Computing Geodesic Paths on Manifolds

Abstract

The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. In this paper we extend the Fast Marching Method to triangulated domains with the same computational complexity. As an application, we provide an optimal time algorithm for computing the geodesic distances and thereby extracting shortest paths on triangulated manifolds.

Full text

Proc. Natl. Acad. Sci. USA
Vol. 95, pp. 8431–8435, July 1998
Applied Mathematics
Computing geodesic paths on manifolds
R. Kimmel* and J. A. Sethian
†
Department of Mathematics and Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720
Communicated by Alexandre J. Chorin, University of California, Berkeley, CA, May 4, 1998 (received for review March 20, 1998)
ABSTRACT The Fast Marching Method is a numerical
algorithm for solving the Eikonal equation on a rectangular
orthogonal mesh in O(M log M) steps, where M is the to-
tal number of grid points. In this paper we extend the Fast
Marching Method to triangulated domains with the same
computational complexity. As an application, we provide
an optimal time algorithm for computing the geodesic dis-
tances and thereby extracting shortest paths on triangulated
manifolds.
1. Introduction
Sethian’s Fast Marching Method (1), is a numerical algorithm
for solving the Eikonal equation on a rectangular orthogonal
mesh in OM log M steps, where M is the total number of
grid points in the domain. The technique hinges on producing
numerically consistent approximations to the operators in the
Eikonal equation that select the correct viscosity solution; this
is done through the use of upwind finite difference operators.
The structure of this upwinding is then used to systematically
construct the solution to the Eikonal equation through an op-
timal ordering of points during the update. The optimal order-
ing is executed using a heap operator to extract the next point
in the update sweep. As such, the technique is a reminiscent
of Dijkstra’s method (2); however, the resulting approxima-
tion is consistent in that it produces the correct shortest path
on an orthogonal grid. For details about Fast Marching Meth-
ods, see refs. 1 and 3.
Barth and Sethian (4) have recently constructed opera-
tors for viscosity solutions to both the Eikonal equation and
Hamilton–Jacobi equations on arbitrary triangulated domains.
These operators exploit the upwind nature to construct the
correct entropy-satisfying weak solutions. In this paper, we
extend these ideas and build a Fast Marching Method for tri-
angulated domains. As an application, we provide an optimal
algorithm for computing the geodesic distances and thereby
extracting shortest paths on triangulated manifolds.
The outline of this paper is as follows. First, we review the
Fast Marching Method for orthogonal grids. Then, for mo-
tivational reasons, we analyze the structure of this method
on a triangulated planar grid constructed directly from an or-
thogonal grid. We then follow with a general procedure for
computing the solution of the Eikonal equation on arbitrary
acute triangulated domains, followed by an extension to gen-
eral (nonacute) triangulations. As an application, we com-
pute geodesic distances and minimal geodesic paths on mani-
folds.
2. The Fast Marching Method on Orthogonal Grids
Here, we briefly review the Fast Marching Method for com-
puting the solution to the Eikonal equation; for details, see
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ref. 1. The goal is to solve the equation
∇T =Fx; y : [1]
The key idea is to build an approximate to the gradient term
that correctly deals with the development of corners and cusps
in the solution. It is well known that the above Eikonal equa-
tion becomes nondifferentiable, and an appropriate weak so-
lution must be built. The appropriate weak solution comes
from satisfying the entropy condition; see, for example, ref. 5.
The crucial point in this (or any such appropriate) numerical
scheme is the correct direction of the upwinding and treat-
ment of sonic points. For details and an extensive review, see
ref. 3. The Fast Marching Method exploits this idea by care-
fully considering the nature of upwind, entropy–satisfying ap-
proximations to the Eikonal equation.
In more detail, consider one particular upwind approxima-
tion to the gradient, given by (see ref. 6)
"
maxD
−x
ij
T; −D
+x
ij
T; 0
2
+
maxD
−y
ij
T; −D
+y
ij
T; 0
2
#
1/2
= F
ij
: [2]
The central idea behind Fast Marching Methods is to sys-
tematically advance the front in an upwind fashion to produce
the solution T . The key is the observation that the upwind dif-
ference structure of Eq. 2 means that information propagates
“one way,” that is, from smaller values of T to larger values.
Hence, the algorithm rests on “solving” Eq. 2 by building the
solution outward from the smallest T value. The algorithm
is made fast by confining the “building zone” to a narrow
band around the front, motivated by the narrow band intro-
duced by Chopp (7), used in recovering shapes from images in
Malladi, Sethian, and Vemuri (8), and analyzed extensively by
Adalsteinsson and Sethian (9). The idea is to sweep the front
ahead in an upwind fashion by considering a set of points in a
narrow band around the existing front, and to march this nar-
row band forward, freezing the values of existing points and
bringing new ones into the narrow band structure. The key
is in the selection of which grid point in the narrow band to
update.
The Fast Marching Method algorithm is as follows: First,
we tag points in the initial conditions as Alive. We then tag as
Close all points one grid point away. Finally, we tag as Far all
other grid points. Then the loop is as follows:
1. Begin loop: Let Trial be the point in Close with the
smallest T value.
2. Add the point Trial to Alive; remove it from Close.
3. Tag as Close all neighbors of Trial that are not Alive;if
the neighbor is in Far, remove it from that list and add
it to the set Close.
4. Recompute the values of T at all neighbors according
to Eq. 2 by solving the quadratic equation, using only
values of points that are Alive.
5. Return to top of loop.
This algorithm works because the process of recomputing
the T values at upwind neighboring points cannot yield a value
*e-mail: ron@math.lbl.gov.
†
To whom reprint requests should be addressed. e-mail: sethian@math.
berkeley.edu. Also, see http://math.berkeley.edu/
~sethian/level
_
set.html.
8431

8432 Applied Mathematics: Kimmel and Sethian Proc. Natl. Acad. Sci. USA 95 (1998)
Fig. 1. Upwind construction of accepted values.
smaller than any of the accepted (Alive) points. This is known
as a “monotone property” and means that we can march the
solution outward, always selecting the narrow band grid point
with minimum trial value for T , and readjusting neighbors,
without having to “go back and correct an accepted value”
(see Fig. 1). Another way to look at this is that each mini-
mum trial value begins an application of Huyghen’s principle,
and the expanding wave front touches and updates all others.
The speed of the algorithm comes from a heapsort technique
to efficiently locate the smallest element in the set Close.If
there are M total points in the computational domain, then,
assuming no work for locating the smallest element in the set
Close, we have a computational complexity of OM. Because
the heap can be re-ordered in Olog M steps, this yields a
computational complexity of OM log M. For more details,
see refs. 1 and 3.
The above technique is strongly reminiscent of Dijkstra’s
method, which allows one to compute minimum costs of paths
on networks. That technique, which is also OM log M), is nu-
merically inconsistent; given two points on the graph, it pro-
duces the network minimum (Manhattan) length, which may
not be optimal. The Fast Marching Method is a consistent
approximation to the continuous partial differential gradient
operator, and thus it provides “sub-grid” resolution and hence
finds the true shortest path.
Before proceeding, it is instructive to say a few words
about the actual update procedure involved in “solving”
Eq. 2. Imagine a uniform square grid and suppose that the
goal is to update the value of T at the center point i; j.
We label the values of T at the surrounding grid points
T
A
= T
i−1;j
, T
B
= T
i+1;j
, T
C
= T
i;j−1
, and T
D
= T
i;j+1
(see Fig.
2). Some of the values may be infinite, corresponding to Far
values.
Standing at the center point i; j, we attempt to solve the
quadratic given by each possibility. For example, we refer to
possible contributors A and C and assume, without loss of
generality, that T
A
 T
C
. We attempt to solve
T − T
A

2
+T − T
C

2
= h
2
F
2
i;j
;
where h is the uniform grid spacing. Two possibilities are as
follows:
• There is a real solution T , with T , T
A
and T , T
C
,to
the quadratic
T − T
A

2
+T − T
C

2
= h
2
F
2
i;j
:
Fig. 2. Construction of upwind solution on orthogonal mesh.
• There is a real solution T , with T , T
A
and T  T
C
,to
the one-dimensional update (degenerate quadratic)
T − T
A

2
= h
2
F
2
i;j
:
For each possible up–down/left–right pair, we construct all
possible real solutions; we then accept as the updated point
the one that produces the smallest value of T . Judicious pro-
gramming and careful attention to Far values makes the above
search extremely fast. This is the Fast Marching Method de-
scribed in refs. 1 and 3.
3. Fast Marching on a Particular Triangulated
Planar Domain
Our goal now is to extend this method to triangular domains.
To do so, we shall build a monotone update procedure on
the triangulated mesh. As motivation, in this section we con-
sider the obvious triangulation of a square grid in the plane.
Imagine the triangulation given in Fig. 3.
If we consider the possible contributors T
A
and T
C
, and look
at the triangle formed by the points i; j, i −1;j and i; j −1, we
can easily write down the equation of the plane determined
by the known values T
A
and T
C
, as well as the unknown value
T , namely

T − T
A
h

x +

T − T
C
h

y +T = z:
Computing the gradient, we then want to select a value of T
such that

T − T
A
h

2
+

T − T
C
h

2
= F
2
ij
:
In other words, we are tilting the plane by a value T at the
center point i; j to have a gradient magnitude equal to F.
We note that this is the exact same construction as the one
produced by the “orthogonal” construction. Note also that in
this case the gradient vector, with origin at the center point,
always points into the triangle from which it is updated. This is
not necessarily the case for an arbitrary triangle. To establish
monotonicity, we will need to verify this condition.
The inability to solve the above quadratic corresponds to an
inability to tilt at an appropriate angle, and the requirement
that the solution T be greater than the contributors means
that the solution is always constructed in an upwind manner.
By using the same update rules as above, and the heap struc-
ture to maintain a list of grid points of Close points, this pro-
vides a method for executing the Fast Marching Method on
such a simple triangulation.
4. Fast Marching Methods on Triangulated Domains
Our goal now is to extend this idea to an arbitrary triangu-
lation. Here, we are essentially following the construction of
upwind approximations to the gradient on triangulated meshes
developed by Barth and Sethian in (4).
Fig. 3. Simple triangulation for building a monotone update op-
erator.

Applied Mathematics: Kimmel and Sethian Proc. Natl. Acad. Sci. USA 95 (1998) 8433
4.1. A Construction for Acute Triangulations. We start
with an acute triangulation and consider the triangulation
around the grid point given in Fig. 4.
A large number of triangles may share the center vertex.
Our procedure, motivated by the simple triangulation in the
previous section, is to compute a possible value for T from
each triangle that includes the center point as a vertex. Be-
cause several triangles can produce admissible values for T,
we must select an appropriate value. There are several possi-
bilities. In our examples we have chosen the one that produces
the smallest new value for T ; this will correspond to an algo-
rithm similar to the one taken above. More elaborate upwind
constructions on triangulated meshes are given in ref. 4.
We now construct a simple update procedure for one non-
obtuse triangle ABC in which the point to update is C.We
should verify that the update is from within the triangle, i.e.,
the altitude h should be inside the nonobtuse triangle ABC.
See Fig. 5.
This means that we search for t = EC such that
t − u
h
= F:
Denote a = BC and b = AC; we have by similarity that
t/b = DF/AD = u/AD, thus CD = b − AD = b − bu/t =
bt − u/u. Next, by the Law of Cosines: BD
2
= a
2
+ CD
2
−
2aCDcos θ; and by the Law of Sines: sin φ =
CD
BD
sin θ: Now,
by using the right angle triangle CBG, we have
h =a sin φ =a
CD
BD
sin θ =
aCD sin θ
p
a
2
+CD
2
−2aCDcos θ
: [3]
We end up with the quadratic equation for t:
a
2
+ b
2
− 2ab cos θt
2
+ 2bua cos θ −bt
+ b
2
u
2
− F
2
a
2
sin
2
θ=0: [4]
The solution t must satisfy u + t, and should be updated
from within the triangle, namely:
a cos θ +
bt − u
t
+
a
cos θ
: [5]
Thus, the update procedure is given as follows:
If u + t and a cos θ +
bt − u
t
+
a
cos θ
;
then T C=minT C;t+ TAy
else T C=minT C;bF+ TA;cF + TB:
It is easy to verify that this equation is a finite difference
approximation to the Eikonal equation. It is monotone by con-
struction, consistent, converges to the viscosity solution, and
can be used to extend the Fast Marching Method to acute
triangulated domains.
Fig. 4. Acute triangulation around center grid point.
Fig. 5. Given the triangle ABC, such that u = TB−T A, find
T C=T A+t such that t −u/h = F.(Left) A perspective view of
the triangle stencil supporting the T  values that form a tilted plane
with a gradient magnitude equal to F.(Right) The trigonometry on
the plane defined by the triangle stencil.
4.2. Extension to General Triangulations. Finally, we note
that we have required an acute triangulation. This is so that
any front entering the side of a triangle has two points to
provide values before the third is computed. In other words,
for monotonicity, we restrict the update to come from within
the triangle, i.e., the gradient of the solution at a grid point
should point into the triangle from which it is updated. The
most straightforward way to enforce this requirement is to
build a nonobtuse triangulation, thus making sure that the
grid captures all incoming fronts.
In the case of an obtuse triangulation, the support may only
include a limited (acute) section of incoming wavefronts. One
approach to handle nonacute triangulations, which we now
describe, is to locally build numerical support at obtuse angles
by splitting these angles in a special way. An obtuse angle
at vertex A can be updated by its neighboring points in a
consistent way only at a limited section of upcoming fronts.
Connecting the vertex to any point in this section splits the
obtuse angle into two acute ones. The idea is to extend this
section by recursively unfolding the adjacent triangle(s), until
a new vertex B is included in the extended section. Then, the
vertices are connected by a virtual directional edge from B to
A (i.e., A may be updated by B). The length of the edge AB
is equal to the distance between A and B on the unfolded
triangles plane (Fig. 6).
We can perform a complexity analysis to check on the cost
of this virtual node capturing. Let h
max
;h
min
be the maximal
and minimal altitudes, respectively, i.e., triangles altitudes with
maximal and minimal length. Let θ
max
be the maximal obtuse
angle, and denote α = π −θ
max
the angle of the extended sec-
tion, let θ
min
be the minimal (acute) angle for all triangles, let
e
max
be the length of the longest edge, and let l be the length
Fig. 6. (Left) The initialization of the construction for the splitting
section. (Bottom) A triangulated surface patch. (Right) The unfolded
patch and the splitting section expansion up to the first vertex B, and
the virtual edge connecting the two vertices AB.

8434 Applied Mathematics: Kimmel and Sethian Proc. Natl. Acad. Sci. USA 95 (1998)
Fig. 7. The smallest possible area of one triangle covered by the
longest possible extended section bounds the number of unfolded tri-
angles. The triangle with a
min
is the bright shaded one.
of the virtual directional edge (AB in the above example).
Furthermore, assume that α; θ
min
are small enough angles so
that sin α 8 tan α 8 α.
Then, the angular width of the narrower section is α = π −
θ
max
, so that we have sin
α
2

e
max
2l
: This relation, for small α
angles, yields l 
e
max
α
: Denote by l
max
= e
max
/α.
The maximal area of the extended sections is bounded from
above by a
max
=
e
2
max
2α
; and the minimal area of an unfolded tri-
angle is bounded from below by a
min
=
h
min
α
2
θ
min
2
(see Fig. 7).
Therefore, the number of triangles that are needed to be un-
folded before a vertex is found in the extended section is
bounded by the ratio of these areas
m =
a
max
a
min
=
e
2
max
θ
min
h
2
min
α
3
: [6]
The accuracy of the first order scheme for acute triangles is
of Oh
max
 8 Oe
max
. The accuracy for the obtuse case with
the above construction becomes Ol
max
=Oe
max
/π−θ
max
,
as expected. In the worst case scenario, the scheme accuracy
depends on the largest edge and the widest angle.
The complexity of this virtual construction does not change
substantially change the operation count of the underlying al-
gorithm. The construction of the virtual directional edges in-
cludes unfolding triangles for each obtuse angle until a ver-
tex is detected in the extended splitting section. Because the
number of unfolded triangles is bounded by a constant, the
construction of the virtual directional edges takes OM, and
running time complexity is still optimal OM log M.
5. Construction of Minimal Geodesics
We can use this algorithm to compute distances on triangu-
lated manifolds, and hence construct minimal geodesics. First,
we solve the Eikonal equation with speed F = 1 on the trian-
gulated surface to compute the distance from a source point;
we then backtrack along the gradient of the arrival time field
by solving the ordinary differential equation
dXs
ds
=−∇T;
where Xs traces out the geodesic path. We use second order
Huen’s integration method on the triangulated surface with a
switch to a first order scheme at sonic points in the gradient.
When integrating within a given triangle, its three neighbor-
ing triangles support the computation and are used to inter-
polate T as a second order polynomial whose six coefficients
are computed from those given T values at the vertices.
Fig. 8 presents a perspective view of the triangulation and
shortest paths for two surfaces. Fig. 8a shows shortest paths
on regular triangulation of the surface given by the function
zx; y=0:45 sin2πxsin2πy  on 0; 1 3 0; 1, for grid
size of 50 3 50. The minimal geodesics are painted on the tri-
angulated surface and projected to the xy plane. Fig. 8b gives
Fig. 8. Computing minimal weighted geodesics on triangulated
surfaces.
a polyhedron example, in which a different speed function F
was assigned to each side, causing a Snell Law effect along the
edges. The speed F is 2 at the close side with the start point,
1 at the second top side, and 4 at the side of the destination
point.
Finally, we show two additional computations to illustrate
the algorithm performance on manifolds with an underlying
nonacute (obtuse) triangulation. Fig. 9 shows the computa-
tion of minimal geodesics on a torus genus one object, in
which some shortest paths in fact cut through the middle.
The equidistance curves are shown, as well as the shortest
paths. In Fig. 10, we compute geodesic distances on a synthetic
head.
We thank T. Barth, L. C. Evans, and A. Vladimirsky for helpful
discussions. All calculations were performed at the University of Cal-
ifornia at Berkeley and the Lawrence Berkeley Laboratory. This work
was supported in part by the Applied Mathematical Science subpro-
gram of the Office of Energy Research, U.S. Department of Energy,
under Contract DE-AC03-76SF00098, and the Office of Naval Re-
search under under Grant FDN00014-96-1-0381.

Applied Mathematics: Kimmel and Sethian Proc. Natl. Acad. Sci. USA 95 (1998) 8435
Fig. 9. Shortest paths on a bead (a genus one two-dimensional
manifold).
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Geometry, Fluid Mechanics, Computer Vision and Material Sci-
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4. Barth, T. & Sethian, J. A. (1998) J. Comp. Phys., in press.
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Fig. 10. Shortest paths on synthetic head.
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7. Chopp, D. L. (1993) J. Comp. Phys. 106, 77–91.
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