Conformal metrics and true gradient flows for curves

Abstract

We wish to endow the manifold M of smooth curves in Rⁿ with a Riemannian metric that allows us to treat continuous morphs (homotopies) between two curves c₀ and c₁ as trajectories with computable lengths which are independent of the parameterization or representation of the two curves (and the curves making up the morph between them). We may then define the distance between the two curves using the trajectory of minimal length (geodesic) between them, assuming such a minimizing trajectory exists. At first we attempt to utilize the metric structure implied rather unanimously by the past twenty years or so of shape optimization literature in computer vision. This metric arises as the unique metric which validates the common references to a wide variety of contour evolution models in the literature as "gradient flows" to various formulated energy functionals. Surprisingly, this implied metric yields a pathological and useless notion of distance between curves. In this paper, we show how this metric can be minimally modified using conformal factors that depend upon a curve's total arclength. A nice property of these new conformal metrics is that all active contour models that have been called "gradient flows" in the past will constitute true gradient flows with respect to these new metrics under specific time reparameterizations.

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Page 1

Conformal Metrics and True “Gradient Flows” for Curves
Anthony Yezzi
Georgia Institute of Technology
Atlanta, GA
Andrea Mennucci
Scuola Normale Superiore
Pisa, Italy
Abstract
We wish to endow the manifold M of smooth curves in
lRnwith a Riemannian metric that allows us to treat contin-
uous morphs (homotopies) between two curves c0and c1as
trajectories with computable lengths which are independent
of the parameterization or representation of the two curves
(and the curves making up the morph between them). We
may then define the distance between the two curves using
the trajectory of minimal length (geodesic) between them,
assuming such a minimizing trajectory exists. At first we
attempt to utilize the metric structure implied rather unan-
imously by the past twenty years or so of shape optimiza-
tion literature in computer vision. This metric arises as the
unique metric which validates the common references to a
wide variety of contour evolution models in the literature as
“gradient flows” to various formulated energy functionals.
Surprisingly, this implied metric yields a pathological and
useless notion of distance between curves. In this paper, we
show how this metric can be minimally modified using con-
formal factors the depend upon a curve’s total arclength. A
nice property of these new conformal metrics is that all ac-
tive contour models that have been called “gradient flows”
in the past will constitute true gradient flows with respect to
these new metrics under specfic time reparameterizations.
1Introduction
Ever since the introduction of snakes by Kass, Witkin,
and Terzopoulos [7], active contours [1] have played a
prominent role in a variety of image processing and com-
puter vision tasks, most notably segmentation.
reasearch on active contours saw the tansition from param-
eterization dependent models to geometric models indepen-
dent of the parameterization of the evolving curve. Next,
there were many efforts to incorporated region based im-
age information to make the active contour depend upon
global information about the image rather than just the tra-
ditional locally computed edge descriptors. In recent years,
the latest trend in active contour research seems to be that
of incorporating global shape priors into the active contour
Early
paradigm. This has brought up non-trivial questions such
as how to define an “average shape” or how to character-
ize “variations in shape”. All of these questions ultimately
lead to a more basic and fundamental question of how to
measure the distance between two given curves.
In this paper we study geometries on the manifold M of
curves. This manifold contains curves c, which we param-
eterize as c : S1→ lRn(S1is the circle). Given a curve
c, we define the tangent space TcM of M at c including in
it deformations h : S1→ lRn, so that an infinitesimal de-
formation of the curve c in direction h will yield the curve
c(θ) + εh(θ).
We would like to define a Riemannian metric on the
manifold M of curves: this means that, given two defor-
mations h1,h2∈ TcM, we want to define a scalar product
?h1,h2?c, possibly dependent on c. The Riemannian metric
would then entail a distance d(c0,c1) between the curves
in M, defined as the infimum of the length Len(γ) of all
smooth paths γ : [0,1] → M connecting c0to c1. We call
minimal geodesic a path providing the minimum of Len(γ)
in the class of γ with fixed endpoints.
Surprisingly, almost two decades of literature on vari-
ational approaches to active contours and active surfaces
(whether they be for image segmentation, stereo reconstruc-
tion [4], or other computer vision tasks) suggests a consis-
tent metric on the space of curves. However, the rather
unanimous suggestion of this common underlying metric
structure is made implicitly, perhaps even unknowingly in
many cases. Many authors have defined Energy Function-
als on curves (or surfaces) and utilized the Calculus of Vari-
ations to derive curve evolutions to minimize the Energy
Functionals; often referring to these evolutions as Gradient
Flows. For example, the well known Geometric Heat Flow
[6], popular for its smoothing effect on contours, is often
referred as the gradient flow for length, or the more general
Geodesic Active Contour model [2, 10] is described as the
gradient flow for a conformally weighted length based upon
image data.
The reference to these flows as gradient flows implies a
certain Riemannian metric on the space of curves; but this
fact has been largely overlooked. We call this metric H0
1
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henceforth. If one wishes to have a consistent view of the
geometry of the space of curves in both Shape Optimiza-
tion and Shape Analysis, then one should use the H0metric
when computing distances, averages, and morphs between
shapes. To our knowledge, however, this consistently sug-
gested metric has not been used for these purposes in the
shape analysis literature.
In this paper we first introduce the metric H0and im-
mediatly remark that, even more surprisingly, it does not
yield a well defined metric structure, since the associated
distance is identically zero1, [8, 11] making it pointless to
even talk about geodesics between curves (optimal morphs
from one curve to another). We propose instead in a class
of conformal metrics Hφthat fix the above problems while
minimally altering the earlier flows: in fact the new gradi-
ent flows will amount to simple time reparameterizations of
the earlier flows. As such, contour evolutions models that
have thusfar been referred to as gradient flows will consti-
tute true (time-reparameterized) gradient flows with respect
to these conformal metrics. In addition the conformal met-
rics that we propose have some nice numerical and com-
putational properties: distances measured between curves
are defined using only first order derivatives (and therefore
the resulting optimality conditions involve only second or-
der derivatives); as a consequence, flows designed to con-
verge towards these optimality conditions are second order,
thereby allowing the use of Level Set methods [9]: we in-
deed show such an implementation and a numerical results
for an experimental example.
As a final preface, we wish to point out that a vast lit-
erature is starting to arise on shape metrics for the purpose
of shape analysis (see for example [3, 5, 12]) that have lit-
tle or no relation to the conformal metrics we are present-
ing here. It is not our intent to make comparisons between
these metrics in terms of their utility and performance for
shape analysis tasks. They are all quite different from each
other and such a comparison would not only be difficult,
but would detract from the primary point we wish to make
in this paper. Namely, while the only metric used thusfar
in shape optimization is unsuitable for shape analysis, cer-
tain conformal modifications of this metric can solve this
problem while still being consistent with prior shape opti-
mization techniques. This consistency is lost in any other
class of metrics.
2 The Unspoken Curve Metric H0
2.1The manifold of curves
We will denote by M the manifold of smooth curves c in
lRn. The tangent space TcM to M at the curve c will con-
sist of all possible infinitesimal deformations of the curve c
1This striking fact was first described in [8]
along the direction of its unit normal N at each point. We
may represent such elements of TcM as vector fields along
the curve c which are also normal to the curve c. The re-
striction to the normal direction is necessary if we wish to
treat curves geometrically (i.e. without regard to their pa-
rameterization). It is well known in curve evolution theory
that deformations of a curve along its tangent direction at
each point do nothing more than reparameterize the curve.
2.2Trajectories on the manifold of curves
Let C : [0,1] × [0,1] → lRn, denote a homotopy (con-
tinuous morph) C(u,v) between two boundary curves c0
and c1where u parameterizes each individual curve in the
homotopy and v parameterizes the homotopy itself.2Such
a “curve of curves” represents a trajectory on the manifold
M between the points c0and c1. In what follows, we will
let T(u,v) = Cu/?Cu? denote the unit tangent vector at a
point C(u,v) along a particular curve in the homotopy, and
we will let L(v) denote the total arclength of a particular
curve in the homotopy.
2.3 Specifying a Riemannian metric
To obtain a Riemannian metric on M, we must define a
scalar product ?h1,h2? for all h1,h2∈ TcM at each point
c in M. We initially propose the scalar product
?
where s denotes the arclength parameter of the curve c.
From now on, when we speak of the H0metric, we will
be implying this last definition.
Once we have specified an scalar product, we are able to
calculate the length of trajectories by integrating the speed
of the trajectory (the norm of its derivative according to the
specified scalar product). For a trajectory C(u,v) of curves
and for the H0metric, we obtain the following expression
for its length (noting that ds=?Cu?du).
?1
Related to the length is also the energy E of the homotopy.
?1
Unfortunately, it has been noted in [8, 11] that the metric
H0does not define a distance between curves, since
?h1,h2?H0 =
c
h1(s) · h2(s) ds
(1)
Len(C).=
0
??1
0
??Cv− (Cv· T)T??2?Cu?du dv
E(C).=
0
?1
0
??Cv− (Cv· T)T??2?Cu?dudv
(2)
inf E(C) = 0.
2i.e. C(u,0) = c0(u) and C(u,1) = c1(u)
2
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In other words, for any ε > 0 it is possible to find a homo-
topy (morph) C between any two simple curves c0and c1
such that E(C) < ε.
2.4The implicit omnipresence of H0
There is a good reason to focus our attention on the prop-
erties of this metric (1) for curves. Namely, this is precisely
the metric that is implicitly assumed in formulating gradi-
ent flows of contour based energy functionals in the vast lit-
erature on shape optimization. Consider for example the
well known geometric heat flow (ct = css or ct = κN)
in which a curve evolves along its inward normal N with
speed equal to its signed curvature κ. This flow is widely
considered to be the gradient descent of the Euclidean ar-
clength functional [6]. Its smoothing properties have led to
its widespread use within the fields of computer vision and
image processing. The only sense, however, in which this
is a true gradient flow is with respect to the H0metric as
we see in the following calculation (where L(t) denotes the
time varying arclength of an evolving curve c(u,t) parame-
terized by the time-independent parameter u ∈ [0,1]).
?
L?(t) =
0
|cu|
= −
0
= −
H0
L(t) =
c
ds =
?1
0
|cu|du
?1
cut· cu
?1
?
du =
?1
0
ctu· csdu
?
ct· csudu = −
?
c
ct· cssds
ct,css
If we were to change the metric then the inner-product
shown above would no longer correspond to the inner-
product associated to the metric. As a consequence, the
above well-known and well-loved curvature flow could no
longer be considered the gradient flow for arclength with
respect to the changed metric!
Similar calculations will show that the geodesic active
contour [2, 10] flow [ct= φκN − (∇φ · N)N] is the gra-
dient descent of the geometric energy?
in this respect. To our knowledge, all other variational ac-
tive contour flows that have been derived as gradient flows
of various other constructed energies functionals suffer the
same problem. They are gradient flows only with respect to
the H0metric!
cφds only for the
H0metric. However, this model does not stand out alone
2.5Pathologies of H0
As we have already pointed out, the infimum of the en-
ergy of all possible homotopies between two curves c0and
c1is zero, and therefore, unfortunately, no minimizing ho-
motopy (geodesic) exists. On the other hand the pathologies
of the H0metric are revealed in an interesting and instruc-
tive manner if we attempt to derive in blissful ignorance a
flow that drives any initial homotopy C between two curves
c0and c1to a homotopy of minimal energy, and therefore
minimal length, according to the H0metric (in other words,
a trajectory shorting flow in the space of curves).
2.5.1Geometric parameters s and v∗
We have denoted by u ∈ [0,1] the parameter which traces
outeachcurveinaparameterizedhomotopyC(u,v)andwe
have denoted by v ∈ [0,1] the parameter which moves us
from curve to curve along the homotopy. Note that both of
these parameters are arbitrary and not related to the geom-
etry of the curves comprising the homotopy. We now wish
to construct more geometric parameters for the homotopy
which will yield a more meaningful and intuitive expression
for the minimizing flow we are about to derive. The most
natural substitute for the curve parameter u is the arclength
parmeter s. We must also address the parameter v, however.
While v as a parameter ranging from 0 to 1 seems to have
little to do with the arbritrary choice of the curve parame-
ter u, the differential operator
prior choice. The desired effect of differentiating along the
homotopy is mixed with the undesired effect of differentiat-
ing along the contour if flowing along corresponding values
of u between curves in the homotopy requires some motion
along the tangent direction. To see the dependence of
u, note that C(u,v) andˆC(u,v) where
∂
∂vdepends heavily upon this
∂
∂von
ˆC(u,v) = C
?
u(1+v),v
?
constitute the same homotopy geometrically, yet∂C
We will therefore introduce the more geometric parame-
ter v∗whose corresponding differential operator
the most efficient transport from one curve to another curve
along the homotopy regardless of “correspondence” be-
tween values of the curve parameters. It is clear that such
a transport must always move in the normal direction to the
underlying curve since tangential motion along any curve
does not contribute to movement along the homotopy. More
preceisely, we define the parameteres s and v∗in terms of u
and v as follows.
∂v?=∂ˆ C
∂v.
∂
∂v∗yields
∂
∂s=
1
?Cu?
∂
∂u
and
∂
∂v∗
=
∂
∂v−?Cv· Cs
?∂
∂s
2.5.2The unstable H0minimizing flow
Suppose we now consider a time varying family of homo-
topies C(u,v,t) : [0,1] × [0,1] × (0,∞) → lRnand com-
pute the derivative of the H0energy along this family. Note
3
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that the H0energy, in terms of the new parameters s and v∗
may be simply expressed as follows
?1
After some tedious calculations, one may show that the
time-derivative of E may be written as
?1
?
In the planar case, Cv∗and Cssare linearly dependent (as
both are orthogonal to Cs) which means that
E(t) =
0
?L
0
??Cv∗
??2dsdv
(3)
E?(t) = −2
Cv∗v∗− (Cv∗v∗· Cs)Cs− (Cv∗· Css)Cv∗+1
0
?L
0
Ct·
2?Cv∗?2Css
?
dsdv
(Cv∗· Css)Cv∗= (Cv∗· Cv∗)Css= ?Cv∗?2Css
and therefore
E?(t) =
?1
by which we derive the minimization flow
−2
0
?L
0
Ct·
??Cv∗v∗− (Cv∗v∗· Cs)Cs
?−1
2?Cv∗?2Css
?
dsdv
Ct= Cv∗v∗− (Cv∗v∗· Cs)Cs−1
which is geometrically equivalent to the following more
simple flow (by adding a tangential component):
2?Cv∗?2Css
Ct= Cv∗v∗−1
2?Cv∗?2Css
(4)
Note that the flow (4) consists of two orthogonal diffu-
sion terms. The first term Cv∗v∗is stable as it represents
a forward diffusion along the homotopy, while the second
term −?Cv∗?2Cssis an unstable backward diffusion term
along each curve.
3 Conformal Versions of H0
Given the pathologies of H0we have no choice but to
propose a new metric if we wish to construct a useful Rie-
mannian geometry on the space of curves. However, we
may seek a new metric whose gradient structure is as sim-
ilar as possible to that of the H0metric. In particular, for
any functional E : M → lR we may ask that the gradi-
ent flow of E with respect to our new metric be related to
the gradient flow of E with respect to H0by only a time
reparameterization. In other words, if c(t) represents a gra-
dient flow trajectory according to H0and if ˆ c(t) represents
the gradient flow trajectory according to our proposed new
metric, then we want
ˆ c(t) = c(f(t))
for some positive time reparameterization f : lR → lR,
˙f > 0. The resulting gradient flows will then be related as
follows.
ˆ ct=˙f(t)ct
The only class of new metrics that will satisfy (5) are
conformal modifications of the original H0metric, which
we will denote by H0
by combining the original H0metric with a positive confor-
mal factor φ : M → lR where φ(c) > 0 may depend upon
the curve c. The relationship between the inner products is
given as follows.
?
Note that for any energy functional E of curves c(t) we
have the following equivalent expressions, where the first
and last expressions are by definition of the gradient and
the middle expression comes from the definition (6) of a
conformal metric.
?
(5)
φ. Such metrics are completely defined
h1,h2
?
H0
φ
= φ(c)
?
h1,h2
?
H0
(6)
d
dtE(c(t)) =
∂c
∂t, ∇φE(c)
? ?? ?
?
H0
Conformal
Gradient
?
H0
φ
=
(7)
= φ
?
∂c
∂t, ∇φE(c)
? ?? ?
Conformal
Gradient
=
?
∂c
∂t, ∇E(c)
? ?? ?
Original
Gradient
?
H0
We see from (7) that the conformal gradient differs only in
magnitude from the original H0gradient
∇φE =1
φ∇E
and therefore the conformal gradient flow differs only in
speed from the H0gradient flow.
∂c
∂t= −∇φE(c) = −
1
φ(c)∇E(c)
As such and as we desired, the solution differs only by a
time reparameterization f given by
˙f =
1
φ(c)
The obvious question now is how to choose the confor-
mal factor. A first requirement is that the distance d(c1,c2)
induced by the new conformal metric should be positive for
curves c0,c1that are different:
Proposition 13Suppose there exists an a > 0 such that
φ(c) ≥ aL(c),
?
L = length(c)
?
(8)
3We thank Prof. Mumford for suggesting this result.
4
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for all curves c. Consider a homotopy C connecting two
curves c0= C(·,0),c1= C(·,1), and its H0
?1
Up to reparameterization, |Cu| = L so we can rewrite the
energy (using the relation (4.10) in [11]) as
?1
??1
The last term is the square of the area swept by the homo-
topy. So if c0 ?= c1, and there do not exist a homotopy
connecting c0,c1with zero area, then d(c1,c2) > 0.
We will therefore look for a conformal factor that satisfy
(8). Moreover we may hope to counteract the unstable ele-
ment (backward curvature flow) in the H0minimizing flow,
by choosing a conformal factor dependent upon arclength L
which will thereby add a forward curvature flow term.
φ–energy
0
φ(C(·,v))
?1
0
|Cv∗|2|Cu|dudv
0
φ
L
?1
0
|Cv× Cu|2dudv ≥ a
?1
?1
0
?1
0
|Cv× Cu|2dudv
≥ a
00
|Cv× Cu|dudv
?2
.
3.1The conformal minimizing flow
We now define the conformal H0
C(u,v) between curves c0and c1(when the conformal fac-
tor φ is a function of the arclength L of each curve) as
?1
After another set of tedious calculations, we may write the
derivative of Eφfor a time-varying family of homotopies
C(u,v,t) as
?1
−2φ(Cv∗· Css)Cv∗+ (φm + φ?M)Css
where
φenergy of a homotopy
Eφ=
0
φ(L)
?L
0
??Cv∗
??2dsdv
(9)
E?
φ(t) = −
0
?L
0
Ct·
?
2φ?Lv∗Cv∗+ 2φCv∗v∗− 2φ(Cv∗v∗· Cs)Cs
?
?L
dsdv
m = ?Cv∗?2
As before, we now consider the planar case in which
Cv∗and Cssare linearly dependent and therefore (Cv∗·
Css)Cv∗= mCss, yielding
?1
+φ?Lv∗Cv∗+1
from which we obtain the following minimizing flow (after
adding a tangential term).
Ct= φCv∗v∗+ φ?Lv∗Cv∗+1
and
M =
0
mds =
?L
0
?Cv∗?2ds.
E?(t) = −2
0
?L
2(φ?M − φm)Css
0
Ct·
?
φ?Cv∗v∗− (Cv∗v∗· Cs)Cs
?
?
dsdv
2(φ?M − φm)Css
(10)
3.2A stabilizing conformal factor
To stabilize the flow (10), we look for a φ such that
φ?M − φm ≥ 0
for all (s,v∗)
(11)
or (assuming M ?= 0)
φ?
φ= (logφ)?≥m
M
for all (s,v∗)
(12)
One way to satisfy this is to choose
(logφ)?= max
s,v∗
m
M
.= λ
(13)
giving us
φ = eλL
(14)
yielding the following flow of homotopies
Ct= eλL?
Note that the choice φ = eλLsatisfies (8), and then in-
duces a non-degenerate distance of curves.
2Cv∗v∗+ 2λLv∗Cv∗+ (λM − m)Css
?
(15)
3.3Level set implementation
We note that the minimizing flow (10) consists of two
stable diffusion terms and a transport term.
we have the option to utilize level set methods in the
implementation of (10).We represent the evolving ho-
motopy C(u,v,t) as an evolving surface S(u,v,t) =
(C(u,v),v,t). We then perform a Level Set Embedding [9]
of this surface into a 4D scalar function ψ such that
ψ?C(u,v,t),v,t?= 0.
The goal is now to determine an evolution for ψ which
yields the evoluton (10) for the level sets of each of its 2D
cross-sections. Differentiating
?
where ∇ψ = (ψx,ψy) denotes the 2D spatial gradient of
each 2D cross-section of ψ, and substituting (10), noting
that N = ∇ψ/?∇ψ?, yields the corresponding Level Set
Evolution.
2ψv
?∇ψ?2(∇ψv·∇ψ)+
−1
2
0
+λLvψv
As such,
d
dt
ψ?x(u,v,t),y(u,v,t),v,t?=0
?
−→ ψt+∇ψ·Ct= 0
ψt=ψvv−
ψ2
v
?∇ψ?4
?
?∇2ψ∇ψ?·∇ψ
∇ ·
?∇ψ?
?
ψ2
v
?∇ψ?2− λ
?L
ψ2
v
?∇ψ?2ds
?
∇ψ
?
?∇ψ?
(16)
Note that for simplicity we have dropped the factor eλL
from (10) since we are guaranteed that this factor is always
positive. As a result, we do not change the steady-state of
the flow by omitting this factor.
5
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4Experimental Results
In this section we show experimental results of using the
time evolution PDE (16) to compute the geodesic homo-
topy between two rather different closed curves c0and c1.
The two boundary curves c0and c1are displayed above the
caption in figure 1, while the geodesic homotopy computed
between these two curves is displayed below (in a left-to-
right then top-to-bottom visualization). In figure 2 we show
the homotopy surface S represented by the zero level-set of
the function ψ(x,y,v,t) after running the evolution (16) to
steady state. Note that the curves visualized in figure 1 we
obtained by taking cross-sections of this homotopy surface
at evenly spaced values of v. The initial condition used by
the evolution equation (16) was a simple linear interpola-
tion of the signed distance transforms of the two boundary
curves c0and c1. We chose λ to satisfy (13) at time t = 0
and found that this stabilized the flow until convergence.
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Figure 1.
motopy.
Visualization of the geodesic ho-
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Proceedings of the Tenth IEEE International Conference on Computer Vision (ICCV’05)
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Page 7

Figure 2. Computed homotopy surface (two
views).
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Proceedings of the Tenth IEEE International Conference on Computer Vision (ICCV’05)
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Authorized licensed use limited to: Georgia Institute of Technology. Downloaded on October 6, 2008 at 20:10 from IEEE Xplore. Restrictions apply.