idea is to look at the histogram of the gray levels of the latter, and
segment the image into functions thresholds on the histogram. We will therefore consider as belonging
to the same region of interest pixels having an intensity within a certain band. The problem with this
type of segmentation is that it does not produce in general a connected region, and is very sensitive
to noise. We therefore seek methods of segmentation sufficiently robust to noise and low contrast
images. Among the existing segmentation methods, except the threshold described above, one can list
the active contours, deformable models, growth models ... We are interested in this part in the active
contour models, of which we present the general principles.
It amounts to move a curve (or a surface for a 3D image) so that it fits the contours of the object
that we want to identify in the image. In order to move Γt , in Lagrangian representation we usually
seek to minimize a functional of the following kind:
F (γ(t)) = α
Z
0
M
|∂r γ(r, t)|dr + β
Z
M
2
|∂rr
γ(r, t)|2 dr
0
−λ
Z
0
M
|∇I(γ(r, t))|2 dr,
(3)
where α ≥ 0, β ≥ 0, λ > 0 are parameters. The first two measure the smoothness of the curve, the last
will move the γ curve to areas of strong gradients of the image.
To write the dynamics of movement, rather than writing a derivative with respect to the curve,
using our velocity field we try to move the curve γ so as to minimize F as t increases. On has
d
F (γ(t)) = dFγ(t) (∂t γ(t)) = dFγ(t) (u(γ(t))) = ∇F (γ(t)) · u(γ(t))
dt
We see that choosing u(γ) = −ρ∇F (γ(t)), where ρ > 0 is a descent parameter, will produce decreasing
F . This is the "snakes" method introduced by Kass, Witkin, Terzopoulos in 1988. The general principle
of these methods is to identify regions that are defined by areas of high variations the gradient of the
image. Furthermore, if there are multiple objects to be captured, if the starting area encircles these
objects, then the contour will always gather the union of these objects and therefore, as is, this method
can not capture multiple objects. This is a serious caveat of the method. More generally we consider
1
a decreasing function g : [0, +∞) → R+ such that g(r) → 0 for r → ∞, for instance g(s) = 1+s
2 . We
consider here the case β = 0 and therefore the energy boils to:
F (γ(t)) = α
Z
0
M
|∂r γ(r, t)|dr + λ
Z
M
g(|∇I(γ(r, t))|)dr,
(4)
0
A remark raised after a few years in the development of the method is the dependence on parameterization of the curve. Noticed by Caselles, Kimmel and Sapiro in 1997, that is explained by the fact that if
we define a new parameterization of the curve by a change of variable r = φ(s) with φ : [0, L] → [0, M ],
φ0 > 0 we get:
Z
Z
M
0
L
|∂r γ(r, t)|dr =
4
0
|∂s γ(φ(s), t)|ds
rs
271
(a)
FIGURE 1: Snakes (Kass, (b)
Witkin, and Terzopoulos 1988)
es (Kass, Witkin, and Terzopoulos 1988) c 1988 Springer: (a) the “snake
which does not depend on parametring, whereas
ly controlling shape; (b) lip
Z Mtracking.
Z L
g(|∇I(γ(φ(s), t))|)φ0 (s)ds
g(|∇I(γ(r, t))|)dr =
0
0
is completely
dependent
on the
(nb: if youittake
square ingradithe first term of
m attractsWhich
the snake
to dark
ridges,
theparametrization
edge term attracts
to astrong
energy as in the articles cited, this term also depends on the setting). The trick used by Caselles,
he term term
attracts
it to
line to
terminations.
In practice,
mostof systems
onlyso that it is
Kimmel
and Sapiro
is simply
write an energy considering
the metric
the curve itself,
invariant under change of parametrization:
which can
either be directly proportional to the image gradients,
Z M
Z L
X
g(|∇I(γ(r, t))|)|∂r γ(r, t)|dr
=
g(|∇I(γ(φ(s), t))|)|∂r γ(φ(s), t)|ds
Eedge =0
krI(f (i))k2 , 0
(5.4)
The general form of the
i energy is now:
Z
M
version of the image Laplacian,F (γ(t)) =
g(|∇I(γ(r, t))|)|∂r γ(r, t)|dr,
(5)
0
X
2
2 form by replacing g by α + λg. Let us compute
since
indeed,
energy
(4)
may be
expressed
Eedge =
|(G
⇤r
I)(funder
(i))|this
.
(5.5)
the flow generated by such an energy. After some elementary calculations, we find:
i
Z M
d
F (γ(t)) =
[g(∇I(γ))κn + (∇(g(∇I))(γ) · n)n] · u(γ)dr
extract edges and then
map to the edges as an alternative
dt use a distance
0
(6)
imes
where npotentials.
is the outward normal, which gives as evolution of γ :
nally proposed
applications, a variety of user-placed
constraints
can also
be added, e.g.,
∂t γ = −ρ[g(∇I(γ))κn
+ (∇(g(∇I))(γ)
· n)n].
(7)
forces towards
anchor
d(i),
The application
of apoints
discretized
version of this algorithm to follow lips motion, in the original article of
1988, is depicted on figure 1. Problems could occur on sharp edges, see figure 2. Algorithm is sensitive
2
to noise, and cannot detect more than one object,
see figure 3.
E
=
k
kf
(i)
d(i)k
,
(5.6)
spring
i
We now translate this expression to Eulerian coordinates. First is a curve is evolving with normal
velocity V , that is:
∂t γthe
= V snakes
n
(“volcano”) forces (Figure 5.2a). As
evolve by minimiz-
ve 1/r
the transport
ont thewhich
level setaccounts
function reads:
hey oftenthen
“wiggle”
andquartions
“slither”,
for their popular name.
∂t φ + V n · ∇φ = 0
snakes being used to track a person’s lips.
ar snakes have a tendency to shrink (Exercise
5 5.1), it is usually better to
FIGURE 2: Snakes, case with sharp edges
FIGURE 3: Snakes, case with two objects and noise
6
which as n is a unit vector field collinear to ∇φ, gives
∂t φ = −V |∇φ|
Thus the equation on φ becomes (for ρ = 1) :
∇φ
+ ∇(g(∇I)) · ∇φ
|∇φ|
∂t φ = g(∇I)|∇φ| div
2.5
2.5.1
Active contours and level set: one step beyond
Computation of volume and surface integrals
The implicit representation can be used to compute approximate surface and volume integrals.
Lemma 2 Let φ : Rd → R be Lipschitz on Rd and g : Rd → R integrable. Assume there exists η0 > 0
such that ess inf |φ|<η0 |∇φ| > 0. Then for η ∈]0, η0 [,
Z
Z ηZ
g(x)dx =
g(x)|∇φ|−1 dσdν
|φ(x)|<η
−η
φ(x)=ν
Proof. In [41], proposition 3 page 118, it is shown under such assumptions that
Z
Z
d
g(x)dx = −
g|∇φ|−1 dσ
a.e. s
ds
φ>s
φ=s
The above result easily follows by setting s = −t and taking φ and −φ in this formula, and then
summing up the two obtained identities after integrating them between 0 and η.
A less rigorous but more intuitive proof amounts to write the volume element in a neighborhood
∇φ
of a point x as dx = dσ × dh, where dh is along the normal |∇φ|
to the level-set of φ through x, and
by remarking that
∇φ
) = φ(x) ± dh|∇φ| + O(dh2 )
ν ± dν := φ(x ± dh
|∇φ|
from which we get dx = |∇φ|−1 dσdν.
From now on we assume the following regularity for φ:
Z
n
(Hφ ) ∀t ∈ [0, T ], ∀f ∈ Cc (R ), s →
f (x)dx est C 1 au voisinage de s = 0
{|φ(x,t)|<s}
Let M(Rd ) be the space of bounded measures on Rd , that is, of linear continuous forms on the
space of bounded continuous functions. Then we have the following:
Proposition 1 Let r → ζ(r) be a continuous cut-off function, that is with support in [−1, 1], and such
that r → 1ε ζ( rε ) converges to δ0 in M(R). Then under assumption (Hφ ), when ε → 0,
1
φ
ζ
|∇φ| * δ{φ=0} in M(Rd ).
ε
ε
Proof. For a continuous g(r), we have by assumption,
Z
1 r
lim
ζ
g(r)dr = g(0).
ε→0 R ε
ε
Applying this, for f ∈ Cc (Rd ), to
g(r) =
Z
f dσ.
{φ=r}
7
We have
lim
Z
ε→0 R
so that
lim
1 r
ζ
ε
ε
Z Z
ε→0 R
{φ=r}
Z
1
ζ
ε
f dσdr =
{φ=r}
Z
f dσ,
{φ=0}
Z
φ
f dσ.
f dσdr =
ε
{φ=0}
As 1ε ζ( φε ) is vanishing outside |φ| < ε, using Lemma 2, we get
Z Z
R
{φ=r}
1
ζ
ε
Z εZ
Z
1
1
φ
φ
φ
f dσdr =
f dσdr =
f |∇φ|dx
ζ
ζ
ε
ε
ε
−ε {φ=r} ε
|φ(x)|<ε ε
Z
φ
1
ζ
f |∇φ|dx
=
ε
Rd ε
which means finally, for all f ∈ Cc (Rd ),
Z
Z
1
φ
lim
ζ
|∇φ|f (x)dx =
f (x)dσ.
ε→0 Rd ε
ε
{φ=0}
We therefore hereby justified that δ{φ=0} can be approximated by
|∇φ| 1ε ζ
φ
ε
for small ε.
Remark 2 This approximation has however to be handled with care. Indeed it amounts to replace a
purely geometrical object, namely the measure supported on a curve, δ{φ=0} (which depends only on
this curve but not on the chosen φ to capture it), by a function that does
on the scale
depend
of φ
2φ
φ
1
1
(for instance {2φ = 0} or {φ = 0} are the same curve, whereas |2∇φ| ε ζ ε 6= |∇φ| ε ζ ε ). This
is the origin of some numerical subtleties in the Level Set method. We will see how redistancing and
renormalization give answer to this problem.
2.5.2
General level set formulation of a gradient based active contour method
Thanks to the above approximation, an Eulerian expression of the (approximated) active contour
energy (5) can be written as
Z
1 φ
Fε (φ) =
g(|∇I|) ζ( )|∇φ|dx
ε ε
Ω
Let us compute, independently of the dimension, the functional derivative of this energy. Using the
transport equation verified by φ et l’équation (??) vérifiée par |∇φ|,
Z
d
1 0 φ
1 φ
Fε (φ) =
g(|∇I|) 2 ζ ( )∂t φ|∇φ| + ζ( )∂t |∇φ| dx
dt
ε
ε
ε ε
Ω
Z
1 0 φ
1 φ ∂t ∇φ · ∇φ
=
g(|∇I|) 2 ζ ( )(−u · ∇φ)|∇φ| + ζ( )
dx
ε
ε
ε ε
|∇φ|
Ω
Z
1 φ
∇φ 1 φ
=
ζ( ) ∂t φdx
g(|∇I|) −u · ∇( ζ( ))|∇φ| − div g(|∇I|)
ε ε
|∇φ| ε ε
Ω
where we performed an integration by parts to get the last expression. Still using the transport
equation, ∂t φ = −u · ∇φ so that
Z
d
1 φ
1 φ
Fε (φ) =
g(|∇I|) −u · ∇( ζ( ))|∇φ| + |∇φ|∇( ζ( )) · u
dt
ε ε
ε ε
Ω
∇φ 1 φ
∇φ 1 φ
+ g(|∇I|) div
ζ( )∇φ · u + ∇[g(|∇I|)] ·
ζ( )∇φ · udx
|∇φ| ε ε
|∇φ| ε ε
8
FIGURE 4: Level-set formulation of Snakes (Caselles et al, 1997). Initial contour / Steady state
contour.
Finally, we got
d
Fε (φ) =
dt
Z Ω
g(|∇I|) div
∇φ
|∇φ|
∇φ ∇φ
1 φ
+ ∇[g(|∇I|)] ·
· u|∇φ| ζ( )dx
|∇φ| |∇φ|
ε ε
which is the level translation of (6), up to the approximation of the surface measure given by |∇φ| 1ε ζ( φε )dx.
Remark 3 In the case where curvature energy has to be taken into account, the level-set formalism
can of course be used and is also dimension independent. Actually the curvature energy can take the
form
Z
1 φ
Gε (φ) =
G(κ(φ))|∇φ| ζ( )dx
ε ε
Ω
where the more standard case corresponds to G(r) = 12 r2 . As above, the time derivative of this curvature
energy corresponds to the power of curvature forces Hε (x, t), since we have
Z
d
Gε (φ) = dGε (φ)(∂t φ) = dGε (φ)(−u · ∇φ) = − Hε (x, t) · udx.
(8)
dt
Ω
After a lengthy computation, we obtain:
1 φ
∇φ
1
0
+
P ⊥ ∇[|∇φ|G (κ(φ))]
ζ( )∇φ.
Hε (x, t) = div −G(κ(φ))
|∇φ| |∇φ| ∇φ
ε ε
2.5.3
Some tests
The following tests were performed using the CREASEG Matlab software from Olivier Bernard, INSA
Lyon. In figure 4, the level-set formulation shows its ability to perform better than the lagrangian
formulation of snakes, for the cas elf the star. Indeed, the corners are far better captured, and the
solution indeed reaches a steady state. However, for noisy images, the algorithm fails to reach a steady
star, and while it seems to perform well at some point, as iterations increase, the curve will finish to
disappear completely of partially (figure 5).
9
FIGURE 5: Level-set formulation of Snakes (Caselles et al, 1997). Best contour / Contour after too
much iterations (no steady state).
2.6
2.6.1
Region-based methods
Idea
In the above methods, one drawback is that the image gradient is involved, which could be a problem
for noisy images. Other methods are built on the principle of capturing regions instead of capturing
contours delimiting them. The pioneering method of Chan & Vese (2001) is based on the minimization
of the so-called Mumford-Shah functional:
Z
XZ
MS
2
E (u, Γ) =
|u − I| dx + µ|Γ| + ν
|∇u|2 dx.
i
Ω\Γ
Ωi
A minimizer of such an energy is therefore regular outside Γ, and try fit the image on the regions Ωi ,
while minimizing the length of Γ. Starting from that energy, Chan & Vese proposed the following
level-set formulation by looking for a minimizer in the class of piecewise constant functions, therefore
canceling the last term:
Z
Z
φ
1 φ
φ
2
CV
2
E (φ, c1 , c2 ) =
|c1 − I| H( ) + |c2 − I| (1 − H( ))dx + µ |∇φ| ζ( )dx
ε
ε
ε ε
Ω
Ω
First note that deriving with respect to ci immediately gives that at optimum,
R
R
φ
φ
Ω I(x)H( ε )dx
Ω I(x)(1 − H( ε ))dx
c1 =
,
c
=
R
R
2
φ
φ
Ω H( ε )
Ω 1 − H( ε )
and the gradient flow associated to this energy can be obtained by:
1 φ
∂t φ = µκ(φ) − (I − c1 )2 + (I − c2 )2 |∇φ| ζ( )
ε ε
with c1 , c2 obtained as above. Typically we take a function 1ε ζ( φε ) > 0 to allow creation of new regions.
10
Some generalizations are the following: first we can seek unknown patterns instead of simple
constants. This amounts to consider functions instead of constants. We define
Z
Z
φ
1 φ
φ
2
CV
2
|u1 (x) − I| H( ) + |u2 (x) − I| (1 − H( ))dx + µ |∇φ| ζ( )dx
E (φ, u1 , u2 ) =
ε
ε
ε ε
Ω
Ω
Z
Z
φ
φ
|∇u2 |2 (1 − H( ))dx.
+ν
|∇u1 |2 H( )dx + ν
ε
ε
Ω
Ω
which gives as Euler-Lagrange equations:
u2 − I = ν∆u2 on φ > 0,
u1 − I = ν∆u1 on φ < 0
with homogeneous Neumann boundary condition on φ = 0 for the ui . The the descent algorithm gives:
1 φ
∂t φ = µκ(φ) − ν − (I − u1 )2 + (I − u2 )2 − ν|∇u2 |2 + ν|∇u1 |2 |∇φ| ζ( )
ε ε
Another extension is to capture more than two regions. Does this need to introduce as much level-set
functions as regions ? Hopefully not, due to the four colors theorem. This states that each partition
of R2 in regions can be colored with 4 colors, so that to contiguous regions are not painted with the
same color. This can be implemented with two level-set functions using the combination of signs of
those two functions. We therefore introduce
Z
φ1
φ2
φ1
φ2
E(φ1 , φ2 , u11 , u12 , u21 , u22 ) =
|u11 (x) − I|2 H( )H( ) + |u12 (x) − I|2 (1 − H( ))H( )
ε
ε
ε
ε
Ω
φ
φ
φ
φ
1
1
2
2
+ |u21 (x) − I|2 (1 − H( ))H( ) + |u22 (x) − I|2 (1 − H( ))(1 − H( ))dx
ε
ε
ε
ε
Z
φ1
φ2
φ1
φ2
φ2
φ1
+ν
|∇u11 |2 H( )H( ) + |∇u12 |2 (1 − H( ))H( ) + |∇u21 |2 (1 − H( ))H( )
ε
ε
ε
ε
ε
ε
Ω
Z
φ
1
φ
1
φ2
φ
2
1
1
+ |∇u22 |2 (1 − H( ))(1 − H( ))dx + µ |∇φ1 | ζ( ) + |∇φ2 | ζ( )dx
ε
ε
ε ε
ε ε
Ω
and we generalize the above descent algorithm.
Exercise 1 Compute the gradient flow associated to that energy and write the active contour algorithm
associated to it.
2.6.2
Tests
The following tests were performed using the CREASEG Matlab software from Olivier Bernard, INSA
Lyon. In figure 6, the Chan-Vese algorithm shows its ability to reach a steady state in the two-objects
noisy test case. In the more complicated three objects noisy case, which is a difficult test case for
contouring method, it perform also well. However, when it comes to contour regions where the image
intensity is not constant, the Chan-Vese algorithm can be fooled (see figure 7 where two regions of
are localized by assuming a mean value, giving as bad segmentation. To localize region with possibly
varying intensity, the extension to patterns presented above is a better choice. This is even more visible
on a nosier image which depicts a spiral. The original Chan-Vese algorithm find two regions which
ignore the natural segmentation, while Li algorithm behaves quite well (figure 8).
2.7
2.7.1
Numerical aspects of level set methods
Reinitialisation or renormalisation of φ ?
Assume a continuous medium of co-dimension one, or the interface between two media is captured by
the zero level set of a function φ that is we advected by a given velocity field. When the problem is
11
FIGURE 6: Region-based contouring of (Chan-Vese, 2001). The two-objects and three-objects cases.
FIGURE 7: Region-based contouring of (Chan-Vese, 2001) / Extension to the pattern matching case
(Li et al).
12
FIGURE 8: Region-based contouring of (Chan-Vese, 2001) / Extension to the pattern matching case
(Li et al).
simply to localize the interface between two media, only the zero level-set of φ for all time is involved,
and the norm of its gradient, provided it does not vanish, does not matter. By contrast when φ is
expected to measure the distance to interface in order to localize a force, we realize that there is no
reason that φ remains a distance function for all time. Besides, it has even been demonstrated in the
case of a co-dimension structure that |∇φ| represents locally the change in length (or area, depending
on dimension) of the surface, so that it is a priori sure that |∇φ| =
6 1 for t > 0. However a distance is
precisely such that the norm of its gradient is 1 wherever it is defined.
Starting from this fact, several workarounds are possible. The first is to redistanciate φ at each
time step. This can be done, for instance by solving the following Hamilton-Jacobi equation:
∂ψ
+ sgn(φ)(|∇ψ| − 1) = 0,
∂τ
ψ(0) = φ,
(9)
to which one looks for a stationary state. In that equation, sgn(φ) is an approximation of the sign of
φ. Practically speaking, due to the hyperbolic nature of the equation, characteristics are emanating
from the zero level-set, so that, in order for φ to recover a unit gradient in a neighborhood of φ = 0.
More precisely, the Hamilton-Jacobi equation can be rephrased as a transport equation:
∂ψ
∇ψ
+ sgn(φ)
· ∇ψ = −sgn(φ).
∂τ
|∇ψ|
(10)
Thus characteristics are emanating from φ = 0 and normal to this interface. One can show that this
equation theoretically does not move the zero-level set of φ, that is {ψ = 0} = {φ = 0} for all τ .
The disadvantage of this approach is first that this additional equation has to be solved, and
second that this numerical computation may induce a small displacement of the interface [38]. It is
also responsible for the volume loss wrongly attributed to the level set method by many authors. The
∇φ
reason is that the equivalent velocity field sgn(φ0 ) |∇φ|
is not divergence free, and this is even more
true while the interface curvature is large. It will therefore locally change the volume and this is likely
to have implications for the conservation of volume within {φ = 0}. A bunch of methods to overcome
this problem have been implemented: add a volume constraint [84, 83], add particles of each side of
the interface [38] or merely raise of lower the level-set function [80, 81] !
13
Another method is to carry φ outside the interface so that we always have |∇φ| = 1. A method,
initially introduced by Osher and his co-authors from an idea Evans Spruck and then taken over by
Gomes and recently by Faugeras Delfour and Zolésio is to determine φ as the solution of
∂t φ(x, t) + (u · ∇φ)(x − φ∇φ(x), t)) = 0
which amounts to the transport equation on the interface, and the solution is always a distance function,
if the original function is. Unfortunately this equation is non local and difficult to solve numerically.
2.7.2
WENO schemes for the transport equation
Le coeur de la méthode présentée ici réside dans une équation de transport d’une fonction (vectorielle) niveau qui va enregistrer les déformations du milieu lors de son déplacement par un champ
de vitesse calculé par ailleurs. Il est donc important de résoudre numériquement cette équation
de manière précise. De nombreux schémas numériques existent pour l’équation de transport : le
schéma amont est le plus classique, mais est assez diffusif. Le schéma de Lax-Wendroff est d’une
part compliqué à écrire en dimension deux et d’autre part se comporte assez mal pour une solution irrégulière. La méthode des caractéristiques souffre du même défaut du fait de l’interpolation
qui l’accompagne. Au milieu des années 80 des schémas non linéaires généralisant le schéma amont
ont été introduits par Harten, Engquist, Osher et Chakravarthy. Ces schémas prennent en compte
la régularité locale de la solution numérique pour déterminer sur quels points les différences finies
doivent être calculées : il s’agit des schémas ENO (Essentially Non Oscillatory). Plus récemment, les
schémas WENO (Weighted Essentially Non Oscillatory), qui consistent à prendre une combinaison optimale des stencils de discrétisation (un polycopié de Shu complet sur ces schémas se trouve à l’adresse
http://citeseer.ist.psu.edu/shu97essentially.html).
To understand their behavior, let us consider the semi-discretized transport equation
φn+1 − φn
+ un φnx = 0.
∆t
(11)
On a node i of a discretization of the spatial domain, this gives:
φn+1
− φni
i
+ uni (φx )ni = 0.
∆t
(12)
The upwind scheme simply amounts to approximate the x derivative of φ at node i by a one-sided
φi −φi−1
φi+1 −φi
finite difference, taking into account the sign of ui . Let (φ−
and (φ+
x )i =
x )i =
∆x
∆x , we write
(omitting n)
(
(φ−
x )i if ui > 0
(φx )i ≈
(13)
(φ+
x )i if ui < 0
since the value taken for ui = 0 does not matter in (12). This scheme is stable under the CourantFriedreichs-Lewy (CFL) condition, but rather diffusive.
∆t <
∆x
max |u|
(14)
The ENO / WENO schemes that we present use values {φi−3 , φi−2 , φi−1 , φi , φi+1 , φi+2 } to find an
+
approximation of (φ−
x )i (resp. {φi−2 , φi−1 , φi , φi+1 , φi+2 , φi+3 } for (φx )i ). Let
v1 =
φi−2 − φi−3
φi−1 − φi−2
φi − φi−1
φi+1 − φi
φi+2 − φi+1
, v2 =
, v3 =
, v4 =
, v5 =
∆x
∆x
∆x
∆x
∆x
After some expansions (cf Shu lecture notes) we can prove that the three quantities defined by
φ1x =
v1 7v2 11v3
−
+
,
3
6
6
φ2x = −
v2 5v3 v4
+
+ ,
6
6
3
14
φ3x =
v3 5v4 v5
+
−
3
6
6
are third order approximation of φ−
x . A ENO scheme of order 3 would choose the best of those
approximations using a criterium minimizing the third order finite differences of φ. In that way, we
obtain a third order scheme. The WENO idea is to find an optimal convex combination of those three
quantities, which reaches fifth order in regions where φ is regular. We therefore choose
1
2
3
(φ−
x )i ≈ ω1 φx + ω2 φx + ω3 φx
where 0 ≤ ωk ≤ 1 are weights such that ω1 + ω2 + ω3 = 1. In regions where φ is regular, the optimal
choice turns out to be ω1 = 0.1, ω2 = 0.6 et ω3 = 0.3, but this choice could become very unsatisfactory
when φ is less regular. In that case an ENO scheme would be better (i.e. one ωk equal to 1 and
canceling the two others). Finally several improvement of this method lead to determine those weights
using regularity indicators of the numerical solution itself. We set
13
(v1 − 2v2 + v3 )2 +
12
13
S2 = (v2 − 2v3 + v4 )2 +
12
13
S3 = (v3 − 2v4 + v5 )2 +
12
S1 =
and then
α1 =
0.1
,
(S1 + ε)2
α2 =
1
(v1 − 4v2 + 3v3 )2
4
1
(v2 − v4 )2
4
1
(3v3 − 4v4 + v5 )2
4
0.6
,
(S2 + ε)2
α3 =
(15)
(16)
(17)
0.3
(S3 + ε)2
where ε > 0 is small (for instance ε = 10−6 max{vk2 } + 10−99 ), and the weights themselves are given
by:
αk
ωk =
, k = 1, 2, 3
α1 + α2 + α3
This choices (several versions exist, corresponding to improvement of this first choice, WENO-Z,
WENO-M, etc) give nearly optimal order of accuracy in regions where φ is regular, and try to reproduce the behavior of an ENO scheme when φ is less regular. These schemes are nonlinear, since the
coefficients depend on the solution itself. Therefore the numerical analysis is still an active research
topic and recent results have been obtain in this direction.
As an example we show on the following figures the simulation of the 1D transport equation for
a distance-like initial condition, that is continuous but with possibly discontinuities on its derivative.
We took in these examples 200 points on the space interval [0, 10], so that ∆x = 0.05, and a CFL of
0.2. The choice of this small CFL is justified by the fact that in coupled problem, one can often not
choose a CFL close to one, and the diffusivity of the advection schemes is increasing when the CFL
number is deceasing.
We use an Euler time stepping scheme, except for the scheme RK3 / space centered (for which
Euler would lead to instability). The WENO scheme is clearly the best, even better that high order
schemes of Fromm and beam-warming, which are built for regular solution. We also remark that the
(W)ENO scheme, as the upwind scheme upon which they are built, do no show oscillations before or
after discontinuities like other schemes do.
15
Schema Amont
Schema de Lax-Wendroff
Approchee
Exacte
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
2
4
6
8
Approchee
Exacte
1
10
0
2
Schema RK3 en temps / Centre en espace
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
2
4
6
8
10
0
2
4
Schema de Fromm
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
4
6
8
10
6
8
Approchee
Exacte
1
0.8
2
10
Schema ENO 2
Approchee
Exacte
0
8
Approchee
Exacte
1
0.8
0
6
Schema Beam-Warming
Approchee
Exacte
1
4
10
0
16
2
4
6
8
10
Schema WENO 5
Approchee
Exacte
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
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