Computer-Aided Design 39 (2007) 1081–1090
www.elsevier.com/locate/cad
Efficiently determining a locally exact shortest path on polyhedral surfaces
Shi-Qing Xin, Guo-Jin Wang ∗
Department of Mathematics and State Key Laboratory of CAD & CG, Zhejiang University, Hangzhou 310027, PR China
Received 17 September 2005; accepted 3 August 2007
Abstract
In this paper, we present an efficient visibility-based algorithm for determining a locally exact shortest path (LESP) from a source point to a
destination point on a (triangulated) polyhedral surface. Our algorithm, of a finitely-iterative scheme, evolves an initial approximately shortest path
into a LESP. During each iteration, we first compute the exact shortest path restricted on the current face sequence according to Fermat’s principle
which affirms that light always follows the shortest optical path, and then optimize the face sequence where the path is not locally shortest on the
polyhedral surface. Since the series of paths we obtained are monotonic decreasing in length, the algorithm gives a LESP which is shorter than
the initial path, at conclusion.
For comparison, we use various methods to provide an initial path. One of the methods is Dijkstra’s algorithm, and the others are the Fast
Marching Method (FMM) and its improved version. Our intention for improvement is to overcome the limitation of acute triangulations in the
original version. To achieve this goal, we classify all the edges into seven types according to different wavefront propagation manners, and
dynamically determine the type of each edge for controlling the subsequent wavefront expansion. Furthermore, we give two approaches for
backtracing the approximately shortest paths directed at the improved FMM. One exploits the known propagation manners of the edges as well as
the Euler’s method. This is another contribution in this paper.
c 2007 Elsevier Ltd. All rights reserved.
Keywords: Computational geometry; Shortest paths; Discrete geodesics; Fast marching method
1. Introduction
The problem of computing the shortest path between two
points s and t on a polyhedral surface S arises naturally
in a wide range of application areas such as robotics,
geographic information systems and navigation [1,2]. Besides
its importance in computational geometry, it is also central
to many computer graphics problems. For example, cutting
a mesh into several charts [3,31] and establishing a surface
distance metric [30,4] often involve computation of shortest
paths; see [5] for a survey.
To solve the exact globally shortest path problem, Sharir and
Schorr [6] first presented an O(n 3 log n) algorithm that runs on
a convex polytope, where n is the complexity of the surface.
The algorithm was subsequently improved by Mount [20]
to O(n 2 log n). For an arbitrary polyhedral surface, Mitchell,
Mount and Papadimitriou [7] (MMP) gave an O(n 2 log n)
algorithm which inherits the paradigm of Dijkstra’s algorithm
∗ Corresponding author. Tel.: +86 571 87951609x8306.
E-mail address: [email protected] (G.-J. Wang).
c 2007 Elsevier Ltd. All rights reserved.
0010-4485/$ - see front matter doi:10.1016/j.cad.2007.08.001
[38]. Later, Chen and Han [8] (CH) provided an O(n 2 )
algorithm based on a key observation of “one angle one split”.
(Also, Kapoor [9] announced a further improvement in 1999.)
Recently, Xin and Wang [10] (XW) proposed two techniques
for improving CH algorithm, and the new algorithm has a much
higher performance than previous algorithms in time and space.
To make a trade-off between computation time and path
accuracy, researchers are also interested in computing an
approximately shortest path at most (1 + )π(s, t), where is a
given tolerance and π(s, t) denotes the exact globally shortest
surface distance between the source point s and the destination
point t. Approximation algorithms for the general 3D shortest
path problem were first studied by Papadimitriou [21], Choi
et al. [22] and Clarkson [23]. For the shortest path problem on
a polyhedral surface, many known algorithms [12,14–16,11,19,
13,1,17,18,5,28] have been proposed.
In this paper, we study how to efficiently determine a
locally exact shortest path (LESP) between two points on a
(triangulated) polyhedral surface. Our motivation is to find an
almost globally shortest path that is also locally shortest in
the exact sense. In fact, the local minima (maxima) makes
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sense in many problems such as gravitational lensing [32].
Just as many known approximation algorithms [13,25,35], we
adopt a two-step scheme for the LESP problem, computing
an approximately shortest path joining the source s and the
destination t followed by evolving the initial path iteratively to
a LESP.
Generally speaking, an accurate initial path more possibly
gives a LESP short enough. Therefore, we use various methods
to provide initial paths for comparison. One of the initialization
methods is Dijkstra’s algorithm [38]. The others are based
on the Fast Marching Method (FMM) [16,2,29,37,36,33,34],
which is a reminiscent of Dijkstra’s method and optimal in
the sense that the computational complexity is O(n log n). This
paper presents a variant of the FMM to overcome the limitation
of acute triangulations in the original version [2], which is
described in detail in Section 3. Our technique is to classify
all the edges into seven types for simulating the wavefront
expansion and update the wavefront around the active vertex
in a wing-like fashion. In addition, we give an equivalent
upwind construction of the FMM on triangulated domains. The
overall process can be described as follows, and the detailed
description can be seen in Algorithm 3.
– While there still exist pending vertices,
. Take out the pending vertex v with the smallest distance
and mark it with “fixed”;
. Determine the propagation type of the edges vi v incident to
v in a wing-like fashion, where vi is already a fixed vertex,
as Fig. 6 shows;
. Update the distances of the two pending vertices which are
incident to v and next to the fixed vertices.
Then Section 4 proposes a finitely-iterative algorithm for
evolving an initial path to a LESP. Our algorithm proceeds like
this:
– While the current path Γ is not a LESP,
/∗ Suppose Γ lies on a face sequence F. ∗ /
. Optimize the face sequence F where the path is not locally
shortest;
. Compute the exact shortest path restricted on the new face
sequence.
In fact, determining the exact shortest path lying on a face
sequence is equivalent to the shortest path problem inside a
simple planar polygon. Although many known algorithms [24]
with low complexity have been presented, they are complicated
and impractical for this specific problem. Therefore we devise a
visibility-based method motivated by Fermat’s principle which
alleges that light always follows the shortest optical path.
Suppose the boundary of the face sequence can “absorb” light
and we place a luminary at the source position. With the
marching of the light, a part thereof will be absorbed by the
boundary. When the light is completely absorbed, we place
another luminary at the vertex that blocks the light. Repeat the
process until the destination is visible. Then the light path to
the destination t, actually which is a 3D polyline, can be proved
to be the shortest path restricted on the current face sequence.
Since the face sequences are reasonably “straight”, the intuitive
algorithm is natural and efficient.
Fig. 1. Definitions of the face sequence F and the edge sequence E. The thick
polyline denotes the shortest path from s to t restricted on F , while the lightgray polygon denotes the boundary of F .
We have made numerous tests on the improved FMM in
Section 3 and the visibility-based algorithm in Section 4, and
provide some comparative results by experiments in Section 5.
The main observations are as follows: (1) the improved FMM
performs better than the original version in [2]; (2) the
visibility-based algorithm possesses faster convergence rate;
and (3) combining the two algorithms together, we obtain an
efficient method for solving the LESP.
2. Preliminaries
Let S be a (triangulated) polyhedral surface in R3 , defined
by a set of faces, edges and vertices. Assume that the surface
S has n faces and s, t are two points on the surface. Our task
is to compute a LESP restricted on S between s and t. We first
borrow some terminologies from [7].
A face sequence F is defined by a list of adjacent faces
f 1 , f 2 , . . . , f m+1 such that f i and f i+1 share a common edge
ei . Then we call the list of edges E = (e1 , e2 , . . . , em ) an edge
sequence. Actually, the face sequence F and the edge sequence
E mean the same, as Fig. 1 shows. Without specification, the
face sequences mentioned later are all simple, i.e., the faces are
different from each other in a face sequence.
In studying the shortest path problem, a technique called
planar unfolding is often used. An edge sequence E =
(e1 , e2 , . . . , em ) is unfolded in this way: rotate f 1 around e1
until its plane coincides with that of f 2 ( f 1 and f 2 are kept
on different sides of e1 ), rotate f 1 and f 2 around e2 until their
plane coincides with that of f 3 , and repeat the process until all
the faces f 1 , f 2 , . . . , f m lie in the plane of f m+1 .
Obviously, a shortest path is, without doubt, locally shortest.
More properties can be seen in [6,20,8]. For example, two
shortest paths from the same source cannot intersect each other
unless at the source point or the destination point or a saddle
vertex. For description of the general form of a shortest path,
we cite a lemma from [7].
Lemma 1. The general form of a locally shortest path is a
path which goes through an alternating sequence of vertices
and edge sequences such that the unfolded image of the path
along any edge sequence is a straight line segment and both the
angles of the path passing through a vertex is greater than or
equal to π . The general form of a shortest path is the same as
that of a locally shortest path, except that no edge can appear
in more than one edge sequence and each edge sequence must
be simple.
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Now consider the shortest path problem restricted on a
simple face sequence F that connects s and t. We can come
to an immediate conclusion from [26,27] as follows.
Lemma 2. There exists a unique locally shortest path from
the source point s to the destination point t on a simple face
sequence F.
3. Improving the FMM
The FMM on triangulated domains was introduced by
Kimmel and Sethian [16] in 1998. Its explanation in physics
is to simulate an advancing wavefront from a source point and
update the distance values of the vertices where the wavefront
arrives. Mathematically, the FMM is a computational technique
for approximating the solution to nonlinear Eikonal equations
of the form |∇T | = 1, where T denotes the distance field
restricted on the surface.
Fig. 2. The distance value d(v) of the vertex v is determined by v1 ’s and v2 ’s
distance values d(v1 ) and d(v2 ), using the upwind construction. There exists
a point paccess , called the access point, on the straight line v1 v2 such that the
dotted line segment vpaccess coincides with the gradient of the distance field
on the face 4vv1 v2 . Note that the condition paccess ∈ v1 v2 is required so that
both the vertices v1 , v2 can provide values for the vertex v.
3.1. The original FMM
Here we give a brief description of the process of the
FMM on triangulated domains. Technically, all vertices can
be classified into three groups: Fixed vertices, whose distance
values are known and fixed; Close vertices, whose distance
values have been calculated, but have not been fixed yet; and
Far vertices that have not been updated and lie ahead of the
wavefront. Initially, we tag the source vertex s as Close and all
other vertices as Far. Then the algorithm proceeds as follows:
– While the destination vertex t has not been fixed,
. Extract the vertex v in Close with the minimum distance
value;
. Tag v as Fixed and its Far neighbors as Close.
. Recompute the distance values of v’s Close neighbors in
an upwind finite-difference scheme as Fig. 2 shows.
The above algorithm computes the approximate shortest
distance from the source s to the destination t at the conclusion.
However, determining the corresponding path relies on the
solution to the differential equation
dX (µ)
= −∇T,
dµ
where X (µ) = 0 represents the geodesic path with respect to
the arc parameter µ. As the triangulation is refined, the resulting
path converges to the exact shortest path [2].
As is shown in Fig. 2, the vertices v1 , v2 can provide a
validatory distance value for the vertex v if the condition
paccess ∈ v1 v2 holds. However, it is not always like
this. So Kimmel and Sethian [16] suggested that non-acute
triangulations should be handled by splitting the obtuse angles
in a special way as shown in Fig. 3. However, their idea does
not work if the wavefront entering the vertex A comes across
the edge D E. Therefore, we next introduce a new approach
for overcoming the limitation of acute triangulations. The main
difference between our approach and the original version lies
in the handling after we retrieve the Close vertex with the
minimum distance value; see Algorithm 3 for details.
(a) Case 1: the front comes
across the edge BG, giving
the shortest distance to the
vertex A.
(b) Case 2: the front comes
across the edge D E, giving
the shortest distance to the
vertex A.
Fig. 3. A non-acute triangulation with 6 B AC > π/2. Kimmel and
Sethian [16] suggested connecting the virtual edge AG in the unfolded
triangulated patch, where 6 C AH1 = 6 B AH2 = π/2 and the vertex G is
located in the shadow region 6 H1 AH2 so that 6 B AG and 6 C AG are acute.
However, this does not work if the wavefront entering the vertex A comes across
the edge D E.
3.2. Various propagation manners
Since the FMM can be regarded as a simulation of an
advancing wavefront [2], we should consider the propagation
manner that the wavefront arrives at vertices or edges. In fact,
a vertex may receive fronts from adjacent vertices or opposite
edges, while an edge may receive fronts from adjacent edges or
opposite vertices. In detail, we can classify all the vertices into
two types, From Vertex and From Edge, for distinguishing how
a vertex v gets its minimum distance value d(v), as is shown
in Fig. 4. Likewise, we classify all the edges into seven types
according to the manner of the incoming wavefront, as Fig. 5
shows.
Technically, the two manners From Vertex and From Edge
can be interpreted by Algorithm 1 and Algorithm 2. Note that as
opposed to the original FMM (Fig. 2), Algorithm 2 (Fig. 4(b))
gives another explanation of the upwind construction of the
FMM on the triangulated domains.
Algorithm 1. If the vertex v 0 has been tagged as Fixed and its
adjacent vertex v is still in Close, and the inequality d(v) >
d(v 0 ) + kvv 0 k holds, then
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respectively d(v1 ) and d(v2 ), and v is on the upper side
of the edge v1 v2 .
(ii) Compute the coordinates (x, y) of v. If x < x1 or x > x2 ,
exit. /∗ x1 ≤ x ≤ x2 implies that the wavefront entering
the vertex v comes across the edge v1 v2 .∗ /
(iii) If y is less than the existing distance value d(v) of v, we
compute the coordinates (x, y) of the access point p with
(a) From Vertex: the shortest path
to the vertex v goes via its adjacent
vertex v 0 .
(b) From Edge: the shortest path to
the vertex v goes across the edge
v1 v2 ; p is the access point.
Fig. 4. The distance value of the vertex v may be computed in two ways,
From Vertex or From Edge, where the thick arrows indicate the advancing
wavefront.
(a) From Left Part (resp.,
From Right Part): the edge v1 v3
(resp., v3 v2 ) receives wavefront
coming from the left (resp., right)
part of edge v1 v2 .
(c) From Both Edges: the edge
v2 v3 receives wavefront coming
from both the edges v2 v1 and
v1 v3 .
(b) From Left Edge (resp.,
From Right Edge): the edge v1 v3
(resp., v3 v2 ) receives wavefront
coming from the entire edge v1 v2 .
(d) From Pseudo Source: any
point p ∈ v2 v3 receives wavefront
from the pseudo-source vertex v1 .
(e) From End To End: the
wavefront propagates from one
endpoint to the other.
Fig. 5. Seven manners of wavefront propagation to the edge e. The thick arrows
mean the wavefront. The fronts entering the point p2 come from the point p1
in (a), (b) and (c), and from the vertex v1 in (d) and (e).
(i) d(v) := d(v 0 ) + kvv 0 k; /∗ Fig. 4(a).∗ /
(ii) Set a From Vertex flag to the vertex v.
Algorithm 2. If a Close vertex v is opposite to the edge v1 v2
whose endpoints are both in Fixed, then the distance value d(v)
of v can be updated as follows. /∗ Fig. 4(b).∗ /
(i) Build an XY coordinate system in the plane of ∆v1 v2 v,
such that the y-components of the points v1 and v2 are
x=x
and
(x − x1 )d(v2 ) + (x2 − x)d(v1 )
.
x2 − x1
Otherwise, exit. /∗ p is the access point.∗ /
(iv) If d(v) > max(d(v1 ), d(v2 ), min(y, d(v))), then /∗ max(·)
is used to guarantee that the vertices are processed in
monotonic increasing order of their distance values.∗ /
(a) d(v) := max(d(v1 ), d(v2 ), min(y, d(v)));
(b) Set a From Edge flag to the vertex v.
y=
But when the wavefront advances towards a (directed) edge
e, things become a bit complicated. (Without specification, the
edges discussed in this paper are all directed.) The wavefront
entering the edge e may come from a pseudo-source, or an edge,
or a part of an edge, or both edges; see Fig. 5 for totally seven
manners. Accordingly, we can label the edge e with one of the
seven flags as follows (see Fig. 5):
– From Left Part, From Right Part (see Fig. 5(a)): the edge
v1 v3 receives wavefront coming from the left or right part
of the edge v1 v2 .
– From Left Edge, From Right Edge (see Fig. 5(b)): the edge
v1 v3 or v3 v2 receives wavefront coming from the entire edge
v1 v2 .
– From Both Edges (see Fig. 5(c)): the edge v2 v3 receives
wavefront coming from both the edges v2 v1 and v1 v3 .
– From Pseudo Source (see Fig. 5(d)): the edge v2 v3 receives
wavefront coming from the single source vertex v1 .
– From End To End (see Fig. 5(e)): the wavefront cannot go
across either the edge vv1 or the edge v1 v, but rather passes
through the undirected edge vv1 from one endpoint to the
other.
These flags are set for controlling the subsequent wavefront
propagation manners. Note that the propagation process in
the exact shortest path problem is far more complicated.
Technically speaking, the FMM simplifies the propagation
process greatly based on the assumption that the fronts are
locally parallel to each other. Our intention is to extend the
FMM by seven propagation manners, instead of one manner.
The discussions here are prepared for Algorithm 3 in the next
subsection.
3.3. Wing-like update
As mentioned above, the overall scheme of our improved
FMM is just the same as the original version, except that we
keep track of the propagation manners and adopt a wing-like
update technique during each iteration. When we take out the
Close vertex v with the minimum distance, we first check how
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3.4. Backtracing paths
At the conclusion of Algorithm 3, it provides a distance
value for each vertex. However, the corresponding paths are still
unknown. This subsection gives two methods for backtracing
the paths. One method utilizes the known propagation manners
of the edges, and the other is Euler’s method.
(a) The Fixed vertex v 0 gives the
distance value of the vertex v.
(b) The edge vl vr gives the
shortest distance value of the
vertex v.
Fig. 6. Wing-like update process: the wavefront propagates ahead from two
sides. The vertices marked with a “•” mark are in Fixed, while the vertices
marked with a “M” mark are in Close. The wavefront advances forward from
two sides until it meets v’s Close neighbors.
3.4.1. Exploiting the known propagation manners
To backtrace the paths using the known propagation manners
of the edges, we make an assumption that the shortest
path divides the successive wavefront segments by the same
proportion. Take Fig. 5(a) for example, we treat p1 ∈ v1 v2
as the access point of p2 ∈ v1 v3 if kv1 p2 k/k p2 v3 k =
kv1 p1 k/k p1 paccess k. It is the same with Fig. 5(c) if
kv2 p2 k/k p2 v3 k = kv2 p1 k/(k p1 v1 k + kv1 v3 k),
p 1 ∈ v2 v1
or
Fig. 7. The table exhibits how we label the edge v2 v3 .
the vertex v gets its distance value for the last time, and then
determine the propagation type of each edge incident to the
vertex v in a wing-like order. Fig. 6 gives an illustration of the
wing-like update process, and the detailed pseudo-code can be
seen in Algorithm 3.
Algorithm 3. Initially, tag the source vertex s as Close with
d(s) := 0, while tag any other vertex v as Far with d(v) :=
+∞. Assign a From End To End label to all the edges. Then
each iteration proceeds as follows.
(i) Extract the Close vertex v with the minimum distance
value d(v). Label v with a Fixed flag, and label its Far
neighbors with a Close flag.
(ii) Update the distance values of all of v’s Close neighbors
with Algorithm 1. If v is the source vertex, stop this
iteration.
(iii) If v carries the From Vertex flag, then
/∗ Suppose vertex v 0 gives d(v); see Fig. 6(a).∗ /
Tag the directed edges vv 0 and v 0 v as From End To End.
/∗ This prevents the approximately shortest paths from
intersecting the undirected edge v 0 v.∗ /
Else
/∗ Suppose edge vl vr gives d(v); see Fig. 6(b).∗ /
(a) Keep the access point of v.
(b) Tag the directed edges vl v and vvr respectively as
From Left Part and From Right Part.
(iv) Label the edges opposite to v in the wing-like order,
following Figs. 6 and 7, until we find v’s Close neighbors.
Then we get two directed edges el and er such that the
edges next to them have a Close endpoint. /∗ Fig. 6(a) and
Fig. 6(b).∗ /
(v) Update the distance values of the two vertices opposite to
the edges el and er exploiting Algorithm 2.
In fact, the above algorithm is motivated by the continuous
propagation of fronts, and therefore is pretty natural. It performs
well even for non-acute triangulations.
kv2 p2 k/k p2 v3 k = (kv2 v1 k + kv1 p1 k)/k p1 v3 k,
p 1 ∈ v1 v3 .
The assumption induces a simple backtracing algorithm.
We introduce a bool variable fIsVertex to indicate whether
the current point p is a vertex or an inner point of an edge, and
use an integer variable of index to denote the corresponding
index of the vertex or the edge. Then the backtracing algorithm
can be described as follows.
Algorithm 4. Initially, set fIsVertex := TRUE and p := t,
where t is the destination vertex. Assign index with the index
of t. Then continue the following iterations.
(i) Keep the position of the current point p.
(ii) If fIsVertex is TRUE
/∗ p is a vertex, say, v.∗ /
If the vertex v coincides with the source s, then
Return the resulting path and stop.
Else if d(v) is given by another vertex v 0 , then
(i) Let p := v 0 .
(ii) Assign the index of v 0 to the variable index.
Else /∗ An edge e gives d(v).∗ /
(i) Assign the index of e to index.
(ii) Set fIsVertex to be FALSE.
(iii) Update p by the access point of v.
Else /∗ p is located on an edge e.∗ /
Update the variables fIsVertex, p and index according to
Fig. 5.
On the whole, the backtracing method here is an inverse
procedure of wavefront expansion. Since the propagation
manners of the edges are already known, it demands only a little
computation. In addition, it can be proved that the resulting
paths from different points to the source vertex do not intersect
each other except at vertices at the conclusion of Algorithm 4.
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(a) Rotating I around the edge
v1 v2 until I is located in the face
of 4v1 v2 v3 .
(b) Building a planar coordinate
system in the face of 4v1 v2 v3 .
(a) Initialization: the luminary is
placed at the source vertex, and
hence the edge e is illuminated.
(b) λ ≤ a: the next edge e receives
all the light from the source image
I0.
(c) a < λ < b: the next edge e
receives part of the light from the
source image I 0 .
(d) λ ≥ b: the next edge e receives
no light from the source image I 0 .
Fig. 8. Unfolding a face sequence.
3.4.2. Euler’s method
Euler’s method for backtracing the paths is to numerically
solve the differential equation dX (µ)/dµ = −∇T . Suppose
the planar coordinate system has been built in the face 4v1 v2 v3 .
Since Euler’s method has order 1, we assume T = ax + by + c
in 4v1 v2 v3 . Then ∇T = (a, b) can be determined by solving
the equation array

T1 = ax1 + by1 + c
T2 = ax2 + by2 + c

T3 = ax3 + by3 + c,
where (xi , yi ) are the planar coordinates of the three vertices,
as Fig. 2 shows. Along the gradient (a, b), we can compute
the access point of any point on the boundary of 4v1 v2 v3 .
Similarly, we can prove that the resulting paths from different
points to the source vertex do not intersect each other except at
vertices.
4. Evolution to a LESP
Given an initial path between the source s and the destination
t, we adopt a two-step scheme to compute a LESP between
them: (1) determining the shortest path Γ restricted on the
current face sequence F and (2) updating F if Γ is not a LESP.
In detail, the first subsection introduces the process of unfolding
a face sequence, and the other subsections propose a method for
computing a LESP.
4.1. Unfolding a face sequence
Suppose F is a face sequence connecting the source vertex
s and the destination vertex t. During the planar unfolding of
F, we will obtain a sequence of images of the source. Take
Fig. 8(a) for example, we now rotate the source image I until I
is located in the face of 4v1 v2 v3 , where 4v1 v2 v3 is a triangle
among F. Assume I 0 is the image of I by rotating around the
edge v1 v2 . Our intention is to denote the images I and I 0 by a
pair.
Let l− and r− be respectively the distances between the
image I and the vertices v1 , v2 . We first build a planar
coordinate system on the plane of 4v1 v2 v3 to accommodate
the new source image I 0 , as shown in Fig. 8(b). Without loss of
generality, we assume v1 is located at (0, 0) and v2 is located
at (b, 0), where b is the length of the edge v1 v2 . (v3 and I 0
should be kept on different sides of the X -axis.) Then the planar
Fig. 9. Suppose the straight line vI 0 divides the edge e by proportion λ. With
the variation of λ, the next edge e0 may receive all the light, part of the light, or
no light from the image I 0 .
coordinates of I 0 (x− , y− ) and v3 (x+ , y+ ) can be represented as
follows:
q
2 )/b + b
(l 2 − r−
2 − x2 ,
,
y− = − l−
x− = −
−
2
and
q
2 − r 2 )/b + b
(l+
+
2 − x2 ,
,
y+ = l+
x+ =
+
2
with which we can compute the length of kI 0 v3k. Thus the new
source image by rotating I 0 around the edge v1 v3 or v3 v2 can
also be determined. Repeat the process until we compute all the
source images.
4.2. The visibility-based algorithm
This subsection presents a visibility-based algorithm for
solving the LESP problem, according to Fermat’s principle
which states that light always follows the shortest optical path.
As Fig. 9 shows, the overall procedure is like this: (1) place
a luminary at the source s and let the light march forward as
much as possible; (2) if the light is completely “absorbed” by
the boundary, then place another luminary at the vertex which
blocks the light; and (3) repeat the process until the destination
t is visible. In Fig. 9, the variables a, b ∈ [0, 1] are used to
denote the two endpoints of a projection segment on an edge,
and vl and vr are used to designate the two vertices that block
the light. Then the visibility-based algorithm can be written as
follows.
Algorithm 5. Initially, the image I is located at the source
vertex s, as Fig. 9(a) shows. Let e denote the edge opposite
to I , and vl and vr denote the two endpoints of e. Set a := 0
and b := 1. Then the algorithm proceeds as follows.
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(a) A luminary is placed at s
initially.
(b) Light marches forward as
much as possible.
(c) Place another luminary at the
vertex that completely blocks the
light.
(d) The destination point is visible
at the conclusion of Algorithm 5.
Fig. 10. The general execution procedure of Algorithm 5.
/∗ Suppose e0 is the edge next to e. Then e0 and e determine a
face f and have a common vertex. Let v be the endpoint of e0
different from the common vertex. Without loss of generality,
assume v is the left endpoint of e0 as Fig. 9(b) shows. Build a
locally planar coordinate system in the plane of f .∗ /
(i) Compute the dividing proportion λ on the edge e0 by the
straight line v I .
(ii) If λ ≤ a/∗ Fig. 9(b).∗ /
(a) Compute the new dividing proportions a, b on the edge
e0 .
(b) Compute the new source image I.
Else if a < λ < b/∗ Fig. 9(c).∗ /
(a) Let vl := v and a := 0.
(b) Compute the new right weight b and the new source
image I.
Else /∗ Fig. 9(d).∗ /
Let I := vr , a := 0 and b := 1.
(iii) Repeat the process until the destination t is visible.
Fig. 10 illustrates a general execution procedure of
Algorithm 5. At the conclusion of the algorithm, we compute
the LESP restricted on the current face sequence. Its correctness
is given by the following theorem.
Theorem 1. Algorithm 5 computes the unique shortest path on
the concerning face sequence.
Proof. At the conclusion of Algorithm 5, luminaries are placed
at certain vertices. Suppose v is the first one except the source
vertex s; see Fig. 11. Since v blocks the light from the source
s at one time, there must exit an edge e such that a part of e
receives light from s while the next edge e0 is entirely invisible
to e. We extend the line segment sv until it intersects the
boundary at a point, say, v 0 . Then we have the line segment vv 0
cutting the domain of the concerning face sequence into two
parts. Thus any path Γ from the source s to the destination t
intersects the line segment vv 0 at one or more points. Let p be
one of the intersection points between Γ and the line segment
S
c can be decomposed into sp
b
pbt.
vv 0 , and then the path spt
b passes through the vertex v, Γ goes via
Since the subpath sp
v definitely. By analogy, we can conclude that all “luminary”
vertices are definitely located on the shortest path restricted
on the face sequence. Together with the observation that the
portion of the path between successive “luminary” vertices is a
straight line segment, we prove the theorem. Fig. 11. Algorithm 5 gives the unique shortest path on the concerning face
sequence.
Fig. 12. The path is locally shortest at vertex v if and only if 6θil ≥ π and
6θir ≥ π .
The best time complexity of Algorithm 5 is O(k), while
the worst time complexity is O(k 2 ), where k is the number
of the edges contained in the current edge sequence. In
our experiments (see Section 5), Algorithm 5 runs in time
O(ck) (c ≤ 2), because most of the edge sequences we face
with it are reasonably “straight”.
4.3. Update of the face sequence
Algorithm 5 computes the shortest path Γ restricted on
a given face sequence F connecting the source s and the
destination t. However, Γ is not necessarily a LESP on the
polyhedral surface. In fact, it is possibly not locally shortest at
vertices where luminaries are placed, and hence we need check
if the path Γ is a LESP every time after Algorithm 5 runs.
In Fig. 12, a path Γ passes through the vertex v. Since Γ
cuts the neighborhood of v into two parts, the angles incident to
v are divided into two groups in accordance, {θil } and {θir }. We
can affirm that Γ is locally shortest at v if and only if 6θil ≥
π and 6θir ≥ π; see Lemma 1. Therefore, we need check
any “luminary” vertex v as follows. If one of the inequalities
6θil ≥ π and 6θir ≥ π does not hold, then remove the edges
incident to v from the concerning edge sequence F, and insert
into F the remaining edges incident to v. (Note that the edges
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S.-Q. Xin, G.-J. Wang / Computer-Aided Design 39 (2007) 1081–1090
(a) Exact algorithms; paths.
(b) Exact algorithms; isolines.
(c) DJ; paths.
(d) DJ; isolines.
(e) FMM; paths.
(f) FMM; isolines.
(g) FMM1; paths.
(h) FMM1; isolines.
(i) FMM2; paths.
(j) FMM2; isolines.
Fig. 13. Testing various initialization methods on the bunny model of 5110 faces. Figures (a), (c), (e), (g) and (i) show the (approximately) shortest paths, while
figures (b), (d), (f), (h) and (j) show the corresponding isolines.
should be arranged in order.) After all the “luminary” vertices
are checked, we obtain a new face sequence F 0 . If F and F 0
are the same, Γ is a LESP on the surface. Otherwise, repeat
such a process. Thus we obtain a sequence of face sequences
F1 , F2 , . . . , and corresponding to it, a sequence of shortest
paths Γ1 , Γ2 , . . . . It can be proved that the lengths of Γi are
strictly monotonic decreasing with regard to i. Taking into
account that there are finitely many face sequences connecting
the source s and the destination t, we conclude that a LESP
between s and t can be computed after finite iterations.
5. Experimental results
In this section, we provide some experimental results of
the initialization algorithms (see Section 3) and the visibilitybased algorithm (see Section 4). Our experiments are made on a
computer with 3.00 GHz Pentium(R) 4 CPU and 1 GB memory.
5.1. Initialization algorithms
This subsection compares various algorithms on the
shortest path problem. Representative exact algorithms include
(1) MMP algorithm [7], (2) CH algorithm [8] and (3) XW
algorithm [10]. The approximation algorithms discussed here
include (1) Dijkstra’s algorithm [38], (2) the FMM [2],
(3) FMM1: computing the distance values using Algorithm 3
and backtracing paths exploiting Algorithm 4; and (4) FMM2:
computing the distance values using Algorithm 3 and
backtracing paths exploiting Euler’s method. Fig. 13 illustrates
a group of experimental results on a bunny model of 5100
faces, and Table 1 lists the experimental data on running
time (Time), memory requirements (Space) and length error
(Error), where Error means the relative error of the length
sum of the approximately shortest paths to all the vertices
to the length sum of the exact globally shortest paths to
all the vertices. From the experimental results, we can
see that FMM2 has a desirable overall performance and
therefore is more appropriate for providing an initial path,
while the FMM is more time-consuming than FMM1 and
FMM2. The inefficiency of the original FMM lies in that
it requires connecting the virtual edges in the preprocessing
step.
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S.-Q. Xin, G.-J. Wang / Computer-Aided Design 39 (2007) 1081–1090
(a) The bunny model of a
non-acute triangulation.
(b) The exact globally shortest
path between s and t.
(c) Initialized by DJ.
(e) Initialized by FMM1.
(f) Initialized by FMM2.
(d) Initialized by FMM.
Fig. 14. Testing various initialization algorithms on the 5k-face bunny model of a non-acute triangulation: the exact globally shortest path between the points s and
t is drawn in a dashed curve; the initial paths are shown in thick curves; and the resulting LESPs are illustrated in thin curves.
Table 1
Testing various 3D shortest path algorithms on the bunny model of 5110 faces
MMP
[7]
CH
[8]
Time
(ms)
Space
(MB)
797
Error (%)
XW
[10]
DJ
[38]
FMM
[2]
FMM1 FMM2
237,593 219
3
46
8
5.42
2.35
0.23
0.04
0.2
0.12
0.12
–
–
–
6.36
2.38
3.12
1.73
16
5.2. Testing the visibility-based algorithm
In this subsection, we test the visibility-based algorithm
by using several initialization algorithms including DJ [38],
FMM [2], FMM1 and FMM2 on the bunny model of a nonacute triangulation. Fig. 14 illustrates the experimental results
of computing the (approximately) shortest path between the
two given points s and t by the visibility-based algorithm. The
globally shortest path, the initial paths and the resulting LESPs
are respectively drawn in dashed curves, thick curves and thin
curves.
We list the experimental data in Table 2, where the notations
have the following meaning:
– “Iteration”: the total number of iterations required;
– “Repetition”: the average visit times of an edge during an
iteration;
– “Error1”: the relative length error of the initial path to the
exact shortest path;
– “Error2”: the relative length error of the resulting LESP to
the exact shortest path.
Numerous experiments show that the visibility-based
algorithm has a high performance. Averagely speaking, the
Table 2
Testing the visibility-based algorithm with various initialization methods on the
bunny model of a non-acute triangulation
Initialized by →
DJ
FMM
FMM1
FMM2
Iteration
Repetition
Error1 (%)
Error2 (%)
4
1.55
5.39
0.007
5
1.49
1.13
0.007
5
1.52
2.67
0.007
4
1.48
0.48
0.007
initial paths given by DJ, FMM, FMM1 and FMM2 respectively
have a relative error of 5.92%, 1.09%, 1.86% and 0.54%
in length, and the corresponding LESPs respectively have
a relative error of 0.097%, 0.031%, 0.059% and 0.017%.
Furthermore, tests on a 2000 K-triangle model illustrate that we
can compute a LESP between any two points in 0.2 s using the
visibility-based algorithm initialized by the FMM2. Based on
the experimental results, we have the following observations.
– Different initial paths may give different LESPs. Wholly
speaking, the FMM2 is a desirable initialization method for
it runs efficiently and the resulting approximately shortest
path is mostly close enough to the exact globally shortest
path.
– When used to solve the LESP problem, the visibility-based
algorithm has a linear-time complexity experimentally since
“Repetition” is always between 1 and 2. What accounts for
the high performance of the visibility-based algorithm is that
the edge sequences involved in the LESP problem are mostly
rather “straight”.
– Provided that the initial path is accurate enough, the
visibility-based algorithm can determine the exact globally
shortest path.
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S.-Q. Xin, G.-J. Wang / Computer-Aided Design 39 (2007) 1081–1090
6. Conclusions
In this paper, we improve the original FMM by classifying
all the edges into seven types according to different wavefront
propagation manners, and dynamically determining the type of
each edge for controlling the subsequent wavefront expansion.
Furthermore, we give two approaches for backtracing the
approximately shortest paths. One exploits the known
propagation manners of the edges as well as the Euler’s method.
The second contribution of the paper is a visibility-based
algorithm that is specially devised for the locally exact shortest
path problem on a mesh. It is based on Fermat’s principle
which affirms that light follows the shortest optical path, and
therefore is natural and intuitive. The visibility-based algorithm
has a high performance because the edge sequences in the LESP
problem are reasonably “straight”. Furthermore, provided that
the initial path is accurate enough, the LESP we obtain is also
globally shortest.
Acknowledgements
This work was supported by the National Basic Research
Program of China (No. 2004CB719400), the NNSF of China
(No. 60673031, No. 60333010) and the NNSF for IRG (No.
60021201).
References
[1] Agarwal PK, Har-Peled S, Karia M. Computing approximate shortest
paths on convex polytopes. In: Symposium on computational geometry.
2000. p. 270–9.
[2] Sethian JA. Fast marching methods. SIAM Review 1999;41(2):199–235.
[3] Sander PV, Wood ZJ, Gortler SJ, Snyder J, Hoppe H. Multi-chart
geometry images. In: Proceedings of the 2003 eurographics/ACM
SIGGRAPH symposium on geometry processing. 2003. p. 146–55.
[4] Peyré G, Cohen L. Geodesic computations for fast and accurate surface
remeshing and parameterization. Progress in Nonlinear Differential
Equations and Their Applications 2005;63:157–71.
[5] Surazhsky V, Surazhsky T, Kirsanov D, Gortler SJ, Hoppe H. Fast exact
and approximate geodesics on meshes. ACM Transactions on Graphics
2005;24(3):553–60.
[6] Sharir M, Schorr A. On shortest paths in polyhedral spaces. SIAM Journal
on Computing 1986;15(1):193–215.
[7] Mitchell JSB, Mount DM, Papadimitriou CH. The discrete geodesic
problem. SIAM Journal on Computing 1987;16(4):647–68.
[8] Chen J, Han Y. Shortest paths on a polyhedron. In: Proceedings of the
sixth annual symposium on computational geometry. 1990. p. 360–9.
[9] Kapoor S. Efficient computation of geodesic shortest paths. In:
Proceedings of the thirty-first annual ACM symposium on theory of
computing. 1999. p. 770–9.
[10] Xin S, Wang G. Improving Chen & Han’s algorithm on the discrete
geodesic problem [submitted for publication].
[11] Har-Peled S. Approximate shortest paths and geodesic diameter on
a convex polytope in three dimensions. Discrete & Computational
Geometry 1999;21(2):217–31.
[12] Hershberger J, Suri S. Practical methods for approximating shortest paths
on a convex polytope in R3. In: Proceedings of the sixth annual ACMSIAM symposium on discrete algorithms. 1995. p. 447–56.
[13] Kanai T, Suzuki H. Approximate shortest path on polyhedral surface
based on selective refinement of the discrete graph and its applications.
In: Proceedings of the geometric modeling and processing 2000. 2000.
p. 241.
[14] Agarwal PK, Har-Peled S, Sharir M, Varadarajan KR. Approximate
shortest paths on a convex polytope in three dimensions. Journal of the
Association for Computing Machinery 1997;44:567–84.
[15] Varadarajan KR, Agarwal PK. Approximating shortest paths on a
nonconvex polyhedron. In: IEEE symposium on foundations of computer
science. 1997. p. 182–91.
[16] Kimmel R, Sethian J. Computing geodesic paths on manifolds.
Proceedings of National Academy of Sciences 1998;95(15):8431–5.
[17] Novotni M, Klein R. Computing geodesic distances on triangular meshes.
In: The 10th international conference in central Europe on computer
graphics, visualization and computer vision’2002. 2002.
[18] Aleksandrov L, Maheshwari A, Sack J. An improved approximation
algorithm for computing geometric shortest paths. In: FCT. 2003.
p. 246–57.
[19] Har-Peled S. Constructing approximate shortest path maps in three
dimensions. SIAM Journal on Computing 1999;28(4):1182–97.
[20] Mount DM. On finding shortest paths on convex polyhedra. Technical
report 1495. 1984.
[21] Papadimitriou CH. An algorithm for shortest-path motion in three
dimensions. Information Processing Letters 1985;20:259–63.
[22] Choi J, Sellen J, Yap CK. Approximate euclidean shortest paths in 3space. International Journal of Computational Geometry and Applications
1997;7(4):271–95.
[23] Clarkson K. Approximation algorithms for shortest path motion planning.
In: Proceedings of the nineteenth annual ACM conference on theory of
computing. 1987. p. 56–65.
[24] Guibas L, Hershberger J, Leven D, Sharir M, Tarjan R. Linear
time algorithms for visibility and shortest path problems inside
simple polygons. In: Proceedings of the second annual symposium on
computational geometry. 1986. p. 1–13.
[25] Martinez D, Velho L, Carvalho PC. Geodesic paths on triangular meshes.
In: Proceedings of the computer graphics and image processing, XVII
Brazilian symposium. 2004. p. 210–7.
[26] Tan X. Fast computation of shortest watchman routes in simple polygons.
Information Processing Letters 2001;77(1):27–33.
[27] Dror M, Efrat A, Lubiw A, Mitchell JSB. Touring a sequence of polygons.
In: Proceedings of the thirty-fifth annual ACM symposium on theory of
computing. 2003. p. 473–82.
[28] Mitchell JSB, Sharir M. New results on shortest paths in three dimensions.
In: Proceedings of the twentieth annual symposium on computational
geometry. 2004. p. 124–33.
[29] Chopp DL. Some improvements of the fast marching method. SIAM
Journal on Scientific Computing 2001;23(1):230–44.
[30] Zigelman G, Kimmel R, Kiryati N. Texture mapping using surface flattening via multi-dimensional scaling. IEEE Transactions on Visualization
and Computer Graphics 2002;8(1):198–207.
[31] Zhou K, Synder J, Guo B, Shum HY. Iso-charts: stretch-driven mesh
parameterization using spectral analysis. In: Proceedings of the 2004
eurographics/ACM SIGGRAPH symposium on geometry processing.
2004. p. 45–54.
[32] Saust AB. A statistical investigation of the contamination of
binary gravitational lens candidates by cataclysmic variables,
http://www.citebase.org/abstract?id=oai:arXiv.org:astro-ph/9606101;
1996.
[33] Yan J, Zhuang T. Applying improved fast marching method to endocardial
boundary detection in echocardiographic images. Pattern Recognition
Letters 2003;24(15):2777–84.
[34] Yatziv L, Bartesaghi A, Sapiro G. O(N ) implementation of the fast
marching algorithm. Journal of Computational Physics 2006;212:393–9.
[35] Wang CCL. CyberTape: An interactive measurement tool on polyhedral
surface. Computers & Graphics 2004;28(5):731–45.
[36] Danielsson PEE, Lin Q. A modified fast marching method. In:
Scandinavian conference on image analysis. 2003. p. 1154–61.
[37] Telea A, Wijk JJ. An augmented fast marching method for computing
skeletons and centerlines. In: Proceedings of the symposium on data
visualisation 2002. 2002. p. 251–9.
[38] Dijkstra EW. A note on two problems in connexion with graphs.
Numerische Mathematik 1959;1:269–71.