Computers & Graphics 74 (2018) 44–55
Contents lists available at ScienceDirect
Computers & Graphics
journal homepage: www.elsevier.com/locate/cag
Special Section on SMI 2018
Surface reconstruction of incomplete datasets: A novel Poisson surface
approach based on CSRBF
Jules Morel a , b , ∗, Alexandra Bac
b , Cédric Véga
c
a French Institute of Pondicherry, 11 Saint Louis Street, Pondicherry 605001, India b Laboratoire des Sciences de l’Information et des Systèmes, 163 Avenue de Luminy, Marseille 13288, France c Laboratoire dInventaire Forestier, Institut National de LInformation Géographique et Forestière, 14 rue Girardet, Nancy 540 0 0, France
a r t i c l e i n f o
Article history:
Received 27 April 2018
Revised 11 May 2018
Accepted 14 May 2018
Available online 17 May 2018
Keywords:
Surface reconstruction
Approximation
Radial basis functions
Poisson surface
Terrestrial LiDAR scanning
a b s t r a c t
This paper introduces a novel surface reconstruction method based on unorganized point clouds, which
focuses on offering complete and closed mesh models of partially sampled object surfaces. To accomplish
this task, our approach builds upon a known a priori model that coarsely describes the scanned object to
guide the modeling of the shape based on heavily occluded point clouds. In the region of space visible
to the scanner, we retrieve the surface by following the resolution of a Poisson problem: the surface is
modeled as the zero level-set of an implicit function whose gradient is the closest to the vector field
induced by the 3D sample normals. In the occluded region of space, we consider the a priori model as
a sufficiently accurate descriptor of the shape. Both models, which are expressed in the same basis of
compactly supported radial functions to ensure computation and memory efficiency, are then blended to
obtain a closed model of the scanned object. Our method is finally tested on traditional testing datasets
to assess its accuracy and on simulated terrestrial LiDAR scanning (TLS) point clouds of trees to assess its
ability to handle complex shapes with occlusions.
© 2018 Elsevier Ltd. All rights reserved.
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1. Introduction
The reconstruction of 3D surfaces from scattered data has re-
ceived increasing attention with the emergence of new close-range
3D acquisition technologies, such as laser scanning devices (Li-
DAR), close-range photogrammetry and time-of-flight cameras. The
dense 3D point clouds thus acquired accurately describe object sur-
faces (e.g., millimeter resolution for laser scanning). Such scanning
processes have a wide range of applications: urban reconstruc-
tion and modeling, architecture, artifacts modeling, quality control
for production, and medical imaging. However, despite their accu-
racy, the data acquired by these sensing technologies share com-
mon constraints, such as non-homogeneous sampling, occlusion
and noise. In view of their characteristics and complexity, dedi-
cated algorithms are required to segment, model and reconstruct
objects of interest from raw point clouds. Terrestrial laser scan-
ning (TLS) technology is broadly used in forest studies. TLS en-
ables 3D forest structures to be acquired as point clouds in record
time [1] , with applications ranging from ecology (allometric re-
∗ Corresponding author at: French Institute of Pondicherry, 11 Saint Louis Street,
Pondicherry 605001, India.
E-mail address: [email protected] (J. Morel).
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https://doi.org/10.1016/j.cag.2018.05.004
0097-8493/© 2018 Elsevier Ltd. All rights reserved.
ationships, 1 and growth modeling carbon storage assessment) to
orestry (forest monitoring, sustainable development) and industry
harvest planning, sawmill optimization). However, the features of
uch data are even more challenging with respect to reconstruct-
ng models and extracting information (e.g., more challenging than
lassic applicative data, such as urban environments or isolated
bjects, because the clouds are extremely dense, inhomogeneous,
oisy and present large occlusions). These constraints arise both
rom the remote sensing technology and from the complexity of
orest environments. The TLS point cloud sampling rate may vary
rom one scanner to another, and the spherical geometry of the
ensor results in irregular sampling density. Moreover, the combi-
ation of TLS geometry and the vegetation itself (branches, leaves,
ow vegetation) results in large and numerous occluded areas that
xpand both in size and number far from the sensor. Noise con-
ributes additional confusion at surface extremities and in foliage.
herefore, data obtained from a given tree have different charac-
eristics from the base up to the crown. Forest measurements from
LS data also suffer from object-specific limitations. Stems bark can
1 Allometry consists of a set of general relations derived from a large compilation
f forest measurements. It provides an estimate of the tree structure according to a
ew given parameters, such as the diameter at breast height (DBH, diameter of the
tem 1.30 m above the ground) and the tree height.
J. Morel et al. / Computers & Graphics 74 (2018) 44–55 45
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e rough and therefore produce highly uneven surfaces. Moreover,
he non-trivial topology of branches and the intricate occlusions
roduced by these branches are highly challenging for point cloud
rocessing. Finally, let us note that LiDAR scanning of trees may
e affected by wind and thus create multiple scan alignment is-
ues. All these artifacts induce specific point cloud distortions and
enerate crooked objects. Therefore, in the context of LiDAR data,
econstruction is necessary to handle partially described objects.
o overcome the data distribution characteristics inherent to TLS
oint clouds, especially for data acquired in forests, our idea was
o rely on a priori knowledge about the forest elements expressed
s geometrical models. This paper presents a novel surface recon-
truction method that is specifically designed for the reconstruc-
ion of partially described objects based on 3D point clouds. We
ddress this challenge by introducing a novel surface reconstruc-
ion method based on a Poisson scheme, building upon sturdy ap-
roximating basis functions. On the basis of this innovation, our
lgorithm lets us integrate a priori models of occluded areas, ex-
ressed using basis functions, to describe partially tubular objects.
his approach provides good estimates of missing data and en-
bles complete reconstruction of forest objects. This algorithm was
ested and validated against “classic” datasets and on occluded
lmost-tubular shapes and tree sections. In Section 2 , we present a
hort literature review of surface reconstruction and tree modeling
rom TLS samples.
. State of the art
Pioneering works on surface reconstruction from raw point
louds date to the late nineties and are usually classified as ex-
licit or implicit according to the underlying model (see [2–4] for
omplete surveys). There are two types of explicit surfaces, para-
etric and meshes, both of which have been investigated in this
ontext. Parametric surfaces, such as B-splines [5,6] and NURBS
7] , entail determining a 2D parameter space together with a set
f associated control points. Such surfaces are controlled by these
oints but not as an approximation or interpolation. Therefore,
omplex surfaces are not easily representable, and parameteriza-
ion is a complex issue for scattered data, especially in the pres-
nce of noise, inhomogeneity and occlusions. In [8] , the authors
efine polynomial splines over locally refined parameter spaces
n any dimension and thus successfully reconstruct sharp features
nd details from point clouds. However, although the approach
erforms well on terrain data, its parametric nature, as well as its
omplexity, limit its application to data from complex scenes. Tri-
ngular mesh reconstruction received substantial attention through
elaunay triangulation, alpha shape reconstruction and Voronoi di-
grams [9–12] . However, scanning devices, such as LiDAR scan-
ers, produce dense, noisy and potentially occluded point clouds
hat cannot be accurately modeled by meshes. The successive re-
eshing required to obtain an adequate model multiplies the cum-
ersome computations. Moreover, the inhomogeneity of sampling
eads to unbalanced meshes, with larger polygons far from the sen-
or, tiny polygons close to it (where the point density can reach 1
illion points / m
2 ) and stretched polygons in occluded areas.
Therefore, the unstructured, nonplanar nature of point clouds
akes implicit surfaces a key modeling tool. Moreover, such mod-
ls structurally smooth the noise by approximating the input
oints and are tolerant to inhomogeneity and limited occlusion.
mplicit surface reconstruction is the process of finding a function
hat best fits the input data. However, the implicit representation
f a surface needs to be post-processed to be visualized. Marching
ube [13,14] is the best-known method to generate a triangulated
urface from the implicit representation of the surface. Because the
urface is extracted as a level set of an implicit function, the result-
ng mesh is guaranteed to be a watertight manifold.
Computing implicit functions from point clouds as an approxi-
ate of the signed distance function has been extensively studied
15] . However, such approaches prove to be unstable in the pres-
nce of nonuniform sampling. The moving least-squares method
introduced in [16] , see [17] for a complete survey) addresses this
roblem but struggles in the presence of missing data, as noted
n [2] : the large spatial support of basis functions required near
oles spoils the reconstruction. Another class of methods, namely,
lobal reconstruction methods, was proposed by Carr et al. [18] .
hese approaches are based on radial basis functions (RBFs) and
ake advantage of their approximating properties. RBFs are posi-
ive definite basis of functions and hence guarantee approximation
easibility (see [19,20] for more details). The benefits of modeling
urfaces with RBFs are broadly recognized [21–24] . However, poly-
armonic RBFs have global support; hence, reconstruction entails
nverting dense ill-conditioned matrices. To mitigate this problem,
urther works focused on compactly supported radial basis func-
ions (CSRBFs) (either used directly [25] or as blending functions
etween local reconstructions [26–28] or both [29] ). The finite sup-
ort of such functions enables faster filling of the interpolation ma-
rix [30] , which simultaneously becomes sparse. Matrix inversion
an thus be accelerated by using a direct sparse matrix solver (see
orse [31] for a summary of the advantages of CSRBF over classic
BF). A different approach takes advantage of both global and local
tting schemes by approximating the field of estimated normals
hrough Poisson reconstruction [32] . Owing to the representation
s a Poisson problem, this method is robust to nonuniform sam-
ling, noise, outliers and to a certain extent, missing data. These
ualities make it a choice method for surface reconstruction from
LS point clouds. However, for computational reasons, implicit sur-
aces are computed and expressed in a basis of functions obtained
y convolution of a box-filter with itself. Unfortunately, this basis
s not positive definite (unlike radial basis functions) and thus it
oes not have sufficient approximation properties to express any
priori information about occluded areas. While these surface re-
onstruction methods have proven to produce sharp models from
oint clouds, none is able to fill the large gaps created by occlusion
n LiDAR point clouds acquired in forests.
To solve this problem and extract the information needed in
orestry, scientists rely on a common assumption: a woody tree
tructure is assumed to be a network of quasi cylinders. Thus,
he tree structure, branching organization and branch size distri-
ution are modeled through so-called quantitative structure mod-
ls that summarize this information by describing the tree compo-
ents in hierarchical order as a stack of elementary building blocks.
his approach has been widely explored. Côté et al. [33] proposed
n architectural model combined with a skeletal curve approach
o retrieve the tree structure and allometric relationship to build
he branching structure and further assess the amount of foliage.
assot et al. [1] introduced a semi-automatic approach to model
ree architecture using cylinders and to estimate tree parameters,
uch as tree volume. Raumonen et al. [34] introduced a method
nvolving clusterization and segmentation of the point cloud, fol-
owed by reconstruction of the tree architecture using cylinders.
hey combined this geometric and hierarchical information into
he concept of quantitative structure modeling (QSM). A similar
ylinder-based tree-reconstruction method was proposed by Hack-
nberg et al. [35] using a sphere-following approach to progres-
ively reconstruct the tree structure from the ground to the apex.
witching from tree-level reconstruction to plot-level reconstruc-
ion is a challenging task. Intermingling crowns and occlusions due
o branches and leaves in the signal path makes it difficult to ac-
urately segment trees. Raumonen et al. [36] used a clusterization
pproach to detect tree bases, followed by a distance-based expan-
ion procedure to allocate the remaining clusters to the detected
rees. Tao et al. [37] used clustering and shortest-path algorithms
46 J. Morel et al. / Computers & Graphics 74 (2018) 44–55
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to detect trees and segment associated crowns. Combining Hough
transform and active contours, Ravaglia et al. [38,39] introduced a
method to automatically detect and reconstruct largely occluded
tubular shapes. This approach bears some similarities with skele-
ton extraction methods such as [40] . However, it mitigates the ef-
fects of noise and longitudinal occlusions which are both highly
present in LiDAR point clouds. While these methods are effective
for extracting structural parameters, they propose discontinuous
models and rely on local cylindrical (and thus purely tubular) ap-
proximations, which has proven to be the best local approxima-
tion but can lead to substantial error. Indeed, the assumption that
a tree is composed of cylinders can be far from reality, especially
when we consider tropical trees. Chave et al. [41] found an error of
50% in the computation of the volume from field measurements at
the tree level by such approximations in tropical forests. Therefore,
even if QSM approaches are well suited to reconstruct the struc-
ture of trees from LiDAR data, they provide only a rough estimate
of the structure. Hence, sharper 3D modeling is required to more
accurately describe the shape and improve the geometric model to
precisely assess tree properties.
In spite of this specific initial context, our approach is actually
general: in order to reconstruct largely occluded, roughly tubular
data, we assumed that a detailed reconstruction can be achieved
only in reasonably sampled areas, but the reconstructed model
should be resilient enough to integrate an a priori model of oc-
cluded (tubular) areas. Such “roughly tubular” shapes, locally well
sampled and partly occluded are quite common in LiDAR data
(such as archeological, urban or natural data for instance). The aim
of our work is to build a model which proves both locally accurate
(in properly sampled areas) and globally coherent (according to a
priori information).
Using a priori models to guide the reconstruction is not new.
Previous works are based on priors on the surface using Bayesian
approaches (works on medical data), global matching of the shape
using priors stored in a database (see [42] for instance) or local
shape priors (see [43] ). However such approaches are based on
the assumption that the shape is globally or locally known a priori ,
and thus focus on matching data and priors. However in our con-
text, we intend to reconstruct the shape as accurately as possible
in visible areas (and the shape is unknown there) and priors are
used only to reconstruct occluded areas in a coherent way.
3. Overview of the approach
Our work introduces two significant innovations. We propose
a new 3D reconstruction method based on a Poisson scheme de-
veloped over CSRBF, whose approximating properties have already
been mentioned and provide the “resilient” setting for further in-
tegration of a priori knowledge. In addition to the reconstruction
efficiency demonstrated in our validation, this choice of basis func-
tion enables the expression and integration of a priori knowledge
in occluded areas.
Our method hence focuses on accurately approximating data
points while managing occlusions to produce a closed surface. To
do so in our application context, the method identifies and sep-
arately processes visible and occluded areas. For the visible por-
tions of the object, a novel Poisson surface algorithm finely re-
constructs the surface. In occluded areas, the tree surface is rep-
resented using a a priori model characterized by a tubular profile
that is adapted to approximate the shape of the tree. The smooth-
ing characteristic of the CSRBF then guarantees the continuity of
shape between models. The following two sections describe both
pre-processing steps, upon which our method builds: the compu-
tation of the point normals, which are required by the Poisson sur-
face solver, and the computation of the tubular shape (or QSM) to
guide reconstruction in occluded areas.
.1. Point normals computation
Let P = { p i } i =1 ... N ⊂ R
3 be the input point cloud. As a prelimi-
ary processing step, Poisson surface reconstruction requires com-
utation of a consistent normal field for each sample. We use the
ethod introduced by Hoppe [15] to compute a normal n i at each
oint p i . Actually, more accurate algorithms exist (local triangula-
ion, quadric fitting...), however, LiDAR point clouds can be made
f billions of points, making the computation time a critical issue.
ollowing [15] , we first study the neighborhood of each point p i to
ompute a normal n i , then we direct consistently the normals by
ropagating their orientation on a minimum spanning tree.
.2. Tubular guide estimation
In our next preliminary processing step, we compute a QSM of
he tree (in the context of our application, we used the Hough
ransform approach introduced by Ravaglia et al. [38] ). From this
iscontinuous stack of cylinders, the global outline of the tree is
odeled by a continuous tubular shape, as introduced in [44] . Fol-
owing this approach, in each slice along the Oz axis, the cylin-
er (centered in c i ) is used as an initialization to fit a cylindrical
uadratic function g i , whose influence lies in the sphere of radiusi . Each radius σ i is thus computed so that the set of spheres S( c i ,i ) covers the whole tree. The tubular model is then expressed as
n implicit function obtained by blending the quadratic approxi-
ations g i together with a CSRBF, namely, Wenland’s CSRBF [30] .
herefore, the a priori model of the tree is expressed as:
f ( q ) =
∑
i
g i ( q ) · �σ i (‖ q − c i ‖ ) , (1)
here i ranges over the indices of slices {Z i } along the Oz axis, g i is
quadratic function and �σ i is the Wendland φ3, 1 CSRBF centered
n c i and of support radius σ i .
. CSRBF Poisson surface reconstruction guided by a model
nown a priori
As illustrated in Fig. 1 , following the previous pre-processing
teps, our method is based on five steps.
• We divide the space into two parts, namely, occluded and visi-
ble areas, by analyzing the angular distribution of points. • We define a space of functions based on a CSRBF with high
resolution near the points and near the surface of the tubular
model and coarser resolution away from them. • In visible areas, we set up and solve the Poisson equation. • In occluded areas, taking advantage of the approximating prop-
erties of the CSRBF, we express the tubular shape in our space
of functions and insert it into the solution. • Finally, we extract an isosurface of the resulting implicit func-
tion.
The details of each steps are provided in the following subsec-
ions.
.1. Mark off the occluded areas
As illustrated in Fig. 1 , the QSM computation described in
ection 3.2 provides a division of space into several slices Z i along
he stem axis. In each subset, an analysis of the angular distribu-
ion of points enables the detection of holes: among the points
ontained in Z i , if two points are separated by an angle larger than
given user-defined threshold, then the portion of space between
hem is considered to be occluded. In this way, we eventually ob-
ain a boundary between the visible and occluded areas for the
hole set of slices.
J. Morel et al. / Computers & Graphics 74 (2018) 44–55 47
Fig. 1. Overview of the method: For preliminary processing, we compute a QSM and the normal of the points. Then, we consider each building block of the QSM indepen-
dently. (left) Data points are distributed over the part of the trunk exposed to the LiDAR beam, dividing the space into occluded and visible areas. (Top) Based on a QSM,
we estimate the tree shape as a tubular model expressed as an implicit surface. (Bottom) In the visible area, we compute a Poisson surface reconstruction based on CSRBF.
(Right) The algorithm approximates the surface of the tree as a Poisson surface in the visible area and closes the surface using the tubular model in the occluded area. The
tree surface is retrieved by blending the models computed for each QSM building block.
Fig. 2. Overview of the discretization in 2D: (a) the first division of the octree is driven by the points distribution: every points should fall into a leaf, then (b) leaves are
divided close to the tubular shape (represented as a grey circle) and finally (c) the octree is refined by allowing a maximum difference of levels equal to one between a leaf
and its neighbors.
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.2. Problem discretization
To efficiently solve the Poisson scheme, our model needs to be
efined close to the points and the tubular model surface. We per-
orm this adaptive subdivision using an octree, which allows us to
djust the resolution locally while providing rapid access to neigh-
orhoods and a low memory footprint. The octree subdivision pro-
ess is illustrated in Fig. 2 . We define the octree as the minimal
ctree so that every point falls into a leaf of a resolution fixed by
he user. Then, we force octree division close to the tubular shape
y analyzing the eight corners of each leaf falling in the occluded
rea; the sign of the implicit function is given in Section 3.2 . Sub-
ivision occurs if at least one corner of a leaf has a different sign.
inally, we refine this octree by allowing a maximum difference of
evels equal to one between a leaf and its neighbors to guarantee
mooth variation in resolution. We denote by O and O
v isible occluded48 J. Morel et al. / Computers & Graphics 74 (2018) 44–55
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the leaves falling, respectively, in visible areas and occluded areas,
as defined in Section 4.1 . The resulting set of leaves O form a par-
tition of space.
4.3. Definition of the function space
Let P denote the input point cloud (set of points of R
3 ). Given
{O 1 , . . . , O N } , a partition of P (in our case, the previously defined
octree subdivision of space), for each subset O i (or cell) we denote
by o c i the center of O i . To build our implicit model, we define a
space of functions as the span of translates of Wendland’s CSRBF:
for every node o ∈ O, we thus set �σo to be the CSRBF centered on
o c i with radius σ o , defined hereafter to ensure a sufficient coverage
of space:
�σo (‖ q − o c ‖ ) = φ
(‖ q − o c ‖
σo .
)(2)
More precisely, we use φ(r) = (1 − r) 4 + (1 + 4 r) , where + represents
(x ) + = x if x > 0 and (x ) + = 0 otherwise. The radii σ o are adjusted
to recover the details of smaller cells while limiting the overlap-
ping of supports for larger cells. With this in mind, we compute
σ o as:
σo = a × Size min +
√
3 / 2 × Size o , (3)
with Size min the size of the smallest octree leaf, Size o the size of
leaf o and a a parameter tuned by the user. Span { �σo } , the set of
independent basis functions �σo is used as a basis to express the
implicit functions computed in the following sections.
4.4. Setup of the Poisson problem
Poisson surface reconstruction methods are motivated by the
computation of the characteristic function χ of a solid M with
surface boundary ∂M . In [32] , the authors highlighted an integral
relation between the gradient ∇χ (where χ has been smoothed to
become at least C 2 ) and the field of oriented point normals. Hence,
χ can be computed as an implicit function χ whose gradient ap-
proximates a vector field V induced by the normals of the sample
points. Given p ∈ ∂M :
∇(χ ∗ F )( q ) =
∫ ∂M
F( q − p ) · n · dp, (4)
where F is a smoothing filter. Because V is rarely integrable, this
variational problem is transformed into a standard Poisson prob-
lem by applying the divergence operator:
�χ = ∇ . V (5)
The definition of the discrete approximation of V is a critical
step. The main point is to discretize and approximate Eq. (5) . First,
following [32] , Eq. (5) is discretized as a sum over data points by
assuming that any point p of normal n describes an elementary
patch of surface A p , which leads to the following formula:
V( q ) =
∑
p ∈P A p · F( q − p ) · n . (6)
Then, for computational efficiency, in order to avoid this integral
over data points, the influence of each sample point and its normal
are distributed in the surrounding octree cells. Given a point p ∈P, a coefficient αo, p is introduced to account for the share of p
attributed to cell o . We define these coefficients using the basis
functions themselves. Let N ( p ) be the neighborhood of a sample
point p , that is, N ( p ) = { o ∈ O v isible , ‖ p − o c ‖ < σo } . We define αo, p
as:
αo, p =
�σo (‖ p − o c ‖ ) ∑
o ′ ∈N ( p ) �σ ′ o (‖ p − o
′ c ‖ )
. (7)
t
Now, before defining the vector field V, let us consider the data
nhomogeneity. As mentioned in [32] , local variations in point den-
ity should be considered to adjust the contribution of data points.
e follow a kernel density estimation approach and estimate the
ocal density as:
( q ) =
∑
p ∈P
∑
o∈N ( p )
αo, p . �1 σo
(‖ q − o c ‖ ) , (8)
ith �1 (r) = 1 − r being a Wendland’s CSRBF of first order.
We can now approximate the vector field defined by Eq. (6) as:
( q ) =
∑
p ∈P
1
W( p )
∑
o∈N ( p )
αo, p �σo (‖ q − o c ‖ ) · n . (9)
.5. Resolution of the Poisson equation
We express χ in the space Span { �σo } as the linear combina-
ion:
( q ) =
∑
o∈O v isible
x o · �σo (‖ q − o c ‖ ) . (10)
By projecting the Poisson equation onto this space of functions,
e obtain:
o ′ ∈ O v isible , 〈 �χ, �σo ′ 〉 = 〈 ∇ · V, �σo ′ 〉 , (11)
here �χ( q ) =
∑
o∈O v isible x o · ��σo (‖ q − o c ‖ ) .
Denote by L the matrix of elements 〈 ��σo , �σo ′ 〉 , v the vec-
or of 〈∇ · V, �σo ′ 〉 elements and x the unknown vector com-
osed of the linear coefficient of χ . Because our basis function
s not orthonormal, we solve the Poisson problem by minimizing
L · x − v ‖ 2 in the least-squares sense. In our case, the matrix L is
elf-adjoint and positive definite owing to the properties of func-
ions �σo and ��σo . Moreover, it is sparse because of the com-
act support of the basis functions; thus, solving this problem di-
ectly is not expensive. Therefore, we use the LDL T Cholesky de-
omposition for sparse matrices implemented in the Eigen library.
n the next section, we provide further details about the efficient
esolution of this Poisson problem with CSRBF. In fact, we make
xplicit the system of equations and deduce the symmetry proper-
ies that enable us to optimize its resolution.
.6. Optimization
The Poisson Eq. (5) can be expressed in its matrix form as:
. . .
〈 ��σo , �σo ′ 〉
. . .
⎤
⎥ ⎦
⎡
⎢ ⎣
. . . x o . . .
⎤
⎥ ⎦
=
⎡
⎢ ⎣
. . . 〈∇ · V, �σo ′ 〉
. . .
⎤
⎥ ⎦
. (12)
The dot product is that of L 2 (R ) : given f : R
3 → R and g : R
3 → , two compactly supported functions, and , the union of their
ompact supports, the dot product can be expressed as follows:
f , g〉 =
∫ R 3
f ( q ) · g( q ) · d q =
∫
f ( q ) · g( q ) d q . (13)
eeping with the definitions introduced earlier in this document,
e denote by L = 〈 ��σo , �σo ′ 〉 the Laplacian matrix and by V the
ector (〈∇ · V, �σo 〉 ) o∈O . A study of the symmetries of this system, presented in detail in
ection Appendix A , allows to reduce the number of computations
equired to solve the system. Table 1 , in particular, illustrates the
rastic reduction in the number of computations for reconstruction
f the Stanford bunny. For each explored radii value, among the
undreds of thousands of matrix entries that need to be estimated,
nly a few dozen are actually computed thanks to the symmetry;
he remaining are retrieved by a simple search in the hash table.
J. Morel et al. / Computers & Graphics 74 (2018) 44–55 49
Table 1
Evolution of the number of integral computations L o,o ′ and D t,o,o ′ to reconstruct
the Stanford bunny model according to the radii σ o of the basis functions (pa-
rameter a in Eq. (3) for an octree resolution of 1 mm).
a Non-zero entries in L Entries actually computed for
〈 ��σo , �σo ′ 〉 〈∇ · V, �σo
〉 1.5 4773,131 14 31
1.6 4781,877 16 38
1.7 4785,120 18 44
1.8 4785,766 19 47
1.9 4786,511 21 52
2.0 4786,627 22 54
4
f
t
a
s⎡⎢⎣
w
t
v
t
a
4
d
i
t
t
fi
t
s
(
t
k
w
b
t
χ
4
a
u
i
o
i
Fig. 3. Smoothing of the final surface in the overlap area by linearly combining the
implicit function solution of the Poisson equation χ and the implicit function of the
tubular model f .
Fig. 4. Reconstruction of the Stanford bunny surface with different methods: (Cen-
ter, white) the reference surface reconstructed by the “zipper” program and pro-
vided by Stanford University. (Right, green) The Poisson surface reconstructed by
Kazhdan’s algorithm implemented in Meshlab for an octree depth of 7. (Left, blue)
The Poisson surface reconstructed by our method for an octree depth of 7. (For in-
terpretation of the references to color in this figure legend, the reader is referred to
the web version of this article.)
a
m
5
p
i
T
N
.7. Injection of the tubular shape
To include the tubular shape defined in Section 3.2 , the implicit
unction f describing the tubular shape must be approximated in
he space Span { �σo } . We compute the function by interpolating f
t collocations { o c , o ∈ O} . To accomplish this task, we consider the
ystem:
. . .
A o,o ′
. . .
⎤
⎥ ⎦
⎡
⎢ ⎣
. . . y o . . .
⎤
⎥ ⎦
=
⎡
⎢ ⎣
. . . f ( o
′ c )
. . .
⎤
⎥ ⎦
(14)
here A o,o ′ = 〈 �σo , �σo ′ 〉 .
According to the properties of Wendland’s CSRBF �σo , this ma-
rix is self-adjoint and positive definite. Thus, the system is in-
erted by using the LDL T Cholesky decomposition implemented in
he Eigen library.
Finally, we obtain the linear decomposition of f in Span { �σo }s:
f ( q ) =
∑
o∈O occluded
y o · �σo (‖ q − o c ‖ ) . (15)
.8. Smoothing between visible and occluded areas
In most cases, the transition between χ , the implicit function
efined in the visible areas, and f , the implicit function defined
n the occluded areas, is handled by the smoothing characteris-
ic of Wendland’s CSRBF. Nevertheless, for complex tree shapes,
he transition might be too rough. To overcome this issue, we de-
ne an overlap portion of space, whose size Θmax is defined by
he user, in which we linearly mix the implicit functions. In this
ub-part of O v isible , called O ov erlap , we have Poisson reconstruction
∑
o x o · �σo ) and a priori model ( ∑
o y o · �σo ). Thus, we consider
he implicit function k defined as:
( q ) =
∑
o∈O ov erlap
[ αo · x o + (1 − αo ) · y o ] · �σo (‖ q − o c ‖ ) , (16)
here αo = 1 − Θo Θmax
is the coefficient of the linear interpolation
etween x o and y o according to the angular position Θo of the cen-
er of o in O ov erlap .
Finally, our resulting implicit function h is expressed as: h =+ k + f, as illustrated in Fig. 3 .
.9. Surface extraction
Following from our modeling choices (namely, Wendland’s RBF
nd quadrics), the implicit function f is C 2 . Hence, it is possible to
se a polygonization algorithm to represent the zero-level set of
mplicit function h . In the space defined by the union of spheres
f radius σ o , h is discretized into a scalar field, whose resolution
s set by the user. From this scalar field, the polygonizer computes
set of triangles approximating the implicit surface h by using a
arching-cube-like algorithm [45] .
. Complexity
In the visible areas, the scheme transforms the integral on the
oints into an integral on octree cells by distributing each point
nfluence in the neighboring cells through the αo, p coefficients.
he complexity thus mainly arises from construction of the sparse
× N matrix, where N is the octree size and the resolution of the
50 J. Morel et al. / Computers & Graphics 74 (2018) 44–55
Fig. 5. Mean Hausdorff distance between the computed mesh and the reference for
the three objects considered.
Fig. 6. Insight of the results obtained on Quasimoto.
Fig. 7. Insight of the results obtained on the gargoyle.
Fig. 8. Insight of the results obtained on the anchor.
e
s
6
i
a
t
d
m
i
associated linear system. Owing to the symmetries presented in
Appendix A , the matrix construction involves at most one hundred
integrals.
In the occluded areas, the complexity originates from the ex-
pression of the a priori model in the CSRBF basis. The main op-
ration is also the inversion of a symmetric and positive definite
parse matrix.
. Results and validation
The quality of our algorithm is assessed based on three exper-
ments: in Section 6.1 , we compare the surface generated by our
pproach to the Poisson surface reconstruction found in the litera-
ure, namely, the un-screened version of the Poisson reconstruction
eveloped in [32] . Then, in Section 6.2 , we compare how the two
ethods address occlusions. Finally, in Section 6.3 , we analyze the
mprovement to tree modeling.
J. Morel et al. / Computers & Graphics 74 (2018) 44–55 51
Fig. 9. Result of classic Poisson surface reconstruction (green mesh) and our approach (blue mesh) on a simulated point cloud (red points) that features an angular occlusion
of 10 °, 15 °, 20 °, 90 ° and 180 °. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
6
s
b
s
d
m
t
F
s
q
m
d
t
m
s
e
F
c
d
a
m
i
6
a
l
e
t
n
o
a
1
Fig. 10. Mean Hausdorff distance between the computed mesh and the input point
cloud for the two methods on the four trees.
a
t
p
c
p
l
s
a
6
e
.1. Comparison without occlusion
To test the robustness of our version of Poisson surface recon-
truction, we compare the surface we generate to the one created
y Kazhdan, which acts as a reference in this field of research. We
olved the Poisson problem on a non-occluded well-known stan-
ard model (the Stanford bunny) and directly produced a surface
odel without injecting any shape in the solution, i.e. , we ran only
he part of the algorithm described in Section 4.4, 4.5 and 4.9 .
ig. 4 presents the reconstructed surfaces of the bunny.
Then, we estimated the distances to the reference for the un-
creened Poisson surface reconstruction and for our method. This
uality assessment was performed by comparing the reference
esh model with the outputs of the algorithms using the Haus-
orff distance implementation available in Meshlab. As an addi-
ional validation step, we evaluate the reconstruction force of our
ethod on the data set provided by Berger et al. [4] . Among the
imulated point clouds available in this benchmark, we consid-
red the ones describing the anchor, the gargoyle, and Quasimoto.
or each of these objects, surfaces were reconstructed on 16 point
louds, for an octree resolution of 0.5 cm. Fig. 5 presents the Haus-
orff distances computed between the surfaces produced by our
lgorithm and the reference, which is also provided in the bench-
ark. Figs. 6–8 gives an insight of the reconstructed surfaces qual-
ty.
.2. Occlusion management
The development of our method ensues from the need to man-
ge holes in surface reconstruction from 3D point samples col-
ected in forests. Indeed, the state-of-the-art approaches in the lit-
rature struggle to address holes. In particular, when considering
rees described by single TLS scans, only the bark facing the scan-
er is described by the point samples: the entire back side remains
ccluded. To compare our approach to traditional PSR in this situ-
tion, we generated point clouds distributed on a cylinder with a
meter radius and a decreasing angular distribution to simulate
widening occlusion. Visual assessment of the surface reconstruc-
ion on the occluded cylinders ( Fig. 9 ) demonstrated how our ap-
roach can handle data gaps over 50% of the surface owing to the
ompletion of the shape by the a priori model. In this case, the
oint samples are distributed only on half of the cylinder, simi-
ar to the conditions of a single scan in a forest. On the opposite
ide, traditional PSR shows shape inconsistencies starting at 15 °nd fails to close the shape when the occluded areas exceed 20 °.
.3. Modeling of trees
The following step of our validation focuses on testing the
fficiency of our reconstruction approach with respect to the
52 J. Morel et al. / Computers & Graphics 74 (2018) 44–55
Fig. 11. Normalized difference in volume between the estimate and the reference
for the two methods tested on the four trees.
Table 2
Hausdorff distance statistics (mm) for the reconstruc-
tion of the Stanford bunny.
Min Max Mean RMS
Un-screened PSR 0.00 2.17 0.21 0.32
CSRBF PSR 0.00 1.74 0.22 0.30
Fig. 12. Illustration of the reconstructed surface recovered from the point cloud de-
scribing the tree without leaves for a Terminalia superba (in blue). The complete
point cloud is plotted as an overlay on the scene. (For interpretation of the refer-
ences to color in this figure legend, the reader is referred to the web version of this
article.)
m
m
s
t
s
o
o
s
g
Q
3
t
n
t
l
c
t
s
g
d
f
b
t
t
a
v
u
t
S
Q
p
a
i
o
b
a
cylindroid fitting method proposed in [44] . To that end, portions of
real trees were used to generate a reference model and associated
volume. The efficiency of the method was assessed by estimating
the volumes and computing the distances between the input point
cloud and the reconstructed surface mesh. We chose four portions
of real trees, whose surfaces and volumes were previously esti-
mated, to serve as a reference. On these four datasets, we ran a TLS
simulator to generate several single-view scans from varying points
of view. To artificially increase the size of our testing sample, we
generated a set of single-point-of-view scans by setting up virtual
scan positions that were evenly distributed around the reference
mesh. The simulated point clouds were used to compute the tree
volume following two approaches: (1) a method based on cylin-
drical quadrics and RBF and (2) our Poisson surface reconstruction
method based on CSRBF. Sixteen simulated point clouds were pro-
duced for each tree, resulting in a validation set of a total of 64
computations of volume for the two methods.
Fig. 12 gives an insight of the level of details restored by our
reconstruction, especially in the branches and on the butressesses.
It highlights the improvement on tree modeling induced by the use
of continuous surface.
7. Discussion
First, in order to demonstrate the generality of our Poisson re-
construction scheme, we compared our method, used without any
a priori information, to the un-screened Poisson surface recon-
struction that is well-known in the computer graphics field of sur-
face reconstruction. Despite the longer computation time required
to compute the matrix terms due to the Monte Carlo integration,
our method proved to be slightly more efficient to rebuild the sur-
face mesh of the Stanford bunny, with less noise, higher accuracy
and no holes. Indeed, the distance between the input point cloud
and the reconstructed surface presented in Table 2 has the same
agnitude for both methods (slightly lower for our approach). Our
ethod produces a slightly more detailed surface owing to the
maller radius of the kernels. The un-screened Poisson reconstruc-
ion surface produces a smoother and less-detailed surface for the
ame octree refinement. As a second validation step without pri-
rs, we estimated the difference in between our reconstructions
n the simulated scans proposed in the benchmark [4] and the
upplied references. At a 5 mm resolution, our method showed
ood results on the rounded objects that are the gragoyle and
uasimoto, with errors close to 2 mm. Errors increased to almost
mm on the more angular object, the anchor, presenting some-
imes artifacts on the sharp edges. Those unstabilities, where the
ormal direction tends to change abruptly, will be tackle by a bet-
er discretization induced by the multiscale approach discussed
ater on.
The next step of the validation was to evaluate the effect of oc-
lusion for simple tubular shapes on both methods. The results of
he experiment presented in Fig. 9 clearly show the limit of un-
creened Poisson surface reconstruction: for an angular occlusion
reater than 20 °, which corresponds to a hole of 35 cm on a cylin-
er of points with a 1 m radius, the method diverges (the sur-
ace represented in Fig. 9 is the extraction of the isosurface in the
ounding box of the cylinder, thus divergence appears as a hole in
he reconstructed surface). On the other hand, our approach solves
his occlusion issue up to 180 ° by guiding the reconstruction using
n a priori shape.
In the last part of the validation, we “measured” the added
alue of our method for modeling trees and computing their vol-
me. The model that our method produces was compared to
he tubular shape based on quadrics and CSRBF described in
ection 3.2 (which was itself 50% more accurate than standard
SM approaches). Overall, the proposed Poisson CSRBF based ap-
roach provided an improved approximation of the object shape
nd volume. The method is particularly well suited to reconstruct
rregular shapes and is thus more flexible to handle the diversity
f shapes found in natural environments.
The method is particularly useful for reconstructing crown
ranches whose shapes can be distorted due to gravity, as well
s tree buttresses, which remain difficult to model and could
J. Morel et al. / Computers & Graphics 74 (2018) 44–55 53
Fig. 13. Close-up of the reconstructed surfaces of a slice of points distributed on the lower part of a tree presenting buttresses from two points of view: in the upper part,
the input point cloud with deleted points in blue, which artificially create occlusion, and the remaining points in green; in the middle part, the surface reconstructed with
the quadrics and CSRBF method; in the lower part, the surface reconstructed with our Poisson reconstruction method based on CSRBF. (For interpretation of the references
to color in this figure legend, the reader is referred to the web version of this article.)
c
F
f
b
o
p
l
o
s
m
d
t
f
a
o
c
l
t
t
T
i
d
w
r
l
t
b
t
l
8
p
c
s
c
c
v
a
q
i
s
e
p
a
p
i
e
A
ontain a significant portion of the tree volume and biomass.
ig. 13 presents an example of surface reconstruction in an un-
avorable case of a point cloud distributed on the buttress at the
ase of a tree. While the tubular shape struggles to fit the points,
ur method correctly approximates the shape of the buttress while
reserving the tubular shape in the occlusion. Figs. 10–11 high-
ights the improvement in tree modeling from the tubular shape to
ur Poisson surface reconstruction based on CSRBF. For the whole
et of trees used in the validation, the tubular shape model re-
ains greater than 1.5 cm on average. Our method reduces this
istance to an average of 6 mm. The remaining distance between
he model and the points is due to the smoothing of the implicit
unction described in Section 4.8 , where we set aside input points
nd the surface to handle complex tree shapes. Moreover, because
f the point-density differences, our results are slightly less ac-
urate on thinner trees, for instance, tree 4. These issues high-
ight several ways to improve the method. First, we will improve
he occlusion detection to more accurately merge the surface from
he resolution of the Poisson equation to the shape known a priori .
hen, we plan to improve the combination of the models produced
n the visible and occluded areas by defining a feedback loop to
eform the a priori model by checking the local curvature. Finally,
e intend to implement a multiscale approach to Poisson surface
econstruction to better handle variations in points density. By fol-
owing the framework defined in [46] where the matrices are par-
ially recomputed at each refinement iteration, we also intend to
uild an adaptive solver for our Poisson equation by taking advan-
fage of the refinability of the function space. This process will also
ead to better memory management and faster computation.
. Conclusion
Our original Poisson surface reconstruction based on CSRBF
roved to be efficient to reconstruct surfaces from dense point
louds. ,Moreover, using this function basis, we can model any
mooth prior. Hence, for tubular shapes, the method addresses oc-
lusion by integrating an a priori model of the shape to reconstruct
losed surface models. For scanned trees,we thus improve both
olume (ie. biomass) and shape estimation but the method actu-
lly applies to any quasi tubular shape (such data actually arise fre-
uently in archeological, urban of natural data). In future work, we
ntend to enhance the method to handle even higher point-density
hifts by following a multiscale approach, which will also accel-
rate the resolution of the Poisson equation. We also plan to im-
rove the control of transitions between the visible and occluded
reas by adding feedback from the Poisson reconstruction to the a
riori model. Moreover, as any smooth prior can be expressed us-
ng proper collocations for CSRBF, we are currently working on the
fficient integration of general priors for occluded areas.
cknowledgments
The authors would like to express their thanks to the Stan-
ord Scanning Repository for their generosity in distributing their
54 J. Morel et al. / Computers & Graphics 74 (2018) 44–55
〈
w
∀
A
(
o
b
L
I
fi
q
v
L
t
t
o
mt
f
p
t
L
L
A
p
L
a
(
∀A
t
o
3D models. The authors would also like to thank the French In-
stitute of Pondicherry for supporting the fellowship of Jules Morel,
Alexandre Piboule at ONF (Office National des Forêts, France) to as-
sist in the development of the Computree plugin and Nicolas Bar-
bier at UMR AMAP (Botanique et Modélisation de l’architecture des
plantes et des végétations, France) for providing test data. Those
unpublished data from Cameroon were collected in collaboration
with Alpicam company within the IRD project PPR FTH-AC Change-
ment globaux, biodiversité et santé en zone forestière dAfrique Cen-
trale . Cédric Véga was supported by the Laboratory of Excellence
for Advanced Research on the Biology of Tree and Forest Ecosys-
tems (ARBRE) and the European Unions Horizon 2020 research and
innovation programme under grant agreement No 633464 (DIA-
BOLO).
Appendix A. Optimization
A1. Evaluation of the integral terms
Let us first compute the coefficients of the matrix L , that is,
〈 ��σo , �σo ′ 〉 . Our basis functions have been chosen to be C 2 ;
therefore, by integrating the compactly supported functions byparts, we obtain
∫ f ′′ . g = − ∫
f ′ . g ′ . Thus, we can express the
terms of the matrix as:
〈 ��σo , �σo 〉 = −⟨∂�σo
∂x , ∂�σo
∂x
⟩−
⟨∂�σo
∂y , ∂�σo
∂y
⟩−
⟨∂�σo
∂z , ∂�σo
∂z
⟩. (A.1)
As noted in Section 4.3 , we use Wendland’s CSRBF to
define our basis, that is, for q (x, y, z) ∈ R
3 , �σo ( q ) =(1 − ‖ q −o c ‖
σo
)4
+
(1 + 4 ‖ q −o c ‖
σo
). The derivative along the first coordi-
nate thus gives:
∂�σo
∂x = − 20
σ 2 o
(1 − ‖ q − o c ‖
σo
)3
+ ( q − o c ) x , (A.2)
where ( v ) w
denotes the coordinate of a vector v along the w axis
(with w, x , y and z ). Let us set q o = q − o c and q o ′ = q − o
′ c . The combination of Eqs
(A.1) and (A.2) gives:
〈 ��σo , �σo ′ 〉 = −
(400
σ 2 o .σ
2 o ′
)∫
(1 − ‖ q o ‖
σo
)3
+
(1 − ‖ q o ′ ‖
σ ′ o
)3
+ 〈 q o , q o ′ 〉 d q .
(A.3)
We then simplify Eq. (A.3) as:
〈 ��σo , �σo ′ 〉 = −
(400
σ 2 o .σ
2 o ′
)L o,o ′ , (A.4)
where
L o,o ′ =
∫
(1 − ‖ q o ‖
σo
)3
+
(1 − ‖ q o ′ ‖
σ ′ o
)3
+ 〈 q o , q o ′ 〉 . d q . (A.5)
Following the same process, we express the projection of the
divergence of V on our base of functions by:
〈∇ · V(q ) , �σo ′ 〉 =
∑
p i ∈P
∑
o∈O αo, p i
∫
∇ · [�σo (‖ q o ‖ ) · n i ]�σo ′ (‖ q o ′ ‖ ) . d q , (A.6)
where
∇ . [�σo (‖ q o ‖ ) · n i ] = 〈∇�σo
(‖ q o ‖ ) , n i 〉
= −(
20
σ 2 o
)(1 − ‖ q o ‖
σo
)3
+ 〈 q o , n i 〉 . (A.7)
Finally, we rewrite Eq. (A.6) as:
∇ · V(q ) , �σo 〉 =
∑
p i ∈P
∑
o∈O αo, p i
(−20
σ 2 o
) ∑
t∈{ x,y,z} D t,o,o ( n i ) t d,
(A.8)
here
t ∈ { x, y, z} , D t,o,o ′ =
∫
(1 − ‖ q o ‖
σo
)3
+ �σo ′ (‖ q o ′ ‖ ) ( q o ) t d q .
(A.9)
2. Study of symmetries
To accelerate and simplify the computation of the terms of Eq.
12) , we study the symmetries of L o,o ′ and D t,o,o ′ and show that
nly a few of these terms need to be computed.
Symmetries for the computation of L o,o ′ . Let us define the function f as f ( x ) = (1 − x ) 3 + . Eq. (A.4) then
ecomes:
o,o ′ =
∫
f
(‖ q − o c ‖
σo
)f
(‖ q − o
′ c ‖
σo ′
)〈 q − o c , q − o
′ c 〉 d q . (A.10)
n this section, we assume σ o and σo ′ are given and fixed. Let usrst show that L o,o ′ depends only on vector o c − o
′ c . We set p =
− o c (and let t be the corresponding translation); by change ofariables in Eq. (A.10) , we obtain:
o,o ′ =
∫
f
(‖ p ‖ σo
)f
(‖ p + o c − o ′ c ‖ σo ′
)〈 p , p + o c − o ′ c 〉 | det (J t ) | d p , (A.11)
where det (J t ) is the determinant of the Jacobian of the transla-
ion t , whose value is 1. As a consequence, L o,o ′ depends only on
he vector o c − o
′ c and radii σ o and σo ′ .
Let us now proceed a step further and show that L o,o ′ depends
n only ‖ o c − o
′ c ‖ . Let us consider the rotation r : q → R q , whose
atrix in the canonic basis is R , and send the vector ‖ o c − o
′ c ‖ e 1
o o c − o
′ c along the x axis, with e 1 = (1 , 0 , 0) . As f is a radial basis
unction, for any vector u , f ( u ) = f (‖R u ‖ ) . Moreover, the scalarroduct of two vectors is clearly invariant under simultaneous ro-ation. Hence, Eq. (A.11) becomes:
o,o ′ =
∫
f
(‖R p ‖ σo
)f
×(‖R p + ‖ o c − o ′ c ‖ e 1 ‖
σo ′
)〈R p , R p + ‖ o c − o ′ c ‖ e 1 〉 d p . (A.12)
Now, under the change of variables u = R p , we obtain:
o,o ′ =
∫
f
(‖ u ‖
σo
)f
(‖ u + ‖ o c − o
′ c ‖ e 1 ‖
σo ′
)
×〈 u , u + ‖ o c − o
′ c ‖ e 1 〉 | det (J r ) | d p . (A.13)
s the rotation R is isometric, it maintains the norms and dotroducts. Therefore, by using | det (J r ) | = 1 , Eq. (A.13) becomes:
o,o ′ =
∫
f
(‖ u ‖ σo
)f
(‖ u + ‖ o c − o ′ c ‖ e 1 ‖ σo ′
)〈 u , u + ‖ o c − o ′ c ‖ e 1 〉 d p . (A.14)
Therefore, Eq. (A.14) proves that L o,o ′ depends only on σ o , σo ′ nd ‖ o c − o
′ c ‖ .
Symmetries for the computation of D t,o,o ′ , t ∈ { x, y, z} The same principle can be used to study the symmetries of Eq.
A.9) :
t ∈ { x, y, z} , D t,o,o ′ =
∫
f
(‖ q ‖
σo
)�σo ′ (‖ q + o c − o
′ c ‖ ) ( q ) t d p .
lthough the results are not shown here, it can be demonstrated
hat D t,o,o ′ depends only on σ o , σo ′ , ‖ o c − o
′ c ‖ and the t -coordinate
f o c − o
′ c .
J. Morel et al. / Computers & Graphics 74 (2018) 44–55 55
A
g
C
t
d
s
o
o
w
o
k
p
i
t
R
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
3. Resolution
The integrals L o,o ′ and D t,o,o ′ are computed with the MISER al-
orithm of Press and Farrar [47] implemented in GSL. This Monte
arlo technique reduces the overall integration error by concen-
rating the integration points in the regions of highest variance.
Moreover, the use of the invariance properties we just proved
rastically reduces the number of integrals to compute. Indeed, we
howed that the integrals in L o,o ′ depend on only σ o , σo ′ and ‖ o c −
′ c ‖ and that the integrals in D t,o,o ′ depend on only σ o , σo ′ , ‖ o c −
′ c ‖ and ( o c − o
′ c ) t .
In practice, we stored the values of the integrals in hash tables
hose hashes were computed from the variables σ o , σo ′ , ‖ o c −
′ c ‖ and the extra ( o c − o
′ c ) t for the integrals in D t,o,o ′ . Those hash
eys can be converted into integer values to avoid floating point
recision problems. For each couple o , o ′ , we thus compute the
ntegrals L o,o ′ and D t,o,o ′ only if the value is absent in the hash
able.
eferences
[1] Dassot M , Constant T , Fournier M . The use of terrestrial LiDAR technologyin forest science: application fields, benefits and challenges. Annal Forest Sci
2011:959–74 . [2] Berger M , Tagliasacchi A , Seversky L , Alliez P , Levine J , Sharf A , et al. State of
the art in surface reconstruction from point clouds 2014;1(1):161–85 . [3] Khatamian A , Arabnia HR . Survey on 3d surface reconstruction. J Inf Process
Syst 2016;12(3):338–57 . [4] Berger M, Levine JA, Nonato LG, Taubin G, Silva CT. A benchmark for surface
reconstruction. ACM Trans Graph 2013;32(2):20:1–20:17. doi: 10.1145/2451236.
2451246 . http://doi.acm.org/10.1145/2451236.2451246 . [5] Cox MG . The numerical evaluation of b-splines. IMA J Appl Math
1972;10(2):134–49 . [6] De Boor C . On calculating with b-splines. J Approx Theory 1972;6(1):50–62 .
[7] Versprille KJ . Computer-aided design applications of the rational b-spline ap-proximation form.. Electric Eng Comput Sci Dissert 1975:1993–2002 .
[8] Dokken T , Lyche T , Pettersen KF . Polynomial splines over locally refined box–
partitions. Comput Aided Geom Des 2013;30(3):331–56 . [9] Bernardini F , Mittleman HRJ , Rushmeier H . The ball-pivoting algorithm for sur-
face reconstruction.. Trans Vis Comput Graph 1999:349–59 . [10] Boissonnat J-D . Geometric structures for three-dimensional shape representa-
tion. ACM Trans Graph 1984;3(4):266–86 . [11] Edelsbrunner H , Mücke EP . Three-dimensional alpha shapes. ACM Trans Graph
1994;13(1):43–72 .
[12] Amenta N , Choi S , Kolluri RK . The power crust. In: Proceedings of the sixthACM symposium on Solid modeling and applications; 2001. p. 249–66 .
[13] Lorensen WE , Cline HE . Marching cubes: a high resolution 3d surface construc-tion algorithm. In: Proceedings of the ACM SIGGRAPH computer graphics, vol.
21. ACM; 1987. p. 163–9 . [14] Newman TS , Yi H . A survey of the marching cubes algorithm. Comput Graph
2006;30(5):854–79 .
[15] Hoppe H , DeRose T , Duchamp T , McDonald J , Stuetzle W . Surface reconstruc-tion from unorganized points. In: Proceedings of the ACM SIGGRAPH 1992;
1992. p. 71–8 . [16] Lancaster P , Salkauskas K . Surfaces generated by moving least squares meth-
ods,. Math Comput 1981:141–58 . [17] Cheng Z-Q , Wang Y-Z , Li B , Xu K , Dang G , Jin S-Y . A survey of methods for
moving least squares surfaces.. In: Proceedings of the volume graphics; 2008.
p. 9–23 . [18] Carr JC , Beatson RK , Cherrie JB , Mitchell TJ , Fright WR , McCallum BC , et al. Re-
construction and representation of 3d objects with radial basis functions. In:Pocock L, editor. Proceedings of the SIGGRAPH. ACM; 2001. p. 67–76 .
[19] Buhmann MD . Radial basis functions: theory and implementations. Cambridgemonographs on applied and computational mathematics. Cambridge Univer-
sity Press; 2003. p. 307–18 .
20] Fasshauer G . Meshfree methods. Handbook of theoretical and computationalnanotechnology, 27. American Scientific Publishers; 2006. p. 33–97 .
[21] Carr JC , Fright WR , Beatson RK . Surface interpolation with radial basis func-tions for medical imaging. IEEE Trans Med Imag 1997;16(1):96–107 .
22] Savchenko VV , Pasko AA , Okunev OG , Kunii TL . Function representation ofsolids reconstructed from scattered surface points and contours. In: Proceed-
ings of the computer graphics forum, vol. 14. Wiley Online Library; 1995.p. 181–8 .
23] Turk G , O’brien JF . Variational implicit surfaces. Tech. Rep.. Georgia Institute ofTechnology; 1999 .
[24] Turk G , O’brien JF . Shape transformation using variational implicit functions.In: Proceedings of the ACM SIGGRAPH 2005 Courses. ACM; 2005. p. 13 .
25] Wendland H . Piecewise polynomial, positive definite and compactly supportedradial basis functions of minimal degree. Adv Comput Math 1995;4(1):389–96 .
26] Wendland H . Fast evaluation of radial basis functions: methods based on par-tition of unity. Approximation theory X: Wavelets, splines, and applications.
Citeseer; 2002. p. 473–83 .
[27] Ohtake Y , Belyaev A , Alexa M , Turk G , Seidel H-P . Multi-level partition of unityimplicits. ACM Trans Graph 2003a:463–70 .
28] Ohtake Y, Belyaev A, Seidel H-P. A multi-scale approach to 3d scattered datainterpolation with compactly supported basis functions. In: Proceedings of the
shape modeling international 2003, SMI ’03. Washington, DC, USA: IEEE Com-puter Society; 2003b. p. 292. ISBN 0-7695-1909-1 . http://dl.acm.org/citation.
cfm?id=829510.830307 .
29] Tobor I , Reuter P , Schlick C . Multiresolution reconstruction of implicit surfaceswith attributes from large unorganized point sets. In: Proceedings of the Shape
modeling international. Italy; 2004. p. 193 . 30] Wendland H . Computational aspects of radial basis function approximation.
Topics in multivariate approximation and interpolation. Studies in Computa-tional Mathematics, vol 12. Elsevier; 2006. p. 231–56 .
[31] Morse B , Yoo TS , Rheigans P , Chen DT , Subramanian KR . Interpolating implicit
surfaces from scattered surface data using compactly supported radial basisfunctions. In: Proceedings of the international conference on shape modeling
and applications; 2001. p. 89–98 . 32] Kazhdan M , Bolitho M , Hoppe H . Poisson surface reconstruction. In: Proceed-
ings of the symposium on geometry processing; 2006. p. 61–70 . [33] Côté J-F , Widlowski J-L , Fournier RA , Verstraete MM . The structural and ra-
diative consistency of three-dimensional tree reconstructions from terrestrial
lidar.. Remote Sens Environ 2009:1067–81 . 34] Raumonen P , Kaasalainen M , Akerblom M , Kaasalainen S , Kaartinen H , Vas-
taranta M , et al. Fast automatic precision tree models from terrestrial laserscanner data. Remote Sens 2013;5:491–520 .
[35] Hackenberg J , Spiecker H , Calders K , Disney M , Raumonen P . Simpletreeanefficient open source tool to build tree models from TLS clouds. Forests
2015;6(11):4245–94 .
36] Raumonen P , Casella E , Calders K , Murphy S , Akerblom M , Kaasalainen M . Mas-sive-scale tree modelling from TLS data. ISPRS Annal Photogram Remote Sens
Spat Inf Sci 2015;2(3):189 . [37] Tao S , Wu F , Guo Q , Wang Y , Li W , Xue B , et al. Segmenting tree crowns from
terrestrial and mobile LiDAR data by exploring ecological theories. ISPRS J Pho-togram Remote Sens 2015;110:66–76 .
38] Ravaglia J , Bac A , R F . Tree stem reconstruction from terrestrial laser scanner
point cloud using hough transform and open active contours.. In: Proceedingsof SilviLaser; 2015. p. 1067–81 .
39] Ravaglia J , Bac A , Fournier RA . Extraction of tubular shapes from dense pointclouds and application to tree reconstruction from laser scanned data. Comput
Graph 2017;66:23–33 . 40] Tagliasacchi A, Zhang H, Cohen-Or D. Curve skeleton extraction from in-
complete point cloud. ACM Trans Graph 2009;28(3):71:1–71:9. doi: 10.1145/1531326.1531377 . http://doi.acm.org/10.1145/1531326.1531377 .
[41] Chave J , Réjou-Méchain M , Búrquez A , Chidumayo E , Colgan MS , Delitti WB ,
et al. Improved allometric models to estimate the aboveground biomass oftropical trees. Global Change Biol 2014;20:3177–90 .
42] Pauly M, Mitra NJ, Giesen J, Gross M, Guibas LJ. Example-based 3d scan com-pletion. In: Proceedings of the third Eurographics symposium on geometry
processing SGP’05. Aire-la-Ville, Switzerland, Switzerland: Eurographics Asso-ciation; 2005. ISBN 3-905673-24-X . http://dl.acm.org/citation.cfm?id=1281920.
1281925 .
43] Gal R, Shamir A, Hassner T, Pauly M, Cohen-Or D. Surface reconstruction us-ing local shape priors. In: Proceedings of the fifth Eurographics symposium
on geometry processing SGP’07. Aire-la-Ville, Switzerland, Switzerland: Euro-graphics Association; 2007. p. 253–62. ISBN 978-3-905673-46-3 . http://dl.acm.
org/citation.cfm?id=1281991.1282025 . 44] Morel J , Bac A , Véga C . Computation of tree volume from terrestrial LiDAR data.
In: The ninth symposium on mobile mapping technology, MMT 2015. Sydney,
Australia: UNSW; 2015 . 45] Bloomenthal J . An implicit surface polygonizer. Graph Gems IV 1994;4:324 .
46] Grinspun E , Krysl P , Schröder P . Charms: a simple framework for adaptive sim-ulation. ACM Trans Graph 2002;21(3):281–90 .
[47] Press WH , Farrar GR . Recursive stratified sampling for multidimensional montecarlo integration. Comput Phys 1990;4(2):190–5 .