Surface Reconstruction Algorithms: Review and Comparison
Renoald Tang1
Setan Halim
Majid Zulkepli
Photogrmmetry and Laser Scanning Research Group, Faculty of Geoinformation and
Real Estate, Universiti Teknologi Malaysia (UTM), 81310 UTM Skudai, Johor,
Malaysia
E-mail:[email protected], [email protected], [email protected]
Abstract. This paper reviews the well-known algorithms used to reconstruct the topology
surface from sample points in three dimensional spaces. The sample points can be generated
from laser scanner, photogrammetry technique or some mathematical function. In most cases,
the sample points will describe the shape and topology surface of the object. By applying the
surface reconstruction algorithm on the sample points, the 3D model of the object can be
generated. The algorithms discussed in this paper will be applied to various models. The
topology surface generated from each algorithm will be analysed. Based on this analysing, the
problem and the idea to develop the new surface reconstruction are proposed.
1. Introduction
Surface reconstruction algorithms are a well studied problem in the computer graphic community. The
input is a set of sample points that describe the shape or topology of the object in three dimensions.
Surface reconstruction algorithms turned these points into a 3D model. These algorithms play an
importance role in 3D applications , for example ,the generating the 3D model in reverse engineering ,
modeling of scatter data in geology and the sciences mathematic community and to name a few (Wang
et al ,2010;Heckel et al,2011;Burke et al ,2010). To generate the 3D model from point clouds that
captured by using laser scanner, the surface reconstruction algorithms can be applied. Most of the
surface reconstruction algorithms discussed in this paper are coming from the computer graphics
community. By reviewing the well-known surface reconstruction algorithm and analysing surface
generated by these algorithms, the problems and direction that can be used for the future improvement
of surface reconstruction algorithms are proposed.
2. Tangent Plane Estimation Method
The first surface reconstruction algorithm based on tangent plane estimation is introduced by
Boissonnat (1984). His algorithm selects the neighbourhood of sample point in the point cloud, then
projected these sample points on tangent plane and compute the Delaunay triangulation in the
projected neighbourhood area. When the surface is smooth and sufficiently dense, the neighbour of a
point in the point clouds should not deviate too much from the tangent plane of the surface at that
point (Figure 1). Hence, by computed the tangent plane for each sample point, the surface of the point
clouds can be estimated (Boissonnat and Teillaud, 2004).
Hoppe et al (1992) proposed another algorithm for reconstructed the surface from unorganized point
by using tangent plane estimation and contour tracing. Same with the concept using by Boissonnat
(1984), the tangent plane is estimated by using principle component analysis. The Riemannian Graph
(David, 1999) is constructed to orient the direction of each tangent plane. This tangent plane will be
used to compute the sign distance function for each sample point. The final 3D surface can be
generated by extracting the zero set of sign distance by using well-known marching cube algorithm
(Newman and Yi, 2006). The main problem of the tangent estimation algorithm is the density of the
point clouds. When the point clouds are not dense enough, the wrong tangent plane will be estimated
and the surface reconstructed is not same with the original object (Figure 2b).
.
Figure 1 The information of
normal or tangent plane can be
used to estimate the local
triangulation of sample points (Pal,
2001).
(a)Original object (b)The reconstructed
surface when the tangent
estimation is wrong
Figure 2 The comparison of original object and reconstructed
surface when the tangent estimation is wrong (Hoppe et al,
1992)
3 Alpha Shape
Edelsbrunner and mucke introduced the alpha shape algorithm in 1994. Alpha shape using the Delaunay Triangulation and the user given radius (the α value) to define the surface bounding by
sample points. The Delaunay Triangulation of sample point defines as simplicial complex, (vertices
,edges ,triangles and tetrahedral). Let O is the sphere whose boundary contains all vertices of .(
Figure 3). The size of is defined to be the square of the radius of O. Then, the subcomplex of
simplices in sample point is the alpha shape if
(a) The size of is less than α value
(b) is the face triangle in the subcomplex.
The α value can be range from 0 to . When the α value is zero, the shape of Alpha shape is the
point. Alpha shape will become convex hull of sample point when α value goes to infinity (Figure 4).
The collection of all possible α shape of sample point P is called the family of alpha shapes of the P
(Bajaj et al, 1995). The complexity of algorithm alpha shape is O (n2), where n is the number of
sample points.
The α value of Alpha shape algorithm must choose randomly since not have any specific α value can
be used to describe all the surface objects. Teichmann and Capps (1998) , Melkemi (1997) , Xu and
Harada (2002) proposed the choosing of alpha value based on density of sample points. But their
algorithm still cannot propose the exact α value that can be applied to all sample points.
Figure 3 The definition of vertices, edge, triangle and tetrahedral. The
alpha shape is the surface inside the blue sphere
4 The Delaunay/Voronoi Based Algorithms
Boissonnat (1984) for the first time, proposed the surface approximation of sample points by using
Delaunay triangulation. The Alpha shape algorithm (Edelsbrunner and mucke, 1994) is a subcomplex
of Delaunay Triangulation with α value. But all these algorithms not come with theoretical guarantee.
Amenta and Bern (1998a), Amenta et al (1998b) proposed the first surface reconstruction algorithm
that comes with theoretical guarantee. Based on their works, the extension of the Delaunay/Voronoi
algorithm: Cocone, Tight Cocone, Robust Cocone and PowerCrust are developed. Compared with
other existing algorithms, Delaunay/Voronoi based algorithm always produces a good approximate
surface when the sample points are good enough (Amenta et al, 1998a &b).
4.1 The Crust Algorithm
Given the sample point in plane, the Crust is the graph that connects every pair of correct edge, as
what show in Figure 5(a & b). In two dimensions, the Crust can be defined by using Voronoi Diagram.
For three dimensions, the Crust can be estimated by using medial axis. The medial axis of a surface S
is a closure of the set of points in the plane which have two or more closet points in S (see Figure 6).
Definitely, the surface of sample point can be estimated by using medial axis. Amenta et al (1998a
&b) has shown that when the sample points are dense enough, the Voronoi vertex which called as
pole, well approximated the medial axis of the surface. By computing the Delaunay Triangulation of
the sample points, then using the pole to remove the unwanted triangle, the surface triangulation that
approximates the original surface of S will obtain.
Figure 4 The change of Alpha shape from α value 0 to 10
The theoretical guarantees for Crust algorithm is defined by using r sampling. The sample point S is
said to be the r sample of the surface S if no point p (p is point of surface S) on S is farther than r.LFS
(p) from a point of S, where LFS(p) is the local feature size of point p. Based on this sampling
definition, the Crust algorithm will guarantee reconstruct the good surface when (Amenta et al ,
1998b):
a.) For r 1, the Delaunay triangulation of P contains the polygonal reconstruction of S
b.) For r 0.40, the Crust of P from Delaunay triangulation contains the polygonal reconstruction of
S c) For r 0.252, the Crust of P from Delaunay triangulation is the polygonal reconstruction of S
where P is the sample points from surface S.
The worst case running time and memory consumption for the Crust algorithm are O (m2), where m is
the size of the pole and Delaunay Triangulation P. The computation time of this algorithm is slow
when the number of sample points is increased. To improve this algorithm, the Cocone algorithm,
which is simpler than Crust algorithm is introduced.
(a) The point. The correct
edges are the point
with colour brown
(b) The Crust, the line
that connect all the
correct edge
Figure 5 The point and the Crust
Figure 6 The medial axis, the
black line beside the body of
Giraffe
4.2 The Cocone Algorithm
The improvement of the Crust algorithm, the Cocone algorithm is proposed by Amenta et al (2000).
The Cocone of point p is the complement of a double cone centered at p with an opening angle 8
3
around the axis aligned with the normal direction (Figure 7). Collection of Cocone triangle in the
Delaunay triangulation approximated the medial axis of sample points. By computing all Cocone
triangles from the Delaunay triangulation, the surface of sample point can be determined.
When the sample points are dense enough (r > 0) , the Cocone surface is homeomorphic with the
surface of the object (Amenta et al , 2000). The complexity of Cocone is O(n2) , where n is the number
of sample points. Compare with Crust algorithm, Cocone is faster since the Delaunay triangulation of
input point p is computed once in the algorithm.
Several improvements for Cocone algorithm have been introduced. Day et al (2001) splits the point
clouds into chunks using octree. The Cocone triangle for sample then is computed on each chunk and
the surface of adjacent octree cells is obtained by duplicating the sample point on the cell’s surface.
His algorithm reduced the time used to compute the Delaunay triangulation. Funke and Ramos (2002)
reduced the time used for computing the Cocone algorithm by avoiding the computation of Delaunay
triangulation. Their algorithm computes nearest neighbour of sample point in all spatial directions.
This neighbour sample point approximated the Voronoi neighbour , and can directly used to compute
the Cocone triangle.
4.3 Tight Cocone and Robust Cocone Algorithm
The Crust and Cocone algorithms will leave the hole on the reconstruct surface when the point clouds
are not dense enough, for example, downsampling. This cause the reconstructed surface becomes
incomplete. Dey and Goswami (2003) proposed the Tight Cocone algorithm which produces the
watertight surface that's free of holes. The concept of the Tight Cocone algorithm is simple: after
compute the Cocone triangle, marking the tetrahedral triangle as “In” or “Out”. All the “Out”
tetrahedral triangles are peeling out and the final surface of point clouds is generated from the
boundary of union “In” tetrahedral triangles. This algorithm runs in O (n2) time and space in the worst
case. Tight Cocone algorithm can only handle some amount of displacements of the point away from
Figure 7. (a) Cocone in 2D. np is the normal
direction of point p. The shaded area is Cocone
region. (b) Cocone in 3D. P+ is the normal
direction of point p. The shaded area is Cocone
and S is Cocone surface (c) Double Cocone in 3D
(Dey et al, 2003)
the surface (Dey, 2007). When the noises in the points are beyond this limit, this algorithm fails to
reconstruct the surface.
Dey and Goswami (2004) proposed the Robust Cocone algorithm, which can be used to reconstruct
the surface from noisy sample points. The union of polar balls approximates the solid bounded by the
sampled surface (Amenta et al, 2001a &b). But this property does not hold in the presence of noise in
point set surface. To resolve this problem, Robust Cocone algorithm using big Delaunay ball or
feature ball to approximate the solid surface bounded by point clouds. As shows by Dey and Sun
(2005a), under some reasonable noise model, some of the Delaunay balls (called as big Delaunay ball
or feature ball) remain relative big and can play the role of the polar ball. Tight Cocone and Robust
Cocone also categorize as “Labeling” algorithms since these two algorithms labelled the triangle
surface as “In” or “Out” before the final surface are generated.
4.4 The PowerCrust Algorithm
The PowerCrust algorithm by Amenta et al (2001a&b) is another “Labeling” algorithm that can
reconstruct the watertight surface (surface free of holes) from sample points. PowerCrust algorithm
approximates the medial axis transform (MAT) of the object by using Voronoi Diagram. Then, using
inverse transform to produce the surface from MAT. The MAT can be estimated by using medial ball .
Different with the Tight and Robust Cocone algorithm, PowerCrust algorithm estimated the MAT of
point clouds by using weighted Voronoi Diagram or Power Diagram (Figure 8). This algorithm
generated two surfaces, Power Crust and Power shape. Power Crust is the boundary between the
power diagram cells belonging to inner and power diagram cells belonging to outer poles. Power
shape is a subset of weighted Delaunay triangulation dual to the power diagram (Amenta et al,
2001a&b). Since the boundary of surface captured the nature shape of the object, Power Crust or
Power Shape can be used to represent the 3D surface of the object.
PowerCrust algorithm fails to reconstruct the surface when the point clouds have noise. This is
because the noise will cause the sample points scatter away from the object surface, and the wrong
medial axis is estimated.
Figure 8 The Power Diagram is a weighted
Voronoi Diagram, where the sample points
are divided according to the power distance
5 Ball Pivoting Algorithm
The Ball Pivoting Algorithm (BPA), where surface generated is sub complex of the alpha shape
algorithm was introduced by Bernardini et al (1999). When the point clouds are dense enough, the p
ball (a ball with radius p) cannot pass through the surface without touching the points set surface.
Hence, the p ball will in contact with at least three sample points from point clouds. By keeping this
ball in contact with two of these initial points, “pivot” the ball until it touches with another points
(Figure 9). Triplet of points that the ball contacts will formed new triangles. The set of triangles
formed while the ball “walks” on the surface will become the mesh or the surface of the sample points.
Compare with Delaunay/Voronoi based algorithm, BPA algorithm is time efficiency because this
algorithm doesn’t need to compute the Delaunay Triangulation or Voronoi Diagram to produce the
mesh surface. But this algorithm will output the surface with the hole when the sample points are not
dense enough.
6 Surface Interpolated Methods
Surface interpolated methods using a specific mathematical function to interpolate the sample points.
The final mesh of sample point is the triangulated surface from interpolated function. Compare with
other existing algorithm, the surface interpolated method can be used to reduce the effect of noise in
sample point (smooth the surface) or simplified the density of sample points to resolve the storage
problem.
Carr et al (2001) using polyharmonic Radial Basis Functions (RBF) to reconstruct the smooth surface
from point cloud data. The zero set of RBF function is used to fit the point clouds data. Then, using
Marching Cube algorithm (Newman and Yi, 2006) to extract the isosurface from RBF function. Levin
(2001) proposed the using of Moving Least Square (MLS) surface for data interpolation. Based on this
work, Alexa et al (2001a & b) introduced how to use MLS surface to represent the point cloud's
surface. Their method is simple: first, computed the local reference domain (tangent plane) for the
sample point. Then, project the sample point to this tangent plane (Figure 10). The point that projected
Figure 9 The ball pivoting
algorithm in 2D.A circle of
radius p pivots from sample
point to sample point,
connecting them with edges
on tangent plane will be the MLS surface of the point clouds. By changing the value of Gaussian
function in MLS surface, the smoothness of surface can be controlled.
The Adaptive Moving Least Square (AMLS) surfaces that come with theoretical guarantee was
introduced by Dey and Sun (2005b). Their AMLS surface is based on the model proposed by Shen et
al (2004). The projection technique on AMLS surface is similar to the projection technique on the
MLS surface by Alexa et al (2001 a& b). They applied the featured ball and local feature function
(the concept introduced by Crust algorithm) to compute the Gaussian functions. This cause the change
of Gaussian function in AMLS surface directly give effect to the point set surface.
7 Other Algorithms
A lot of algorithms had been developed in the computer graphic community. In this paper we
only consider the well-known surface reconstruction algorithms that always refer by other
researcher when developed or proposed new algorithms. For algorithms that are the extension
of the well-known algorithm, for example the improvement of original algorithm will only
give a brief explanation in Table 1. The details of algorithm can refer to the book by
Boissonnat and Teillaud (2007).
Figure 10 The MLS projection procedure. First, a local
reference domain H for the point r is generated. The projection
of r onto H defines its original q. Then a local polynomial
approximation g to the height fi of point pi over H is computed.
In both case, the weight for each pi is a function of the distance
to q. The projection of r onto g (the blue point) is the result of
MLS projection (Alexa et al, 2001 a&b)
8 Experiments
Since the limited space and scope of study, only five well-known algorithms (Alpha shape, Cocone,
PowerCrust, BPA and interpolated method) are chosen to reconstruct the surface from point clouds in
this paper. Table 2 shows these algorithms and their implementations. The object and the number of
point clouds for each model are shown in Table 3. The point clouds of face, vase and box are captured
by using Vivid910 laser scanner. The rest is the free model that can be downloaded from the
homepage aims@shape project . All the 3D model reconstructed from the algorithms will be viewed
by using Open source, MeshLab (MeshLab, 2012). The analysing of each 3D model generated from
algorithms are based on:
1) The topology surface,
2) Number of surface and
Surface reconstruction
algorithms
Concept Reference
Greedy algorithm Incrementally reconstructs an
oriented surface by selecting
triangles from Delaunay
Triangulation of sample point
and stitching to surface
Steiner and Da(2004)
Natural neighbour The surface of sample points
is computed from a subset of
Delaunay Triangulation from
implicit function
Boissonnat and Cazals(2000
& 2001)
Wrap The stable manifolds of
sample points are
approximated by sub-
complexes of Delaunay
Triangulation and subset of
the maxima of flow relation
Edelsbrunner (2003)
Convection algorithm Using evolution equations to
construct the surface that
minimizes the energy
functional by deforming a
good initial enclosing
approximation of the surface.
Chaine(2003)
Regular interpolant The triangle contributing to
the interpolant are the Gabriel
triangle minimizing the
granularity at the vertices.
Petitjean and boyer (2001)
,Attene and
Spagnuolo(2000)
Morse theory Obtaining a topologically
correct mesh based on
sweeping through the family
of surface f(x,y,z)=α for
varying parameters α and
watching the critical point
where the topology changes
Stander and Hart(1997)
Siersma(1999)
Table 1 The concept and reference for surface reconstruction algorithms
3) The time used by algorithm to compute the surface
9 Results and Analysis
Figures 11 to 15 show the 3D model reconstructed by using Alpha shape (Figure 11), BPA (Figure
12), PowerCrust (Figure 13), Cocone (Figure 14) and Interpolated method (Figure 15). For
interpolated method, first compute the AMLS surface (Dey and Sun, 2005b), then applied the
PowerCrust algorithm to define the 3D surface. The α value of the Alpha shape algorithm is chosen
based on point cloud density (MeshLab, 2010).
Algorithm Implementation
Alpha shape MeshLab
BPA MeshLab
PowerCrust MATLAB
Cocone Open source
Interpolated method(AMLS) Open source
Model Number of points
Face 11642
Box 10757
Engine 10755
Vase 9649
Flower 7916
Sphere 5724
Table 2 Surface reconstruction algorithm and their implementations
Table 3 The model and number of point clouds
(a) (b) (c)
(d) (e) (f)
Figure 11 Surface created from Alpha shape algorithm
(a) (b) (c)
(d) (e) (f)
Figure 12 Surface created from Ball Pivoting Algorithm
(a) (b) (c)
(d) (e) (f)
Figure 13 Surface created from PowerCrust Algorithm
(a) (b) (c)
(d) (e) (f)
Figure 14 Surface created from Cocone Algorithm
The comparison from Figures 11 to 15 had shown that the topology surfaces created by each algorithm
are based on the density of input point. For point clouds with good sampling density, for example the
face, vase and box , almost all algorithm we used have successfully reconstruct the true surface (refer
(b), (e), (f) for Figures 11 to 15. The extra step needs to remove the noise in the model, but this not
discuss here since beyond the scope of this paper). Alpha shape, BPA and Cocone algorithm will
reconstruct the incomplete surface (surface have unwanted hole, see (a), (c) and (e) from Figure 11 to
15) when the sample point are not dense enough.
PowerCrust algorithm created the surface without hole, even the point clouds are not dense enough
(Figure 13). But the problem is, this algorithm also closes the “inside” surface of the 3D model. For
example, the engine model. The true topology surface of this model should be same with Figure 11 (d)
12 (d) and 14 (d). This is because the PowerCrust algorithm computed the surface using inside
(outside) boundary, the inside surface will not define (this type of surface called as watertight surface).
The advantage of interpolate method (Figure 15) is the removal of noise effect on 3D surface.
Compare with other algorithm, interpolated method interpolates the true surface based on
mathematical functions, for example the Adaptive Moving Least square (AMLS) surface. The final
surface or point clouds from this method are a smooth surface and same topology with the original
model. Figure 16 compared the point clouds before (colour red) and after (colour blue) the AMLS
algorithm is applied. Please mention that these methods just interpolate the true point but not adding
the extra point on the point clouds.
(a) (b) (c)
(d) (e) (f)
Figure 15 Surface created from Interpolated (AMLS) method
Table 4 compared the number of surfaces for each model based on algorithms. The table had shown
that the alpha shape algorithm computed more surfaces that other algorithm. Among the algorithms we
had applied in this paper, BPA or ball pivoting algorithm only creates a small collection of surface
from point clouds. Even for same object and model, the surfaces created by different algorithms are
not same. Alpha shape, PowerCrust and Cocone can be categorized as Delaunay/Voronoi based
algorithm. This type of algorithm will compute more surface that other existing algorithm.
Model Alpha
shape
BPA PowerCrust Cocone Interpolate
method
Face 53713 22917 23182 22975 23208
Box 33877 20197 21492 21414 21494
Engine 48189 19851 21906 21143 21524
Vase 50774 18011 18648 17812 18718
Flower 29120 14493 15838 15440 15844
Sphere 10980 10504 11456 11433 11432
Figure 16 The comparison of point clouds before and after applied the AMLS
algorithm. The point cloud with colour red is original point and colour blue is point
after AMLS algorithm is computed. This Figure show how these two point cloud
deviated to each other
Table 4 The number of surface of each model based on the algorithms
Number of surface
Figure 17 shows the time used by each algorithm to compute the surface based on the number of point
clouds. From the graph, the time used to compute the surface is directly proportional to the number of
input points. More time is needed to compute the surface when the number of point clouds is
increasing. Delaunay/Voronoi based algorithm (Alpha shape, Cocone and PowerCrust) using more
time to compute surface compare with other algorithm. This is because the computation of Voronoi
Diagram or Delaunay Triangulation is time consuming (Bernardini, 1999; Dey, 2007).The interpolated
method, for example, the computation of Moving Least square surface, involving the using of matrix
in the algorithm. When the number of point clouds increases, the dimension of matrix will also
increase, the high memory storage is needed. Indirectly, the complexity time for an algorithm will also
increase. The BPA algorithm not involved any computation in Voronoi Diagram or Delaunay
Triangulation. Hence, this algorithm is faster than other algorithms.
10 The Problems in Surface Reconstruction Algorithms
The choosing of surface reconstruction algorithms that can be applied for all datasets still is an open
problem in computer graphics community (Boissonnat and Teillaud, 2004). Tangent plane estimation
proposed by Boissonnat (1984) and Hoppe et al (1992) will fail to reconstruct the surface when the
point clouds have noise. In real time scanning, the noise cannot be avoided since the measurements
always contain errors. The tangent plane estimation method will estimate the wrong tangent plane.
Hence, the surface with noise will be created.
The Delaunay/Voronoi based algorithms (Alpha shape, Crust, PowerCrust and Cocone) are surface
reconstruction algorithm come with theoretical guarantee. But this theoretical guarantee only met
under some conditions. If these conditions are not met, their behaviour is not specified. From our
experience, it is not easy to meet these conditions in real time scanning. For example, the point clouds
are not dense enough to describe the real object. This will cause the algorithms reconstruct the wrong
surface.
The time complexity is another factor should be considered in surface reconstruction algorithms.The
real time scanning project always produce the huge number of point clouds. The computation of
5000 6000 7000 8000 9000 10000 11000 120000
2
4
6
8
10
12
14The comparison of Time for surface reconstrcution based on algortihms
Number of point clouds
Tim
e(i
n s
eco
nd
)
BPA
alpha shape
powercrust
cocone
Interpolate method
Figure 17 The comparison of time for surface
reconstruction based on algorithms
Voronoi Diagram and Delaunay Triangulation (for example, the Delaunay/Voronoi based algorithm)
on these point clouds are time consuming. The time used to generate the 3D model must be considered
in real time scanning project. This is because the time used by algorithm to generate 3D model from
point clouds will affect the progress of scanning project.
11 Conclusions
This paper compared and analysis the surface generated by well-known surface reconstruction
algorithms. The comparison had shown the topology surface generated by these algorithms is based on
the density of point clouds. When the density of point clouds is not good enough, Alpha shape, BPA
and Cocone algorithm will reconstruct the incomplete surface. PowerCrust algorithm can reconstruct
the surface from low density point clouds, but the surface created is not smooth. This problem can be
resolved by represented the surface as AMLS or Adaptive Moving Least Square surface. The number
of surfaces generated by BPA algorithm is less than the number of surfaces generated by
Delaunay/Voronoi based algorithms. Hence, the generating of 3D surface by using BPA algorithm
will faster than Delaunay/Voronoi based algorithms. These algorithms (except interpolated method)
fail to reconstruct the smooth surface when the point clouds had noise. Based on this analysing, the
new surface reconstruction algorithm will be proposed.
12 The Proposed of new Surface Reconstruction Algorithm
Two factors will be considered in the new surface reconstruction algorithm:
1.) Time complexity of algorithm and
2.) How the algorithm can deal with error or noise in point clouds
For the first factor, the idea is to simplify the number of point clouds. The simplify direction should be
as follows: the algorithm will simplify the point clouds at plane region, since for plane area; just few
numbers of points are enough to describe the true shape of the object. For areas with high curvature,
the simplification process should be maintained the geometry and topology surface of the object.
Overall, the simplification algorithm should not change the shape and topology surface of the point
clouds.
For surface reconstruction, the using of Voronoi Diagram or Delaunay Triangulation in the algorithm
is avoided. So, the solution is extracted the boundary of the point clouds as a line, as what proposed by
Kalogerakis et al (2008). Several researches had been proposed for reconstructing the surface from
noisy sample (Kolluri et al, 2004; Mederos et al, 2005; Ohtake et al, 2005). These ideas will apply in
the new surface reconstruction algorithm.
These two algorithms (simplify the point clouds and extract the boundary of point clouds as a
line) will be combined to become new surface reconstruction algorithm. Hence, the new
surface reconstruction algorithm should be able to simplify the point and at the same time
reconstructed the 3D surface (Figure 18). The detail of this algorithm will discuss in next
paper.
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Open sources and web page:
Meshlab, http://meshlab.sourceforge.net/
Cocone and Tight Cocone, http://www.cse.ohio-state.edu/~tamaldey/surfrecon.htm
Free 3D model, http://shapes.aimatshape.net/