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remote sensing
Article
Combined Rule-Based and Hypothesis-Based Method for
Building Model Reconstruction from Photogrammetric
Point Clouds
Linfu Xie 1,2,3, Han Hu 4, Qing Zhu 4, Xiaoming Li 1, Shengjun Tang 1, You Li 1, Renzhong Guo 1, Yeting Zhang 2
and Weixi Wang 1,*
Citation: Xie, L.; Hu, H.; Zhu, Q.; Li,
X.; Tang, S.; Li, Y.; Guo, R.; Zhang, Y.;
Wang, W. Combined Rule-Based and
Hypothesis-Based Method for
Building Model Reconstruction from
Photogrammetric Point Clouds.
Remote Sens. 2021,13, 1107. https://
doi.org/10.3390/rs13061107
Academic Editor: Ben Gorte
Received: 19 January 2021
Accepted: 11 March 2021
Published: 14 March 2021
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1Research Institute for Smart Cities, School of Architecture and Urban Planning, Shenzhen University & Key
Laboratory of Urban Land Resources Monitoring and Simulation, MNR & Guangdong Key Laboratory of
Urban Informatics & Shenzhen Key Laboratory of Spatial Smart Sensing and Services,
Shenzhen 518060, China; linfuxie@szu.edu.cn (L.X.); lixming@szu.edu.cn (X.L.);
shengjuntang@szu.edu.cn (S.T.); liyou@szu.edu.cn (Y.L.); guorz@szu.edu.cn (R.G.)
2State Key Laboratory of Information Engineering in Surveying Mapping and Remote Sensing,
Wuhan University, Wuhan 430079, China; zhangyeting@263.net
3Department of Land Surveying & Geo-Informatics, The Hong Kong Polytechnic University, Hung Hom,
Kowloon 999077, Hong Kong
4Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University,
Chengdu 611756, China; han.hu@swjtu.edu.cn (H.H.); zhuq66@263.net (Q.Z.)
*Correspondence: wangwx@szu.edu.cn; Tel.: +86-13590317811
Abstract:
Three-dimensional (3D) building models play an important role in digital cities and have
numerous potential applications in environmental studies. In recent years, the photogrammetric
point clouds obtained by aerial oblique images have become a major source of data for 3D building
reconstruction. Aiming at reconstructing a 3D building model at Level of Detail (LoD) 2 and even
LoD3 with preferred geometry accuracy and affordable computation expense, in this paper, we
propose a novel method for the efficient reconstruction of building models from the photogrammetric
point clouds which combines the rule-based and the hypothesis-based method using a two-stage
topological recovery process. Given the point clouds of a single building, planar primitives and their
corresponding boundaries are extracted and regularized to obtain abstracted building counters. In
the first stage, we take advantage of the regularity and adjacency of the building counters to recover
parts of the topological relationships between different primitives. Three constraints, namely pairwise
constraint, triplet constraint, and nearby constraint, are utilized to form an initial reconstruction
with candidate faces in ambiguous areas. In the second stage, the topologies in ambiguous areas
are removed and reconstructed by solving an integer linear optimization problem based on the
initial constraints while considering data fitting degree. Experiments using real datasets reveal that
compared with state-of-the-art methods, the proposed method can efficiently reconstruct 3D building
models in seconds with the geometry accuracy in decimeter level.
Keywords: building models; 3D reconstruction; point clouds; photogrammetry
1. Introduction
Three-dimensional (3D) building models play an important role in constructing digital
cities and have numerous environmental applications in areas such as urban planning [
1
],
smart city [
2
], environmental analysis [
3
], and other civil engineering [
4
]. In recent years,
with the rapid development of aerial vehicles, cameras, and image-processing technologies,
aerial oblique images have become a major data source for 3D city modeling [
5
,
6
]. Due to
the time-consuming and labor-intensive nature of manual modeling processes, researchers
in the photogrammetry, computer vision, and graphic communities have developed auto-
matic building model reconstruction methods [
7
–
10
]. Due to the presence of occlusions in
Remote Sens. 2021,13, 1107. https://doi.org/10.3390/rs13061107 https://www.mdpi.com/journal/remotesensing
Remote Sens. 2021,13, 1107 2 of 23
complex city scenes and noise in forward-intersecting point clouds, some building features
(e.g., planes, edges) are degraded or even missed in the collected point clouds, which leads
to unreliable recovery of the topological relationships [11].
To address this problem, various data- and model-driven methods (or their combina-
tion) have been proposed in recent years and substantial improvements in quality have
been achieved [
9
,
12
–
14
]. However, for complex buildings with imperfect point coverage, it
is difficult to automatically and efficiently reconstruct building models with desired Level
of Detail (LoD) [
15
,
16
], hindering environmental simulation, spatial analysis, and other
model-based applications. To solve this problem, in this paper, a novel framework which
combines the rule-based and the hypothesis-based method is proposed for the efficient
reconstruction of high-quality polygonal building models. Starting with the point clouds
of a single building, the planar primitives and corresponding boundaries are extracted and
regularized to obtain abstracted building counters, followed by a two-stage reconstruction.
In the first stage, the regularity and adjacency of the building counters are used to recover
the topological relationship between different primitives and produce an initial reconstruc-
tion. In the second stage, the topologies of ambiguous areas are removed and reconstructed
by solving an integer linear optimization problem based on the initial reconstruction.
The major contributions of this paper are as follows:
(1)
A novel framework for 3D building reconstruction which combines the efficiency of
traditional rule-based methods and the integrity of recently developed hypothesis-
based methods.
(2)
A method for robust topology estimation that integrates the regularity and adjacency
relationships between building primitives in 3D.
(3)
An effective solution that enforces initial reconstruction results and constraints to
eliminate topological ambiguities.
The remainder of this paper is organized as follows. Section 2provides a brief re-view
of existing work on 3D building reconstruction from point clouds. In Section 3, the details
and key steps of the proposed approach are presented. The performance of the proposed
approach is evaluated in Section 4using real photogrammetric point clouds from aerial
oblique images. Discussion about the method and the experimental results are given in
Section 5. Finally, we draw our conclusions in Section 6.
2. Related Works
According to the City Geography Markup Language (CityGML) model format adopted
by the Open Geospatial Consortium, building models are divided into five LoDs, from
LoD0 to LoD4 [
17
]. With the rapid developments in point-cloud collection and processing
technologies, automatic building reconstruction methods for LoD0 and LoD1 building mod-
els are now relatively mature [
18
–
21
]. In past decades, researchers in the photogrammetry
and computer vision communities have expended great effort on the (semi-)automatic
reconstruction of LoD2 and even LoD3 building models [7,22–24].
Point clouds used for city reconstruction are almost obtained by Light Detection
and Ranging (LiDAR) technology [
25
–
28
] or photogrammetry [
5
–
7
]. LiDAR point clouds
usually have more precise coordinates and less noise compared with those generated by
image matching. But the density of point clouds obtained through the Structure-from-
Motion (SfM) and Multi-View Stereo (MVS) pipeline is higher in areas with sufficient
textures. In general, methods for reconstructing 3D building models from point clouds
can be categorized as data-driven [
2
,
7
,
29
], model-driven [
30
,
31
], or a combination of the
two (also called hybrid-driven methods) [
12
,
32
,
33
]. Comparisons about them could be
found in References [
34
,
35
]. As summarized in several previous works, model-driven
methods that adopt top-down strategies require pre-defined hypothetical models, which
hinders their application in free-form building reconstruction [
35
]. In contrast, data-driven
methods that require top-down approaches have the potential to reconstruct complicated
buildings. This kind of reconstruction pipeline normally includes three crucial steps:
primitive extraction, regularization, and topology estimation. Given a point set, linear
Remote Sens. 2021,13, 1107 3 of 23
primitives such as planes [
10
], cylinders, spheres, and line segments [
36
] are first extracted
using methods based on model-fitting (e.g., RANdom Sample Consensus (RANSAC) [
37
] or
Hough transform [
21
,
38
]), region-growing [
39
], or clustering [
40
]. Then, these primitives or
their boundaries are regularized using Manhattan hard or soft constraints [
41
–
43
]. Finally,
at their core, data-driven methods [
44
] involve estimation and refinement of the adjacency
relations between different primitives to construct final models without any topological
conflicts [
10
,
12
]. Moreover, the hybrid-driven methods combine the primitive extraction
step from the data-driven methods at the first step, then these primitives are used to form
building models with pre-defined combination solutions.
As CityGML LoD2 building models mainly concern rooftop structures, air-borne
laser scanning point clouds have become the major data source for those automatic recon-
struction methods because of their accurate altitude measurements and fewer top-view
occlusions [
22
,
45
,
46
]. By projecting 3D rooftops onto a two-dimensional (2D) horizontal
plane, the topological relations between rooftop primitives are estimated by detecting ridge
edges and jump edges [
8
,
23
,
33
], and these relations are maintained by a binary space parti-
tioning (BSP) tree [
23
], an adjacency matrix [
16
,
47
], or a roof topology graph (RTG) [
13
,
48
].
Then, the subsets of primitives are used to form building models based on pre-defined
rules such as graph edit operations [
13
], planar partitioning, and merging [
23
]. These are
the so-called rule-based methods, which could be either data-driven or hybrid-driven.
In practice, photogrammetric point clouds acquired through SfM and MVS pipelines are
inherently noisier than those collected by laser scanning technology [
49
]. Cameras mounted
on unmanned aerial vehicles are inevitably hampered by occluded areas during the image
collection process, especially on the lower parts of buildings [
50
], which leads to unreliable
geometric accuracy or missing data in the photogrammetric point clouds. Although some
previous works have reported impressive results in the automatic reconstruction LoD2
building models, they may not be suitable for LoD3 building reconstruction from pho-
togrammetric point clouds due to the inferior data quality and difficulty in representing
more complicated topological relations in real 3D space [13,23,47,51].
Recently, researchers have tried to convert the model reconstruction problem into
an optimal subset selection problem with hard constraints [
10
,
14
,
52
]. Given primitives
detected from original point clouds, the object space is first divided into several segments
to form a candidate pool. Then, different hypotheses about the building models are quan-
tized by energy functions that measure their data fitness and rule fitness, with additional
topological constraints, such as watertight. After that, by searching for the maximal (or
minimal) values of the object functions, segments in the candidate pool are labeled as
either selected or not to establish the final building models. These kinds of methods are
referred as hypothesis-based methods in this paper, which could be either data-driven
or hybrid-driven methods. One virtue of these kinds of methods is that they are robust
when data are partially missing. In addition, with help from integer linear programming,
manifold and watertight hard constraints can be embedded to avoid topological conflicts
in the output models [
10
]. Hence, these methods have the potential to reconstruct LoD3
building models that contain more detailed structures [7].
However, with tile views, building façades are visible in the photogrammetric point
clouds generated by aerial oblique images, which result in more planar primitives. The
direct adoption of hypothesis-based reconstruction methods in such scenes may result in
unreasonable computation costs when solving integer linear programming problems [
10
].
As noted by Wang et al. [
35
], global regularities between different primitives in buildings
may reveal their topological relations. If this information could be properly explored and
utilized to estimate building topologies, even partially, the recovered topological relations
could stabilize the solution to decrease artifact and accelerate the problem-solving process
by reducing the size of the candidate pool.
In order to utilize the intrinsic structure of buildings in architectural design to produce
building models with preferred geometry accuracy while eliminating topological conflicts
Remote Sens. 2021,13, 1107 4 of 23
efficiently, a two-stage building reconstruction method that combines the traditional rule-
based and the recently arisen hypothesis-based methods is proposed.
3. Method
3.1. Overview of the Proposed Approach
Starting from the photogrammetric point clouds of aerial oblique images for single
buildings, Figure 1shows the overall workflow of the proposed method. The photogram-
metric point clouds could be produced from images with sufficient overlaps by existing
SfM and MVS pipeline, and single buildings could be extracted from the point clouds
manually or clipped according to existing 2D footprints.
Figure 1. Overall workflow of the proposed method.
In the pre-processing steps, the planar primitives in the point clouds are extracted with
simple parallel and orthogonal constraints using the existing RANSAC-based methods.
Then, the extracted 3D planes which share the same normal orientations are grouped
together. For each plane group, 3D points are projected to 2D space by translating their
centroid to the origin point and rotating the normal of the plane to the positive direction
of the Z-axis, and the consecutive boundary points of each plane are extracted using
alpha-shapes. After that, the boundary points of an individual plane are simplified by
shifting them along their refined normal vector to resist noise, and then grouped into
piecewise smooth segments. Finally, the orientations of each segments in the same plane
group are softly regularized to be parallel (or perpendicular) with each other or the normal
orientation of the other group planes. So, the initial point clouds are abstracted by a set of
Remote Sens. 2021,13, 1107 5 of 23
planar polygons with mutual regularity. For details of the pre-processing step, please refer
to the work by Xie et al. [43].
Si=1,2,···n={Pi,πi,Polyi}(1)
In Equation (1), S
i
represents the detected planar segments (nis the total number of
segments), P
i
is the point set which belong to segment S
i
,
πi
is the regularized base plane
of Si, and Polyiis a regularized boundary polygon of Piin plane πi.
Then, a two-stage reconstruction method is implemented. In the first stage, the adja-
cency relations between different polygons are estimated on the basis of spatial consistency
and mutual regularity rules, even if only partial, to gain an initial reconstruction of the
3D polygonal building model. For areas not reconstructed in the first stage, inspired by
the work of Nan and Wonka [
10
], hypotheses are posed regarding the final model based
on the pairwise intersection in finite distances, followed by the selection of the optimal
combination of candidates by solving a binary linear programming problem.
3.2. Adjacency Detection between Multiple Primitives
In this stage, the robust adjacency relations between different polygons are recovered
in areas with sufficient data support. Specifically, as shown in Figure 2, two types of
topological relations are identified: (1) the intersection of two planar primitives, which
indicates an edge in the model, and (2) the intersection of three planar primitives, which
indicates a vertex in the model.
Figure 2.
Graphic illustration of the two types of topological relations. In polygons A,B, and C, if
edge (Va
1
,Va
2
) matches edge (Vc
1
,Vc
2
), then the polygon pair (A,C) is considered to be pairwise
adjacent. If vertices Va
1
,Vb
1
, and Vc
1
are matched with each other, then polygon triplet (A,B,C) is
considered to intersect at the common point of the three supporting planes.
Although photogrammetric point clouds can be noisy or partly missing, large pla-
nar structures that are well sampled are still reliable for fitting planes and recovering
boundaries. Similar to the work by Arikan et al. [
53
], vertex–vertex matches (VVMs) and
vertex–edge matches (VEMs) between two polygons are searched to determine their pair-
wise adjacency. To obtain robust estimations, the maximal search radius, as described by
Arikan et al. [
53
], should be relatively low (in this paper, it is set to be twice the average
point spacing).
Edge–edge matches are derived from VVMs and/or VEMs. In this work, the adjacency
between two non-parallel polygons, Poly
i
and Poly
j
, is verified by finding at least one pair
of edges from the two polygons that satisfy the following criteria:
(1)
The two edges are parallel or collinear.
(2)
Two VVMs, or one VVM and one VEM, or two VEMs are found for them.
Remote Sens. 2021,13, 1107 6 of 23
Note that, if more than two edges are found to satisfy the first criteria, all of the
possible combination pairs should be tested if they satisfy the second criteria of edge–edge
matches. Then, to detect plane triplets of interest, an undirected graph G(V,E), as shown
in Figure 3, is generated by setting each polygon as a vertex (the blue dots), with the
edge between two vertices indicating matching relationships of the two polygons. As the
intersection of three non-parallel polygons can be defined without ambiguity, the shortest
closed cycles are searched in the graph G(V,E) [
51
], and those with the shortest walk of the
three indicate an underlying intersection of a triplet of polygons if none are parallel (the
red triangles).
Figure 3.
Graphic illustration of polygons and their matching relations. The blue dots represent
polygons while the black lines between two dots indicate that they are matched with each other. The
red triangles indicate the intersections of three non-parallel polygons.
3.3. Building Model Reconstruction with Initial Topology Constraints
In the previous stage, the adjacency relations between different planar primitives
were partially recovered. A set of adjacent planar polygon pairs and a set of intersecting
non-parallel polygon triplets are obtained. So, the topology relationship between different
planar polygons is divide into two parts: the confident part and the ambiguous part.
In this stage, we incorporate these relations to produce candidate faces and constraints
for generating the final building model. Unlike previous work which generates candidate
faces by simply intersecting detected planar segments, we embed the recovered topological
relations in this process to (1) generate more purposeful candidate faces, and (2) reduce the
size of unknown parameters in the energy function. After that, the recovered topology is
used to guide the candidate selection process as soft energy functions and hard constraints
to obtain better reconstructed models with less artifact and in remarkable running time.
3.3.1. Candidate Deduction with Topological and Spatial Hints
Given a set of boundary polygons with base planes, use of their pairwise intersections
is a simple strategy for generating redundant candidate faces. However, the drawback
is that the number of candidate faces increases substantially with increases in the initial
number of detected polygons. In addition, artifactual faces that are obviously invalid may
survive in the reconstructed models. Instead of pairwise intersection of detected planar
segments purely, in this paper, we conduct the candidate generation process based on three
assumptions:
(1)
For adjacent polygon pairs, the candidate faces in each polygon plane might be
bounded by their intersecting lines.
(2)
For adjacent non-parallel polygon triplets, the candidate faces in each polygon plane
might be bounded by the two other intersecting planes.
(3)
The potential intersection points of different polygons might not be far away from
their boundaries.
Remote Sens. 2021,13, 1107 7 of 23
The overall workflow of this process is shown in Figure 4. The planar primitives are
first pairwise intersected with each other within the scope of the enlarged bounding box to
form over-redundant candidate faces. Then, the candidate faces in each plane are processed
separately to eliminate part of them according to the detected topological relations in the
previous steps and the spatial hints. After that, the union set of invalid candidates are
removed. Finally, the remaining candidates in all planes are merged and those that do not
satisfy the two-manifold rules are treated as outliers.
Figure 4. Overall workflow of the proposed candidate face deduction method.
To swiftly eliminate invalid faces in a plane,
π
, we divide the candidate faces, F
P
,
in
π
into three categories, which are marked by green, orange, and gray solid circles
in
Figure 5
. The first category (F
PCover
) includes faces that share common areas with the
detected primitives in the 2D space, these faces are highly confident candidates. The second
category (F
PNear
) includes faces which are not far from the covering area of the detected
primitives and are treated as potential candidates. The rest of the faces (F
PInvalid
) in this
plane are labeled as invalid and should be rejected.
FP=nFCover
P,FNear
P,FInv alid
Po(2)
Remote Sens. 2021,13, 1107 8 of 23
Figure 5.
Illustration of the three constraints used to reduce the number of candidate faces. (
a
): Pairwise Constraint; (
b
):
Triplet Constraint; (
c
): Nearby Constraint. The red lines depict a simplified boundary polygon Ain its supporting plane
πA
,
and the blue lines which separate the plane into several candidate faces are the intersection of other planes in this plane.
The green, orange, and gray solid circles indicate the status of the occupied faces as Cover, Near, and Invalid, respectively.
Based on the above assumptions, three types of constraints are used to classify the
candidate faces obtained by brutal intersection, namely the pairwise constraint (PC), triplet
constraint (TC), and nearby constraint (NC).
Pairwise Constraint: All matched polygon pairs, in which the current one involved
should go through this process. As shown in Figure 5a, consider a polygon Awith support-
ing plane
πA
. If a polygon pair (A,B) is matched in the previous steps, e
A
is the matched
edge in polygon A, and the two supporting planes intersect at line l
AB
. Theoretically, if
the two vertices of e
A
are both on the convex hull of polygon A, the candidate faces in
πA
should be bounded by l
AB
. Then, the candidate faces in
πA
are labeled as covered (green
dot) and invalid (gray dot) respectively, according to their intersection relationship with
polygon A. Note that the same process would be done for polygon Bin its supporting plane
πB
. If both polygon Aand polygon Bare bounded by the intersection line l
AB
, a sharp edge
is implicitly reconstructed in the building model.
Triplet Constraint: All matched polygon triplets, in which the current one involved
should go through this process. As shown in Figure 5b, consider a polygon Awith
supporting planes
πA
, and a detected polygon triplet (A,B,C), whereby the projections
of supporting planes of polygon Band Cin
πA
are two lines, l
AB
and l
AC
. Thus, the
intersection of l
AB
and l
AC
divides
πA
into four parts. Ideally, if polygon Ais located in
only one of the four parts, we could bound the candidate faces in
πA
by the boundaries
of the intersecting lines. In practice, due to the presence of noise in point clouds and the
errors accumulated in the processing steps, we consider polygon Ato be in only one of the
four parts when the area percentage of Ain this part is larger than a given threshold (e.g.,
95%). Then, the candidate faces in
πA
are restricted to being in this part only. The same
Remote Sens. 2021,13, 1107 9 of 23
verification procedure is also performed for polygons Band Cin their supporting planes,
respectively. The best situation is that all three polygons are bounded by the two other
planes, and a corner point at which all three planes intersect is implicitly reconstructed to
bound the building model.
Nearby Constraint: As shown in Figure 5c, for a given polygon Awith supporting
plane
πA
, a haphazard pairwise intersection may result in large numbers of invalid can-
didate faces. Firstly, faces that share common areas with polygon Ain the 2D space (
πA
)
are labeled as Covered. Then, the remained faces that share at least one edge with the
candidate faces in the first category or at least one vertex that is close to (a distance lower
than a given threshold, e.g., 2 m) the vertices of all of the polygons in the first category are
labelled as Near. The rest of the faces in this plane are labeled as Invalid.
The union set of invalid candidate faces are rejected from the candidate pool and those
that do not satisfy the two-manifold rules are also labeled as Invalid, so the recovered
adjacency information in the Section 3.2 is embedded, so that the candidate faces in the
confident part are concise and the redundant candidates are mainly in the ambiguous part.
3.3.2. Face Selection with Initial Constraints
In this step, a subset of candidate faces is selected to establish the final building
model, which prefers some properties and satisfies certain constraints. To find the optimal
configuration of candidate faces, similar to the PolyFit framework [
10
], we use a binary
linear programming approach to quantify the favored properties of the reconstructed
model while imposing certain hard constraints. The optimization model contains the
binary variables shown in Equation (3) below, which was first developed by Nan and
Wonka [10]:
xf,e,es ∈{0, 1},f∈F,e∈E,es ∈E(3)
where xfand xeindicate faces in the face pool (F) constructed by all of the candidate faces
or an edge in the edge pool (E) that includes all of the associated edges of the faces in the
face pool that is selected or not, and x
es
indicates whether the edge is sharp or not. In this
work, four properties are favored:
Property 1: Faces supported by a large number of points are favored.
Epts =1−
∑
f∈F
numhnp
projplanef(p)∈f,dis(p,planef)<εoi
num[p](4)
In Equation (4), pstands for the points in detected primitives, fdonates a face of the
candidate face pool (F), plane
f
represents the regularized 3D plane that supports face f,
proj
plane-f
(p) is the 2D projection of a 3D point ponto the supporting plane of polygon A(
πA
),
dis(p,plane
f
) is the unsigned perpendicular distance from point p to
πA
,
ε
is a pre-defined
distance tolerance, and num(•) is the size of the set.
Property 2: Faces with a high proportion of their area covered by supporting points
are favored.
Ecover =∑
f
(1−area(f∩P)
area(f))(5)
This property is quantified by the ratio of overlapped area in candidate face with
the boundary polygon of the supporting points [
10
]. In Equation (5), Pis a 2D polygon
consisting of the boundary of the supporting points for face fand area(
•
) is the area of the
closed polygon. Both of the areas are calculated in 2D space (the supporting plane of face f).
Equations (4) and (5) are used to determine the degree to which the data fit the candidate
faces, whereby the lower the values, the better the data fit.
Property 3: Intersecting faces in a PC or TC are preferably selected.
Θ(f) = 1+η,f∈PC ∪TC
1, otherwise (6)
Remote Sens. 2021,13, 1107 10 of 23
In Equation (6),
Θ
(
•
) is the confidence coefficient of face fand
η
is a non-negative con-
stant (set as 1 in all the experiments in this paper) that increases the confidence coefficient
values for certain faces.
Property 4: Sharp edges associated with two polygons in a PC or TC are favored.
Eedge =∑
e
xes ·(1−length(e∩eP1∩eP2)
length(e))(7)
This property is measured by the overlapped ratio of the sharp edge with the share
section of the two matched edges. In Equation (7), estands for an edge which connects two
faces, e
P1
and e
P2
are the projections of two matched edges (if they matched as PC or TC)
on the intersecting line of their corresponding planes, and length(
•
) represents the unsigned
length of the segments. Longer overlapping ratio is preferred. Hence, the following energy
function is defined:
E=Eplane +ωedge ·Eedge (8)
Eplane=xf·(ωpts ·Epts +ωcover ·Ecover)/Θ(f)(9)
The first term in Equation (8) measures the degree to which the properties fit the
candidate faces, and the second term measures that for their related edges,
ω
stands for
the corresponding weight. As given in Equation (9), the data fitting degree of a face is
composed of two parts, one for point number (E
pts
), the other one for point coverage
(E
cover
). Larger number of inlier points and bigger area of face coverage is preferred. The
denominator
Θ
(f) are used to give larger opportunities for faces involved in a PC or TC. In
the paper, the weights of the three energy forms, Eedge,Eptd , and Ecovrt, are set as identical.
In addition, three constraints are imposed to strengthen the topologic and geometric
features of the building model:
Constraint 1: An edge must be selected when one of its associated faces is selected.
∃xf∈asso(ei)=1⇒xei=1 (10)
Constraint 2: When an edge is selected, it connects only two candidate faces.
xei=1⇒num[nxf=1
f∈asso(ei)o]= 2 (11)
Constraint 3: The edges associated with two polygons in a PC or TC should be sharp.
xei=1, ei∈PC ∪TC ⇒xesi=1 (12)
Properties 1 and 2 and Constraints 1 and 2 are similar to those reported by Nan
and Wonka [
10
] to ensure a watertight polygonal building model, whereas Properties 3
and 4 and Constraint 3 enforce the recovered topology described in Section 3.2 to realize
favorable models. The above objectives and constraints are formulated as a binary linear
programming problem that can be minimized using existing solvers [
54
]. Lastly, the
selected faces are combined to yield the final building model.
4. Experimental Analysis
4.1. Test Data Description and Experimental Settings
To test the performance of the proposed methods, we made qualitative and quanti-
tative comparisons of the reconstruction quality and computational cost of our method
with those of some state-of-the-art (SOTA) methods. The photogrammetric point clouds
of the aerial oblique images of typical buildings in Shenzhen, China, were incorporated
in the experiments. The original images are captured from five directions (one vertical
and four tilt views), so the building façades are (partially) visible. The image overlap is
approximately 60% to 80%. Image orientation and dense image matching are accomplished
by existing solutions in Context Capture. After that, point clouds of single buildings are
Remote Sens. 2021,13, 1107 11 of 23
manually extracted by drawing 2D bounding boxes. The major components of buildings
are visible, and the roof structures are well-sampled. But there are also some missing and
imperfect areas due to occlusions and unfavorable lighting conditions in the lower parts of
the buildings. The point clouds are shown in Figure 6, while Table 1lists basic information
related to the test data.
Figure 6.
Photogrammetric point clouds used in the experiments. The row numbers correspond
to the building IDs. From left to right, the columns show the original point clouds, the segmented
planar primitives, and the extracted outer boundaries of each primitive.
Remote Sens. 2021,13, 1107 12 of 23
Table 1. Basic information related to the test data.
Building ID Number of
Points
Average
Spacing (m)
Footprint Area
(m2)Detected Planes
1 44,034 0.21 234 18
2 60,675 0.15 432 20
3 203,317 0.09 720 17
4 523,233 0.11 2124 33
5 611,982 0.16 672 37
6 548,766 0.20 5978 45
Starting with the single point clouds of a building, we used the region-growing
method to identify the planar segments. Then, the boundaries of the segments were
traced and regularized to concisely abstract the initial building. As plane detection and
regularization were beyond the scope of this study, we set the planar segments and their
regularized boundaries as inputs in our experiments, which could be generated by an
existing method [
43
]. Figure 6shows the original point clouds, the detected planes, and
the regularized boundaries. As shown in Figure 6, the regularized boundaries of detected
planar primitives could preferably represent building outlines at large primitives which
are well-sampled. Meanwhile, in areas with data insufficiency (e.g., under sampling or
noisy), the recovered boundaries are rather ambiguous, and even some gaps occurred (as
pointed out by the circles).
Because PolyFit [
10
] is the method most closely related to ours, we first qualitatively
and quantitatively compared our method with PolyFit with respect to geometric quality
and computational efficiency. Since the ground truth 3D models are not available in
this area, geometric quality of reconstructed models was determined based on a visual
comparison, cloud-to-mesh (C2M) distance statistics, and mesh-to-cloud (M2C) distance
statistics. C2M distances are calculated by projecting the original points to their nearest
faces in the 3D model, and M2C distances are estimated by first sampling the 3D model
and then calculating the cloud-to-cloud distances between the sampled and original point
clouds. For C2M distances, a lower value means better data fitting degree, meanwhile,
for M2C distances, a lower value means fewer artifacts in the model. For both of the two
values, the lower is the better. Besides, computational efficiency is determined by the
recorded running time of each step in generating 3D building models. Comparisons with
two other SOTA building model reconstruction methods are also presented.
4.2. Comparison with PolyFit
Figures 7–12 show the intermediate candidates generated by the proposed and PolyFit
methods, as well as the final reconstructed models. Table 2provides basic information about
the candidates and final results, along with the computational costs of the reconstruction
process. At first glance, we can observe that the candidate faces generated by the proposed
method are more concise than those produced by PolyFit. Some sharp edges and even some
building corners were already reconstructed by the proposed adjacency-based topology
detection method. As shown in Table 2, the number of candidate faces generated by PolyFit
ranged from 1163 to 15,885, whereas those generated by the proposed method ranged
between 138 and 2964, which again indicates the efficiency of the process of generating
candidate faces by our method. Although the proposed method has to first estimate the
adjacency relations before computing the candidate faces, the running time in this step
(denoted as ADT in Table 2) is relatively fast compared with the computing times of
candidate generation (CGT) and model generation (MGT). The total running times (TT) of
PolyFit and our proposed method, as shown in Table 2for all six tested buildings, reveals
that our method is substantially faster. For simple buildings with fewer planar faces (e.g.,
Buildings #1, 2#, and 3#), although our method used only about 84.2% to 35.6% of the
running time that PolyFit did, the computing times are both in the second level. As a
building became more complicated, the running time for PolyFit increased dramatically
Remote Sens. 2021,13, 1107 13 of 23
because of the geometric growth in the number of candidate faces. For Building #5, PolyFit
used more than 10 min, whereas our method only took 26.428 s, only 4.2% of the total time
that PolyFit took. And for Building #6, PolyFit took about 37 min, whereas our method
accomplished this task in 37.383 s, nearly 60 times faster than PolyFit.
Figure 7.
Comparison of candidate faces and resulting models generated by the proposed method and PolyFit [
10
] for
Building #1.
Figure 8.
Comparison of candidate faces and resulting models generated by the proposed method and PolyFit [
10
] for
Building #2.
Remote Sens. 2021,13, 1107 16 of 23
Table 2.
Quantitative statistics on data and computation cost. BID: Building ID; Can. No.: Number
of Candidate faces; Res. No.: Number of resulting faces; ADT: Time for adjacency estimation; CGT:
Time for candidate generation; MGT: Time for resulting Model Generation; TT: Total time.
BID Method Can. No. Res. No. ADT (s) CGT (s) MGT (s) TT (s)
#1 Our 138 99 0.018 0.577 0.049 0.644
PolyFit 1190 114 - 0.646 1.164 1.810
#2 Our 242 163 0.034 0.777 0.040 0.851
PolyFit 1584 169 - 0.752 0.989 1.741
#3 Our 196 159 0.019 2.953 0.043 3.015
PolyFit 1163 159 - 3.106 0.475 3.581
#4 Our 689 533 0.085 11.668 0.019 11.772
PolyFit 6809 578 - 12.014 60.983 72.997
#5 Our 1187 707 0.222 14.266 11.940 26.428
PolyFit 8117 784 - 13.425 619.913 633.338
#6 Our 2964 1489 0.660 18.500 18.886 37.386
PolyFit 15,885 1558 - 17.041 2210.079 2227.120
In addition, in Buildings #2, #4, and #5, there are obvious artifacts in the final build-
ing models generated by PolyFit as compared with the original point clouds shown in
Figure 6. This is a side effect of candidate generation based on the intersections of all
potential planar segments. In contrast, by utilizing constraints derived from adjacency
information, these artifacts are avoided or alleviated in both candidate generation and
candidate selection stages. To verify the objective data fidelity of the reconstructed building
models quantitatively, we calculated the mean values of the C2M and M2C distances, as
shown in Figure 13. The mean C2M distances for the 3D models generated by PolyFit
and the proposed method are all less than 0.30 m, and those for the same buildings are
similar. However, for Buildings #2, #4, and #5, the M2C distances generated by PolyFit
are significantly greater than those generated by the proposed method. Because greater
M2C distances indicates greater data distortion in the final building models, we can infer
that the 3D models generated by our method have better geometric accuracy, which also
accords with the visual judgments shown in Figures 8,10 and 11.
Figure 13.
Mean C2M and M2C distances from Buildings #1 to #6 generated by three-dimensional (3D) models from
PolyFit [10] and the proposed method.
Remote Sens. 2021,13, 1107 17 of 23
4.3. Comparison with Other SOTA Methods
To further evaluate the proposed method, we also conducted an experiment with two
SOTA 3D building reconstruction methods, namely the 2.5D dual contouring method (2.5D
DC) [
55
] and the structuring method [
56
]. As shown in Figure 14, all three methods were
able to reconstruct the overall structure of buildings from point clouds. The structuring
method also preserved some sharp features, such as the edges and corners of buildings in
the models, although some artifactual meshes appeared because of insufficient sampling
and data noise. The 2.5D DC method was able to reconstruct building roofs and protect
planar structures to a certain degree, but it failed to reconstruct Building #6, which has a
typical 3D structure. The mean values of M2C and C2M distances are shown in
Figure 15
.
From Building #1 to Building #4, the C2M distances for all three methods are almost at
the same level, ranging from 0.05 to about 0.15 m. Meanwhile, the C2M distance of the
proposed method in Building #5 is larger than the two others. The reason is that some
small primitives in the original point clouds are not detected and recovered. For Building
#6, the C2M value of 2.5D DC is distinctively large since the façades of the building are not
well-reconstructed. The mean values of M2C of the three methods are between 0.2 to 0.5
m. Note that, in these three methods, the proposed method is the only one which imposes
watertight constraints, so some holes in the point clouds are filled in the output models (e.g.,
the bottom of the building), which inevitably leads to a higher value of M2C statistics. The
situations in Figure 15 reveal that the models recovered by structuring and the proposed
method could represent true 3D structures of buildings. Compared with those of the 2.5D
DC and structuring methods, the 3D models produced by our proposed method were more
concise, with the building surfaces briefly represented by several planar faces. As given in
Table 3, the models generated by the proposed method are composed of less faces, over
90% to 99% of faces are reduced when compared with those generated by 2.5D DC and
structuring, yielding a concise representation of the building structure which would benefit
subsequent applications such as computational simulation in reducing computational costs.
Although some small details were smoothed, the models generated by our method have
better potential for further interactive editing, if needed, since the main structuring of
buildings have been concisely represented without topological conflicts.
Table 3.
Numbers of faces in the reconstructed models by our method, 2.5D DC method [
55
], and
structuring method [56].
Building ID Our 2.5D DC Structuring
1 99 1452 10,972
2 163 2913 14,086
3 159 3797 126,674
4 533 18,170 32,077
5 707 30,527 128,315
6 1489 28,075 138,474
Remote Sens. 2021,13, 1107 19 of 23
Figure 15.
Mean C2M and M2C distances from Buildings #1 to #6 generated by 3D models from 2.5D DC method [
55
],
structuring method [56], and the proposed method.
5. Discussion
As shown in Figures 7–12, the candidates generated by PolyFit are quite superfluous,
which makes it hard to recognize the building features (edges, corners) from the candidate
faces; meanwhile, the candidates generated by the proposed method have incorporated
some recovered topological information, so some of the building features could be visually
discovered. Consider the information in Figure 14 and Table 3, one virtue of the proposed
method is the concise representation of the original building point clouds with preferred
geometry accuracy. In fact, this is also the virtue of PolyFit. Since our method combines the
initial recovered topological information, which is obtained by the rule-based method, in
the candidate generation and selection process, some unwanted artifacts are alleviated or
avoided. The M2C values in Figure 13 also reveal this virtue. Besides, the running time of
our method is substantially faster than the traditional hypothesis-based methods, such as
PolyFit, especially when the complexity of the building increases. The main difference lies
in the MGT process, where a lower ambiguity needs fewer computation times. For a dataset
with lower noise and better coverage, it is more likely to extract intact building primitives
and to obtain the high-quality regularized boundaries. In these cases, topology relations
between primitives are easier to be detected, leading to better initial reconstructions and
fewer candidates (see the examples in Figures 7–9). Thus, the running times in the candidate
selection process are less. In extreme cases with favorable data coverage and sampling
density, if all the topological relations of building primitives have been recovered, the
candidate faces generated accordingly should equal to the final output model, because
there is only one selection combination from which conform all the properties in the
energy equations and constraints. Conversely, if the building primitives are not well
detected because of data quality (or some other reasons), the adjacency detection process
may only work in a small part of areas. Then, the number of generated candidates of
our algorithm would increase, leading to longer MGTs and, maybe, some unexpected
artifactual structures in the building models. In the most unfortunate cases where no
Remote Sens. 2021,13, 1107 20 of 23
topology has been recovered in the first stage of our algorithm, the proposed method
would degenerate to a pure hypothesis-based reconstruction method, similar to PolyFit.
6. Conclusions
In this work, we proposed a novel method for efficient reconstructing building models
from photogrammetric point clouds obtained from aerial oblique images in a two-stage
topology recovery process which combined the rule-based and the hypothesis-based meth-
ods. Given the point clouds of a single building, planar primitives and their corresponding
boundaries are extracted and regularized to obtain abstracted building counters. In the
first stage, the adjacency relations between different polygons are estimated based on their
spatial consistency and mutual regularity to form initial reconstruction. In the second
stage, the candidate faces of the building model are generated by the pairwise planar inter-
sections along with three constraints, namely pairwise constraint, triplet constraint, and
nearby constraint, derived from the recovered topology. Lastly, the optimal combination of
candidate faces is selected by solving a binary linear programming problem that shapes the
favored properties under certain constraints. An experimental comparison of our method
with three SOTA methods revealed that the proposed method could efficiently reconstruct
3D building models in several seconds and produce concise models with preferred data
fidelity and geometric accuracy at the decimeter level. Detailed comparison with PolyFit
indicated the high efficiency of the proposed method in reconstructing complex building
models and showed promising prospect for this method in 3D city environmental applica-
tions. The advantage of the proposed methods is the ability to handle point clouds with
various noises in the reconstruction process. However, when the point clouds have large
holes or extensively missing areas, the recovered models by our method may degenerate
to those constructed by PolyFit, which is a disadvantage. Besides, if the density of the
input point clouds is too low, the boundaries extracted from the building primitives may
become unstable as well as the initial topology estimation. In future works, methods
for incorporating multi-scale primitives and 2D features from oriented images should be
developed to handle the situation with low point density and to achieve more detailed
reconstruction.
Author Contributions:
Conceptualization, L.X. and W.W.; methodology, L.X. and Q.Z.; software, L.X.
and H.H.; validation, S.T., Y.L. and Y.Z.; formal analysis, Q.Z. and X.L.; supervision, R.G. All authors
wrote the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding:
This work was supported by National Key R&D Program of China (2019YFB210310,
2019YFB2103104), the National Natural Science Foundation of China (No.42001407, No.41971341,
No. 41971354, 41801392), the Open Fund of Key Laboratory of Urban Land Resources Monitoring
and Simulation, MNR (KF-2019-04-042, KF-2019-04-019), the Guangdong Basic and Applied Basic
Research Foundation (2019A1515110729, 2019A1515010748, 2019A1515011872), and the Open Re-
search Fund of State Key Laboratory of Information Engineering in Surveying Mapping and Remote
Sensing, Wuhan University (No. 20E02).
Acknowledgments:
We would like to thank Liangliang Nan, Florent Lafarge and Qianyi Zhou for
providing the source codes or executable programs for experimental comparison.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
LoD Level of Detail
CityGML City Geography Markup Language
LiDAR Light Detection And Ranging
SfM Structure-from-Motion
MVS Multi-View Stereo
RANSAC RANdom Sample Consensus
Remote Sens. 2021,13, 1107 21 of 23
BSP Binary Space Partitioning
RTG Roof Topology Graph
VVM Vertex–Vertex Match
VEM Vertex–Edge Match
PC Pairwise Constraint
TC Triplet Constraint
NC Nearby Constraint
C2M Cloud to Mesh
M2C Mesh to Cloud
References
1.
Tan, Z.; Lau, K.K.-L.; Ng, E. Planning strategies for roadside tree planting and outdoor comfort enhancement in subtropical
high-density urban areas. Build. Environ. 2017,120, 93–109. [CrossRef]
2.
Wang, S.; Cai, G.; Cheng, M.; Junior, J.M.; Huang, S.; Wang, Z.; Su, S.; Li, J. Robust 3D reconstruction of building surfaces from
point clouds based on structural and closed constraints. ISPRS J. Photogramm. Remote Sens. 2020,170, 29–44. [CrossRef]
3.
Badach, J.; Voordeckers, D.; Nyka, L.; Van Acker, M. A framework for air quality management zones—Useful GIS-based tool for
urban planning: Case studies in Antwerp and Gda ´nsk. Build. Environ. 2020,174, 106743. [CrossRef]
4.
Biljecki, F.; Stoter, J.; LeDoux, H.; Zlatanova, S.; Çöltekin, A. Applications of 3D city models: State of the art review. ISPRS Int. J.
Geo-Inf. 2015,4, 2842–2889. [CrossRef]
5.
Toschi, I.; Ramos, M.M.; Nocerino, E.; Menna, F.; Remondino, F.; Moe, K.; Poli, D.; Legat, K.; Fassi, F. Oblique photogrammetry
supporting 3D urban reconstruction of complex scenarios. ISPRS Arch. 2017,XLII-1/W1, 519–526. [CrossRef]
6.
Yu, D.; Ji, S.; Liu, J.; Wei, S. Automatic 3D building reconstruction from multi-view aerial images with deep learning. ISPRS J.
Photogramm. Remote Sens. 2021,171, 155–170. [CrossRef]
7.
Liu, X.; Zhang, Y.; Ling, X.; Wan, Y.; Liu, L.; Li, Q. TopoLAP: Topology recovery for building reconstruction by deducing the
relationships between linear and planar primitives. Remote Sens. 2019,11, 1372. [CrossRef]
8.
Awrangjeb, M.; Gilani, S.A.N.; Siddiqui, F.U. An effective data-driven method for 3D building roof reconstruction and robust
change detection. Remote Sens. 2018,10, 1512. [CrossRef]
9.
Jarz ˛abek-Rychard, M.; Borkowski, A. 3D building reconstruction from ALS data using unambiguous decomposition into
elementary structures. ISPRS J. Photogramm. Remote Sens. 2016,118, 1–12. [CrossRef]
10. Nan, L.; Wonka, P. PolyFit: Polygonal Surface Reconstruction from Point Clouds. In Proceedings of the 2017 IEEE International
Conference on Computer Vision (ICCV), Venice, Italy, 22–29 October 2017; pp. 2372–2380.
11. Verdie, Y.; Lafarge, F.; Alliez, P. LOD generation for urban scenes. ACM Trans. Graph. 2015,34, 1–14. [CrossRef]
12.
Xiong, B.; Elberink, S.O.; Vosselman, G. A graph edit dictionary for correcting errors in roof topology graphs reconstructed from
point clouds. ISPRS J. Photogramm. Remote Sens. 2014,93, 227–242. [CrossRef]
13.
Xu, B.; Jiang, W.; Li, L. HRTT: A hierarchical roof topology structure for robust building roof reconstruction from point clouds.
Remote Sens. 2017,9, 354. [CrossRef]
14.
Ochmann, S.; Vock, R.; Klein, R. Automatic reconstruction of fully volumetric 3D building models from oriented point clouds.
ISPRS J. Photogramm. Remote Sens. 2019,151, 251–262. [CrossRef]
15.
Gruen, A.; Schubiger, S.; Qin, R.; Schrotter, G.; Xiong, B.; Li, J.; Ling, X.; Xiao, C.; Yao, S.; Nuesch, F. Semantically enriched high
resolution LoD3 building model generation. ISPRS Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci.
2019
,XLII-4/W15, 11–18.
[CrossRef]
16.
Wen, X.; Xie, H.; Liu, H.; Yan, L. Accurate reconstruction of the LoD3 building model by integrating multi-source point clouds
and oblique remote sensing imagery. ISPRS Int. J. Geo-Inf. 2019,8, 135. [CrossRef]
17. Gröger, G.; Plümer, L. CityGML–Interoperable semantic 3D city models. ISPRS J. Photogramm. 2012,71, 12–33. [CrossRef]
18.
Zhang, K.; Yan, J.; Chen, S.-C. Automatic construction of building footprints from airborne LiDAR data. IEEE Trans. Geosci.
Remote Sens. 2006,44, 2523–2533. [CrossRef]
19.
Zhao, Z.; Duan, Y.; Zhang, Y.; Cao, R. Extracting buildings from and regularizing boundaries in airborne LiDAR data using
connected operators. Int. J. Remote Sens. 2016,37, 889–912. [CrossRef]
20.
Chen, Q.; Wang, L.; Waslander, S.L.; Liu, X. An end-to-end shape modeling framework for vectorized building outline generation
from aerial images. ISPRS J. Photogramm. Remote Sens. 2020,170, 114–126. [CrossRef]
21.
Widyaningrum, E.; Gorte, B.; Lindenbergh, R. Automatic building outline extraction from ALS point clouds by ordered points
aided hough transform. Remote Sens. 2019,11, 1727. [CrossRef]
22. Vosselman, G. Building reconstruction using planar faces in very high density height data. ISPRS Arch. 1999,32, 87–94.
23.
Sohn, G.; Huang, X.; Tao, V. Using a binary space partitioning tree for reconstructing polyhedral building models from airborne
LiDAR data. Photogramm. Eng. Remote Sens. 2008,74, 1425–1438. [CrossRef]
24.
Yang, B.; Huang, R.; Li, J.; Tian, M.; Dai, W.; Zhong, R. Automated reconstruction of building LoDs from airborne LIDAR point
clouds using an improved morphological scale space. Remote Sens. 2016,9, 14. [CrossRef]
25.
Kurdi, F.T.; Awrangjeb, M.; Munir, N. Automatic filtering and 2D modeling of airborne laser scanning building point cloud. Trans.
GIS 2021,25, 164–188. [CrossRef]
Remote Sens. 2021,13, 1107 22 of 23
26.
Kurdi, F.T.; Awrangjeb, M. Automatic evaluation and improvement of roof segments for modelling missing details using Lidar
data. Int. J. Remote Sens. 2020,41, 4702–4725. [CrossRef]
27.
Song, J.; Xia, S.; Wang, J.; Chen, D. Curved buildings reconstruction from airborne LiDAR data by matching and deforming
geometric primitives. IEEE Trans. Geosci. Remote Sens. 2021,59, 1660–1674. [CrossRef]
28.
Kulawiak, M.; Lubniewski, Z. Improving the accuracy of automatic reconstruction of 3D complex buildings models from airborne
LiDAR point clouds. Remote Sens. 2020,12, 1643. [CrossRef]
29.
Kim, K.; Shan, J. Building roof modeling from airborne laser scanning data based on level set approach. ISPRS J. Photogramm.
Remote Sens. 2011,66, 484–497. [CrossRef]
30.
Poullis, C.; You, S. Photorealistic large-scale urban city model reconstruction. IEEE Trans. Vis. Comput. Graph.
2008
,15, 654–669.
[CrossRef]
31.
Henn, A.; Gröger, G.; Stroh, V.; Plümer, L. Model driven reconstruction of roofs from sparse LiDAR point clouds. ISPRS J.
Photogramm. Remote Sens. 2013,76, 17–29. [CrossRef]
32.
Lafarge, F.; Descombes, X.; Zerubia, J.; Pierrot-Deseilligny, M. Automatic building extraction from DEMs using an object approach
and application to the 3D city modeling. ISPRS J. Photogramm. Remote Sens. 2008,63, 365–381. [CrossRef]
33.
Cao, R.; Zhang, Y.; Liu, X.; Zhao, Z. 3D building roof reconstruction from airborne LiDAR point clouds: A framework based on a
spatial database. Int. J. Geogr. Inf. Sci. 2017,31, 1359–1380. [CrossRef]
34. Tarsha Kurdi, F.; Landes, T.; Grussenmeyer, P.; Koehl, M. Model-driven and data-driven approaches using Lidar data: Analysis
and comparison. ISPRS Arch. 2007,XXXVI W49A, 1682–1687.
35.
Wang, R.; Peethambaran, J.; Chen, D. LiDAR point clouds to 3D urban models: A review. IEEE J. Sel. Top. Appl. Earth Obs. Remote
Sens. 2018,11, 606–627. [CrossRef]
36.
Lin, Y.; Wang, C.; Cheng, J.; Chen, B.; Jia, F.; Chen, Z.; Li, J. Line segment extraction for large scale unorganized point clouds.
ISPRS J. Photogramm. Remote Sens. 2015,102, 172–183. [CrossRef]
37.
Schnabel, R.; Wahl, R.; Klein, R. Efficient RANSAC for point-cloud shape detection. Comput. Graph. Forum
2007
,26, 214–226.
[CrossRef]
38.
Tarsha Kurdi, F.; Landes, T.; Grussenmeyer, P. Hough-transform and extended RANSAC algorithms for automatic detection of
3D building roof planes from LiDAR data. In Proceedings of the ISPRS Workshop on Laser Scanning 2007 and SilviLaser 2007,
Espoo, Finland, 12–14 September 2007; Volume 36, pp. 407–412.
39.
Lari, Z.; Habib, A. An adaptive approach for the segmentation and extraction of planar and linear/cylindrical features from laser
scanning data. ISPRS J. Photogramm. Remote Sens. 2014,93, 192–212. [CrossRef]
40. Lu, X.; Yao, J.; Tu, J.; Li, K.; Li, L.; Liu, Y. Pairwise linkage for point cloud segmentation. ISPRS Ann. 2016,III-3, 201–208.
41.
Sampath, A.; Shan, J. Building boundary tracing and regularization from airborne LiDAR point clouds. Photogramm. Eng. Remote
Sens. 2007,73, 805–812. [CrossRef]
42.
Li, M.; Nan, L.; Smith, N.; Wonka, P. Reconstructing building mass models from UAV images. Comput. Graph.
2016
,54, 84–93.
[CrossRef]
43.
Xie, L.; Zhu, Q.; Hu, H.; Wu, B.; Li, Y.; Zhang, Y.; Zhong, R. Hierarchical regularization of building boundaries in noisy aerial
laser scanning and photogrammetric point clouds. Remote Sens. 2018,10, 1996. [CrossRef]
44.
Chen, D.; Wang, R.; Peethambaran, J. Topologically aware building rooftop reconstruction from airborne laser scanning point
clouds. IEEE Trans. Geosci. Remote Sens. 2017,55, 7032–7052. [CrossRef]
45.
Chen, D.; Zhang, L.; Mathiopoulos, P.T.; Huang, X. A Methodology for automated segmentation and reconstruction of urban 3D
buildings from ALS point clouds. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 2014,7, 4199–4217. [CrossRef]
46.
Benciolini, B.; Ruggiero, V.; Vitti, A.; Zanetti, M. Roof planes detection via a second-order variational model. ISPRS J. Photogramm.
Remote Sens. 2018,138, 101–120. [CrossRef]
47.
Sampath, A.; Shan, J. Segmentation and reconstruction of polyhedral building roofs from aerial lidar point clouds. IEEE Trans.
Geosci. Remote Sens. 2010,48, 1554–1567. [CrossRef]
48.
Elberink, S.O.; Vosselman, G. Building reconstruction by target based graph matching on incomplete laser data: Analysis and
limitations. Sensors 2009,9, 6101–6118. [CrossRef]
49.
Hu, H.; Chen, C.; Wu, B.; Yang, X.; Zhu, Q.; Ding, Y. Texture-aware dense image matching using ternary census transform. ISPRS
Ann. 2016,III-3, 59–66.
50.
Wu, B.; Xie, L.; Hu, H.; Zhu, Q.; Yau, E. Integration of aerial oblique imagery and terrestrial imagery for optimized 3D modeling
in urban areas. ISPRS J. Photogramm. Remote Sens. 2018,139, 119–132. [CrossRef]
51.
Perera, G.S.N.; Maas, H.-G. Cycle graph analysis for 3D roof structure modelling: Concepts and performance. ISPRS J. Photogramm.
Remote Sens. 2014,93, 213–226. [CrossRef]
52. Bauchet, J.-P.; Lafarge, F. Kinetic shape reconstruction. ACM Trans. Graph. 2020,39, 1–14. [CrossRef]
53.
Arikan, M.; Schwärzler, M.; Flöry, S.; Wimmer, M.; Maierhofer, S. O-snap: Optimization-based snapping for modeling architecture.
ACM Trans. Graph. 2013,32, 6. [CrossRef]
54.
Gamrath, G.; Anderson, D.; Bestuzheva, K.; Chen, W.; Eifler, L.; Gasse, M.; Gemander, P.; Gleixner, A.; Gottwald, L.; Halbig, K.;
et al. The SCIP Optimization Suite 7.0; Optimization Online; Zuse Institut: Berlin, Germany, 2020.
Remote Sens. 2021,13, 1107 23 of 23
55.
Zhou, Q.-Y.; Neumann, U. 2.5D Dual Contouring: A robust approach to creating building models from aerial LiDAR point clouds.
In Constructive Side-Channel Analysis and Secure Design, Proceedings of the 3rd International Workshop, COSADE, Darmstadt, Germany,
3–4 May 2012; Springer: Berlin, Germany, 2012.
56. Lafarge, F.; Alliez, P. Surface reconstruction through point set structuring. Comput. Graph. Forum 2013,32, 225–234. [CrossRef]
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