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A modified method of discontinuity trace mapping using three-dimensional point
clouds of rock mass surfaces
Keshen Zhang, Wei Wu, Hehua Zhu, Lianyang Zhang, Xiaojun Li, Hong Zhang
PII: S1674-7755(20)30041-X
DOI: https://doi.org/10.1016/j.jrmge.2019.10.006
Reference: JRMGE 649
To appear in: Journal of Rock Mechanics and Geotechnical Engineering
Received Date: 23 May 2019
Revised Date: 9 August 2019
Accepted Date: 8 October 2019
Please cite this article as: Zhang K, Wu W, Zhu H, Zhang L, Li X, Zhang H, A modified method of
discontinuity trace mapping using three-dimensional point clouds of rock mass surfaces, Journal of Rock
Mechanics and Geotechnical Engineering, https://doi.org/10.1016/j.jrmge.2019.10.006.
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A modified method of discontinuity trace mapping using three-dimensional point clouds of rock mass surfaces
Keshen Zhang
a
, Wei Wu
a,
*, Hehua Zhu
a
, Lianyang Zhang
b
, Xiaojun Li
a
, Hong Zhang
c
a
Department of Geotechnical Engineering, Tongji University, Shanghai, 200092, China
b
Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ 85721, USA
c
Department of Hydraulic Engineering, Tongji University, Shanghai, 200092, China
Abstract: This paper presents an automated method for discontinuity trace mapping using three-dimensional point clouds of rock mass surfaces.
Specifically, the method consists of five steps: (1) detection of trace feature points by normal tensor voting theory, (2) contraction of trace feature
points, (3) connection of trace feature points, (4) linearization of trace segments, and (5) connection of trace segments. A sensitivity analysis was then
conducted to identify the optimal parameters of the proposed method. Three field cases, a natural rock mass outcrop and two excavated rock tunnel
surfaces, were analyzed using the proposed method to evaluate its validity and efficiency. The results show that the proposed method is more efficient
and accurate than the traditional trace mapping method, and the efficiency enhancement is more robust as the number of feature points increases.
keywords: rock mass; discontinuity; three-dimensional point clouds; trace mapping
1. Introduction
Discontinuity trace mapping plays an important role in characterizing
rock masses. Discontinuities have a significant effect on the strength,
deformability and permeability of rock masses (Zhang and Einstein, 1998;
Zhang, 2017), which are often characterized based on the information
from discontinuity trace mapping (Mauldon, 1998; Zhang and Einstein,
2000; Li et al., 2014; Zhu et al., 2014). Discontinuity trace is one of the
seven major parameters suggested by the International Society for Rock
Mechanics (ISRM, 1978) to quantitatively describe rock discontinuities.
The information of discontinuity traces is obtained traditionally through
conducting geotechnical field survey with a tape and a geological
compass (Franklin et al., 1988). However, the traditional field survey is
time-consuming in conjunction with safety concerns, and has its biases
and cannot be easily accomplished correctly or completely (Crosta, 1997;
Priest and Hudson, 1981; Kulatilake, 1993; Vöge et al., 2013). Recently,
emerging non-contact methods, including photogrammetry and light
detection and ranging (LIDAR) technology, have been introduced to
obtain discontinuity information via images and three-dimensional (3D)
point clouds of rock mass surfaces. These technologies can greatly
improve the efficiency of in situ data collection due to convenient
operation. The uniform data format and mapping algorithm also make it
possible to extract more accurate, objective and credible information of
discontinuities.
The two-dimensional (2D) image method of discontinuity trace
mapping is based on the gray-level variation and distribution of pixels
(Crosta, 1997; Franklin et al., 1988; Reid and Harrison, 2000;
Hadjigeorgiou et al., 2003). However, the method shows strong
dependence on image quality, which is easily influenced by dust,
illumination, noise and production constraints, and often generates
meaningless segments or excessive fragmentation (Lemy and
Hadjigeorgiou, 2003; Ferrero et al., 2009). In addition, the images
produced by uncalibrated cameras suffer from lens distortion and
projective distortion which are hard to rectify (Li et al., 2016).
Recently, many researchers have studied discontinuity trace mapping
using 3D point clouds of rock mass surfaces (Roncella and Remondino,
2005; Gigli and Casagli, 2011; Li et al., 2016; Ge et al., 2018; Guo et al.,
2018). The 3D point clouds of rock mass surfaces can be obtained using
either LIDAR or photogrammetry (Kraus and Pfeifer, 1998; Chandler,
1999; Lane et al., 2000). Although LIDAR is more convenient than
photogrammetry in acquiring point cloud directly, it is difficult to cover
*Corresponding author. E-mail address: weiwu@tongji.edu.cn
all relevant viewing directions and achieve good registration of scans and
sufficient resolution for steep terrains or surfaces with vegetation. The
photogrammetry is in principle more flexible because the image scale and
viewing direction can be more easily adapted to the need (Roncella and
Remondino, 2005). Two types of methods have been used to detect
discontinuity traces from 3D point clouds of rock mass surfaces. The first
one considers a discontinuity trace as an intersection line between two
adjacent fitting planes of rock mass surfaces. Gigli et al. (2011) presented
a 3D trace recognition method which projects 2D traces obtained from
image processing on the intersections of corresponding 3D discontinuity
fitting planes. However, the effectiveness of trace mapping is heavily
dependent on the accuracy of both the 2D traces on images and the 3D
fitting planes, which results in the difficulty to precisely recognize the 3D
spatial locations of traces. The second type detects discontinuity traces
from vertices which constitute the rock mass surface and are located
around the real traces. In Umili et al. (2013)’s method, the feature vertices
consisting of the traces were first recognized using principal curvature
values, and then the traces were expressed as straight lines after
connection and segmentation. Similarly, in Li et al. (2016)’s method, the
feature vertices were recognized by the normal tensor voting (NTV)
theory (Page et al., 2002) and the traces were detected based on a growth
algorithm. In the method of Ge et al. (2018), 3D point clouds data were
first converted to grid data, and then the traces were detected using a
modified region growing (MRG) algorithm.
Currently, the second method based on feature vertices is more widely
used to detect discontinuity traces from the 3D point clouds model (Umili
2013; Li et al., 2016; Zhu et al., 2016; Ge, 2018). However, since each of
the feature vertices for a trace needs to be calculated, judged and selected,
the process is very time-consuming. In addition, the intersection of
multiple natural discontinuities with the rock surface and the damage due
to construction blasting and disturbance may separate a natural
discontinuity trace into trivial pieces, making it more difficult to detect
traces. Therefore, the aim of this paper is to improve the efficiency and
accuracy of discontinuity trace mapping from 3D point clouds of rock
mass surfaces by proposing an automated mapping approach.
2. Methodology
Rock mass discontinuities include joints, bedding planes, faults, and
other types of fracture planes (Kemeny and Post, 2003). In this paper, a
trace is defined as the intersection line of two adjacent discontinuity
planes which belong to different sets (Fig. 1).
Fig. 1. Sketch figure of traces. This figure shows a conceptual model of a rock mass
body. Gray plane and yellow plane represent two different sets of discontinuity
planes, and the traces are three blue intersection lines of the two adjacent
discontinuity planes.
The proposed automated method for discontinuity trace mapping
includes five steps (Fig. 2):
(1) Identification of trace feature points with mesh vertices around a
trace;
(2) Contraction of trace feature points together on their point cloud
skeleton;
(3) Connection of trace feature points belonging to the same trace;
(4) Linearization of trace segments parted into linearized segments; and
(5) Connection of trace segments belonging to the same trace.
In the above phases, step (1) is the same as proposed by Li et al.
(2016), step (2) is based on the theory proposed by Cao et al. (2010) and
some modifications are made, and steps (3)–(5) are the improvements
proposed in this paper.
Before applying the 5-step procedure, the 3D point clouds data need
to be preprocessed to consider the disturbances and errors due to various
factors such as vegetation, fragmentation, and dust (Slob, 2007). The
preprocessing includes: resampling the point clouds using a minimum
distance of 3 cm to reserve the rock mass geometry features, performing
denoising using the moving least squares method (Alexa et al., 2003) to
reduce noise, and acquiring a 3D surface model of the rock mass using the
Delaunay triangulation algorithm (Li et al., 2016) and Halcon (MVTec
Software GmbH, 2012). As a result of the preprocessing, triangular
meshes of the 3D point clouds of rock mass surface are obtained.
2.1. Trace feature point detection
2.1.1. Normal tensor voting (NTV) method
Traces on triangulated meshes refer to as the skeleton of feature points
composed of mesh vertices on edges and corners. Edges can be detected
by surface normal variation within a neighborhood because the surface
normal has an abrupt change across edges (Sun et al., 2002). Normal
voting scheme, which is extended from tensor voting, can be performed to
achieve robust detection. The voting scheme can be simply regarded as
the eigenvalue analysis of a set of surface normals (Medioni et al., 2000).
The NTV method can recognize sharp features and show robustness to
noisy data.
Fig. 2. Procedure for discontinuity trace mapping.
Given a triangulated mesh is denoted by ,
is the set of vertices, E is the set of edges and
is the set of faces. Each vertex
is represented
using Cartesian coordinates, denoted by
. The NTV of
a vertex is defined as
f i
T
( )
( )
µ
∈
=
∑
i
v fi fi fi
f N v
T n n
(1)
where
is the unit normal vector of
,
is the
one-ring neighbor face index set of
(Fig. 3), and
is a weight
coefficient given by Kim et al. (2009):
max
( ) exp /3
µσ
−
= −
fi
i
fi
A
A
c v
f
(2)
where
is the area of triangle
,
!"#
is the maximum area of
, $
is the barycenter of triangle
, and % is the edge length of a
cube that defines the neighbor space of each vertex.
In Eq. (1), &
'
is a symmetric positive semi-definite matrix and can
be represented as
T T T
1 1 1 2 2 2 3 3 3
λ λ λ
= + +
v
T e e e e e e
(3)
where (
, (
and (
)
are the eigenvalues of &
'
and (
* (
* (
)
* +,
and ,
, ,
and ,
)
are the corresponding unit eigenvectors.
According to the eigenvalues (Kim et al., 2009), vertices can be
classified into three types: face type, sharp edge type, and corner type.
The classification rules are as follows:
(1) Face type: (
is dominant, and (
and (
)
are close to 0;
(2) Sharp edge type: (
and (
are dominant, and (
)
is close to 0; and
(3) Corner type: (
, (
and (
)
are approximately equal.
Feature points consist of both sharp edge type vertices and corner type
vertices.
Fig. 3. An example of one-ring neighbor points and
. Red point represents
point
, blue points are one-ring neighbor points of
, and blue numbers are the
corresponding one-ring neighbor face indices of
. Therefore,
= {1, 2, 3, 4,
5, 6}.
2.1.2. Detecting feature points
There are two thresholds, - and ., defined to control the recognition
accuracy of corner type and edge type points, respectively. Threshold
value α should be large enough to avoid extracting over many false
corners. Threshold value . is a fine-tuning coefficient around a value to
find a tradeoff between detecting weak features and an extra number of
noisy points (Wang et al., 2012). The definition of - and . depends on
visual evaluation of the number of recognized edge type and corner type
vertices (Li et al., 2016).
2.2. Trace feature point contraction
This step is based on the idea that traces are regarded as curve
skeletons through adjacent feature points. Curve skeletons of point clouds
can be extracted via a Laplacian-based contraction algorithm (Cao et al.,
2010). The contraction algorithm can aggregate feature points on their
skeletons using local Delaunay triangulation and topological thinning.
Therefore, locations of 3D traces are obtained.
2.2.1. Trace feature point grouping
To reduce the mutual interference of contraction of different traces,
feature points are grouped as follows: If
and
/
are two vertices of
the same triangulated mesh, then they are divided in one group.
2.2.2. Feature point contraction
Each group of the feature points is contracted as follows.
Given that a group of feature point is 0
1
, the
corresponding contracted set of point 2 is obtained by solving the
following system:
L
H H
L′=0WV
W W V
(4)
where 3
4
and 3
5
are the diagonal matrices that balance the
contraction and attraction constraints, respectively (Au et al., 2008).
Eq. (4) is solved by the least-squares method, which is equivalent to
minimizing the following quadratic energy E
q
(Sorkine and Cohen-Or,
2004):
2 2
2
q L H ,
′ ′
= + −
∑
i i i
i
EW LV W v v
(5)
where L is the n×n curvature-flow Laplace operator with elements
computed by Eq. (6). Because that the point clouds skeleton generated by
one iteration of Eq. (4) is linear enough for trace mapping and time-saving,
both 3
4
and 3
5
are defined as unit diagonal matrices in this step.
, , ,
,
( , )
cot if ( , )
( ) if
otherwise
0
i j i j k
k
k
i j
i j
i j
i j
ω α
ω
∈
=∈
= − =
∑
∑
E
E
L
(6)
where 6
/
is the cotangent weight,
/
is one of the contraction
neighbors of
(Fig. 4a), and -
/7
is the kth opposite angles
corresponding to the 89:8
/
(Fig. 4b). The edge set and
contraction neighbors are defined as follows:
(1) Perform 3D Delaunay triangulation on all feature points , and edge
set is obtained;
(2) Calculate the distance between each feature point and its Delaunay
neighbors;
(3) Define a contraction radius ;
<
(2.5 times the average edge length of
triangular mesh, in Section 3.3), as shown in Fig. 4a; and
(4) If the distance between a feature point and its Delaunay neighbor
exceeds =;
<
, then delete the neighbor point. Through iteration, the
remaining Delaunay neighbors of a feature point are defined as
contraction neighbors.
Through solving Eq. (4) once,
and its contraction neighbors can
aggregate on their skeleton, as shown in Fig. 4c.
(a) (b) (c)
Fig. 4. Trace feature point contraction algorithm: (a) Positions of feature points before contraction (red circle represents the contraction scope, green points are the
contraction neighbors of the red point, and the other feature points are plotted in blue); (b) Computation of =6
/
; and (c) Result of contraction (the positions of feature points
before contraction are plotted in black).
2.3. Trace feature point connection
The above procedures generate contracted feature points which
aggregate on their point skeletons. This step is connecting the points to
generate trace segments. Connection neighbors of contracted feature
points are defined as Delaunay neighbors within connection radius.
Because the contraction can enlarge intervals of the feature points that
belong to the same trace segment, connection radius is defined larger than
contraction radius, which is 3 times the average edge length of triangular
mesh by data test.
Connection starts with a randomly selected point (pink point in Fig. 5a)
and chooses the nearest points of its connection neighbors as the possibly
next point of the trace segment.
As shown in Fig. 5b, given that the start point
>
of the segment is in
the pink circle, the end point
?
is in the green circle, the one-third point
of the segment
)
@
is in the black circle and the link point
A
is
possibly in the red circle. The connection rules are defined to control the
connection direction as follows:
(1) Angle B
?
A
C
C
C
C
C
C
C
C
C
D
>
?
C
C
C
C
C
C
C
C
C
D
E should be larger than 0° and smaller than 90°
to ensure that the entire trace segment is linearly stretched; and
(2) Angle B
?
A
C
C
C
C
C
C
C
C
C
D
)
@
?
C
C
C
C
C
C
C
C
C
C
C
C
C
D
E should be larger than 0° and smaller than
θ
to control the local direction of connection.
If
A
satisfies the connection conditions, it will be selected as the
new end point (green point in Fig. 5c); otherwise, the nearest point of the
remaining connection neighbors of end point will be judged. The
connection will end if none point of the remaining connection neighbors
satisfies the connection conditions.
(a) (b) (c)
Fig. 5. Trace feature point connection algorithm: (a) Trace feature points before connection; (b) The criterion of trace feature point connection (the start point of the trace
segment is plotted in the pink circle, the points added during connection are plotted in green, red points are the connection neighbors of the end point which is plotted in the
green circle, and the 1/3 point
)
@
of the segment is plotted in black circle); and (c) Trace feature points after connection.
2.4. Trace segment linearization
Because the algorithm of feature point connection in Section 2.3
might falsely connect feature points of different traces (Fig. 6a), this step
performs linear partition on trace segments to generate linearized
segments which are composed only of the feature points of same traces.
The method we used for trace segment linearization is called principal
component analysis (PCA) (Pearson, 1901). Given a set of point
0
1
, PCA analysis starts from calculating the covariance
matrix:
T
p 0 0
1
1[( )( ) ]
=
= − −
∑
n
i i
i
n
M v c v c
(7)
where F is the number of points, and
<
is the centroid coordinate of the
point clouds. Because the matrix
A
is symmetric and positive, it can be
decomposed by eigenvalue as
3T
p1
( )
λ
=
′=
∑
i i i
i
M e e
(8)
where its eigenvalues are (
* (
* (
)
. Then we use parameter u to
measure the degree of linearization, which is defined as
1
1 2 3
λ
λ λ λ
=+ +
u
(9)
For each segment, the procedures of linearization are as follows:
(1) Compute u using PCA algorithm;
(2) If u is smaller than linearization threshold G, the trace segment is
divided into two small segments containing same number of feature
points; and
(3) For each divided segment, perform Steps (1) and (2) iteratively until
all the u values of divided segments are greater than or equal to G, as
shown in Fig. 6b.
(a) (b)
Fig. 6. Trace segment linearization algorithm: (a) The trace segment before linearization; and (b) The result of the segment linearization in (a) and each color represents a
linearized trace segment.
2.5. Trace segment connection
Through the above analyses, we obtained discrete and linearized trace
segments. In this section, the trace segments were connected to form
continuous traces. Because that short linearized segments are affected
more easily by noisy segments, the connection algorithm gives the long
linearized trace segments the priority to be connected. As shown in Fig.
7a, given the blue segment needs to be connected and the four red
segments are the ones that have not been connected. As shown in Fig. 7b,
point
H
that needs to be connected is plotted in green, and possible
connect points
I
are plotted in yellow. Both
A
and
J
are one third
points of corresponding segments and are plotted in pink and red,
respectively.
(a) (b) (c)
Fig. 7. The trace segment connection algorithm: (a) The segments before connection (the blue segment is the one needed to be connected and the red segments are the ones
that have not been connected); In (b), endpoint needed to connected is plotted in green, endpoints which have not been connected are plotted in yellow, and pink and red
points represent the 1/3 points of their belonging trace segments; (c) The connection result of (a), which is the stretched blue segment.
The connection rules are defined as follows:
(1) Axial distance 9
7
K
H
I
C
C
C
C
C
C
C
C
C
D
L MNO B
H
I
C
C
C
C
C
C
C
C
C
D
J
H
C
C
C
C
C
C
C
C
C
D
E K is smaller than
threshold P to control the axial extension, and 9
7
is smaller than
P
<
Q
(which is defined as 3 times the average edge length of triangular
mesh by data test) if =
H
I
C
C
C
C
C
C
C
C
C
D
L
J
H
C
C
C
C
C
C
C
C
C
D
B +;
(2) Radial distance R
7
K
H
I
C
C
C
C
C
C
C
C
C
D
L OST B
H
I
C
C
C
C
C
C
C
C
C
D
J
H
C
C
C
C
C
C
C
C
C
D
E K is smaller than
threshold ;
to control the radial extension;
(3) Angle U
7
VWMMNO=0K
H
I
C
C
C
C
C
C
C
C
C
D
L
J
H
C
C
C
C
C
C
C
C
C
D
KXY
H
I
C
C
C
C
C
C
C
C
C
D
YY
J
H
C
C
C
C
C
C
C
C
C
D
Y1 is smaller than
threshold U
to control the extent of trace curvature; and
(4) Given that the set S is composed of the segments which satisfy the
above three conditions, the segment which should be connected is the
one in S that has the minimum radial distance =R
!Z[
.
The segment with =R
!Z[
is selected to be the next connect segment
because it is the nearest one to the extension cord of the under connect
segment (as the blue segment shown in Fig. 7a) and reflects the extension
trend of trace most. Fig. 7c shows the connection result of Fig. 7a.
In the algorithm, the longest segment is connected first and the
connection continues until none of the other segments satisfies the
connection rules, and the longest segment of the remaining ones that have
not been connected is the next to be connected. The algorithm proceeds
until all segments are longer than threshold \
<
(5 times the average edge
length of triangular mesh by data test) connected.
3. Applications
3.1. Case A
3.1.1. Data description
The point cloud data of this case study were obtained from an
available Rockbench repository (Lato et al., 2013). The natural rock mass
outcrop of a road cut slope is located in Ouray, Colorado, USA. The
scanning was carried out by an Optech Ilris3D scanner and obtained
1,515,722 points with the resolution of about 2 cm. The rock mass of this
case is shown in Fig. 8 and the region under analysis is in the red frame.
Fig. 8. Real road cut slope analyzed in case study A. Image from Rockbench
repository. Analyzed region is in the red frame.
3.1.2. Trace mapping procedure
The feature points detected by NTV are shown in Fig. 9. For clear
observation, feature points without triangular meshes are shown in Fig.
10a. Fig. 10b-e shows each step of the proposed method and Fig. 10f
shows both the feature points and the detected traces.
3.1.3. Mean trace length calculation
The mean trace length is calculated through the circular window
sampling method (Zhang and Einstein, 1998) with an automated trace
sampling procedure (Umili et al., 2013). Firstly, the detected traces were
projected orthogonally on the sampling plane (X-Y plane). Then, the
centers of nine circular windows with five different radii were placed
symmetrically on the sampled region. The radii are 10%, 15%, 20%, 25%,
30%, and 40% of the short edge length of sampled region. The mean trace
length is calculated by (Umili et al., 2013):
]
^_
`
a
(10)
where r denotes the radius of circular window, m denotes the number of
traces with endpoints inside the circular window, and F
b
denotes the
number of intersections which are between traces and the bounding
circular scanline. Fig. 11 shows the sampled plane of circular window
method in case A.
3.2. Case B
3.2.1. Data description
The data of this case were obtained from two excavation faces of a
highway rock tunnel in Yuexi County, Anhui Province, China (Li et al.,
2016). The tunnel was 7.548 km long and was excavated using
drill-and-blast method. The point cloud was obtained using overlapping
photographs (Roncella and Remondino, 2005; Haneberg, 2008;
Sturzenegger and Stead, 2009) to create 3D surfaces. Fig. 12 shows the
excavation face and the 3D point clouds reconstruction region under
analysis is in the red frame.
3.2.2. Trace mapping of case B(a)
The feature points detected by NTV are shown in Fig. 13. For clear
observation, feature points without triangular meshes are shown in Fig.
14a. Fig. 14b-e shows each step of the proposed method and Fig. 14f
shows both the feature points and the detected traces.
3.2.3. Mean trace length calculation
The method of mean trace length calculation was the same as that
depicted in case A (Section 3.1.3). Fig. 15 shows the sampled plane of
circular window method.
Fig. 9. Trace feature points. The triangular meshes are plotted in black.
(a) Trace feature points. (b) Trace feature point contraction.
(c) Trace feature point connection. (d) Trace segment linearization.
(e) Trace segment connection. (f) Effect of trace mapping.
Fig. 10. Trace mapping of case A. (a-e) shows the trace mapping procedures of case A. Each color represents a trace. In (f), the blue segments are detected traces and the red
points are feature points detected by NTV.
Fig. 11. Trace projection of case A on the sampled plane of circular window
method.
3.3. Sensitivity analysis and calibration
Threshold contraction radius =;
<
, threshold angle =U
<
, linearization
threshold G, threshold distance =P, threshold distance =;
, and threshold
angle U
are important parameters for the proposed method for
discontinuity trace mapping. Therefore, the sensitivity analysis will be
performed on these parameters by applying the proposed method to case
A. Besides, the sensitivity analysis is dimensionless and the default set of
these parameters is selected as ;
<
= 2.5L
0
, U
<
= 65°, U = 0.98, P =
17.5L
0
, ;
= 5L
0
and U
= 30°. When one of the parameters changes
during the analysis, other parameters remain default values. Therefore,
there are actually 42 combinations of these parameters that have been
really tried.
The parameters are classified into seven levels respectively as shown
in Table 1. The mean trace length was calculated through the circular
window sampling method (Zhang and Einstein, 1998) with an automated
trace sampling procedure (Umili et al., 2013) which was the same as
depicted in Section 3.1.3.
(a) (b)
Fig. 12. Tunnel excavation faces of case B. (a) shows the excavation face in ZK21+697.9 mileages and (b) shows the excavation face in ZK21+672.3 mileages. Analyzed
region is in the red frame.
Fig. 13. Trace feature points of case B(a). The triangular meshes are plotted in black.
(a) Trace feature points. (b) Trade feature point contraction.
(c) Trace feature point connection. (d) Trace segment linearization.
(e) Trace segment connection. (f) Effect of trace mapping.
Fig. 14. Trace mapping of case B(a). (a-e) shows the trace mapping procedures of case B(a). Each color represents a trace. In (f), the blue segments are detected traces and
the red points are feature points detected by NTV.
The trace projection is the same as Fig. 11. As shown in Fig. 16a, for
threshold ;
<
, the local maximum values of the mean trace are at levels 1,
4 and 7. ;
<
is defined to control the contraction range of feature points.
The detected skeletons of feature points are coarse (Fig. 17a) if ;
<
is too
small, and the details of traces will be obscure and even lost (Fig. 17b) if
;
<
is too large. In addition, the width of traces obtained by NTV is
mostly 1 to 2 ring-neighbors (Fig. 4a). Therefore, the optimal ;
<
is
defined between 2 and 3 times the average edge length of triangular mesh.
As shown in Fig. 16a, for U
<
, the overall trend is that mean trace length
increases obviously from level 4 to level 5 and stabilizes comparatively at
other levels. Threshold U
<
is defined to control the connection directions
of feature points, especially at joints of skeletons, and to generate linear
trace segments. The generated trace segments will not be necessarily
linear and too short (Fig. 17c) if U
<
is too small, and will falsely connect
feature points that belong to different traces if U
<
is too large (Fig. 17d).
Therefore, the optimal U
<
is defined between 60° and 70° to ensure that
the mean trace length is comparatively large and the false connection
caused by large U
<
is reduced. As shown Fig. 16a, the mean trace length
is relatively stable with the variation of threshold G. G is defined to
control trace segments linearization and is fundamentally used to separate
the segments which belong to different traces whereas being connected
falsely. Segments that are connected falsely cannot be separated if G is
too small (Fig. 17e) and can be separated satisfactorily if G is defined a
relatively large value (Fig. 17f). Therefore, the optimal G is defined
between 0.96 and 0.98 by data test.
Fig. 15. Trace projection of case B(a) on the sampled plane of circular window
method.
Table 1. Threshold parameter level. Parameter \
<
denotes the average edge length of triangular mesh.
Parameter level
;
<
U
0
(°) U
D
;
U
(°)
1 1L
0
20 0.8 5L
0
2L
0
10
2 1.5L
0
30 0.85 10L
0
4L
0
20
3 2L
0
45 0.9 15L
0
6L
0
30
4 2.5L
0
60 0.92 20L
0
8L
0
40
5 3L
0
70 0.94 25L
0
10L
0
50
6 3.5L
0
80 0.96 30L
0
12L
0
60
7 4L
0
90 0.98 35L
0
14L
0
70
(a) (b)
Fig. 16. Mean trace length with different threshold parameter levels.
(a) ;
<
\
<
. (b) ;
<
c\
<
.
(c) U
0
d+e. (d) U
0
f+e.
(e) G +g. (f) G +fg.
Fig. 17. Trace mapping effect under different threshold parameters of ;
<
, U
0
and G. Each color represents a trace.
As shown in Fig. 16b, the mean trace length increases as thresholds of
P,=;
or U
increase. Because the above three thresholds define the axial,
radial and curve extents of trace segment connection, the larger they are
defined, the more easily trace segments will be connected. However, too
large values of P, ;
and U
will falsely connect trace segments which
belong to different traces (Fig. 18a, c and e) and segments that belong to
the same trace cannot be connected effectively if they are defined too
small (Fig. 18b, d and f). Through data tests based on cases A and B, the
optimal value of P is between 15 and 20 times the average edge length
of triangular mesh. The optimal ;
is between 6 and 8 times the average
edge length of triangular mesh, and U
is between 30° and 45°.
(a) P h\
<
. (b) P ih\
<
.
(c) ;
j\
<
. (d) ;
dc\
<
.
(e) U
d+e. (f) U
f+e.
Fig. 18. Trace mapping effect under different threshold parameters of P, ;
and U
. Each color represents a trace.
4. Discussion
Li et al. (2015) proposed a growth method of discontinuity trace
mapping on 3D digital surface model (DSM). Based on feature points
detected by NTV, they detected traces by the procedures of trace feature
point grouping, trace segment growth, trace segment connection, and
redundant trace segment removal. For comparison, both the growth
method and our method were employed to detect traces of case A in this
section. Both the methods started with the same feature points generated
by NTV and ended with traces finally detected. Mean trace length were
calculated using the circular window method which was the same as
depicted in Section 3.1.3.
4.1. Comparison of trace mapping effect
As shown in Figs. 19-21, the shortcomings of traces detected by the
growth method can be summarized as follows: (1) traces are coarse; (2)
shapes of traces are easily affected by noisy points; and (3) some trace
segments that belong to different traces are connected falsely.
(a) (b)
(c) (d)
Fig. 19. Comparison of trace mapping effect of case A between different methods: (a, c) The traces detected by the growth method; and (b, d) The traces detected by our
method. In (a, b), each color represents a trace. In (c, d), blue segments represent detected traces while red points represent feature points.
(a) (b)
(c) (d)
Fig. 20. Comparison of trace mapping effect of case B(a) between different methods: (a, c) The traces detected by the growth method; and (b, d) The traces detected by our
method. In (a, b), each color represents a trace. In (c, d), blue segments represent detected traces while red points represent feature points.
(a) (b)
(c) (d)
Fig. 21. Comparison of trace mapping effect of case B(b) using different methods: (a, c) The traces detected by the growth method; and (b, d) The traces detected by our
method. In (a, b), each color represents a trace. In (c, d), blue segments represent detected traces while red points represent feature points.
Comparatively, traces detected by our method were smoother and
linear because they were composed of contracted feature points that
aggregated on skeletons instead of separate feature points that were
scattered around traces. Therefore, the contraction algorithm (in Section
2.2) was robust to noisy points and could reflect the principal position of
traces accurately. In addition, it can be seen that the false connection of
trace segments belonged to different real traces were decreased because
the connection algorithm (in Section 2.5) reflected the principal extension
trends of trace segments more accurate.
4.2. Comparison of trace mapping efficiency
Both the growth method and our method were programed by
MATLAB (2017a) software and performed on an Intel Core I7-8700k and
16 GB DDR4 RAM. For comparison of algorithm efficiency, we used
different point clouds resolution to sample the analyzed regions of both
cases B(a) and B(b). As shown in Table 2, the running time of our method
is shorter than that of the growth method. As shown in Fig. 22, with the
number of feature points increases, the running time of the growth method
increases significantly faster than the time increment of our method.
Table 2. CPU time. For clearer comparison, cases B(a)-1 and B(a)-2 are generated from case B(a) using different point clouds sampling resolutions, which is the same as
cases B(b)-1 and B(b)-2.
Case
Number of initial
feature
points
Running time (s) Ratio of time (%) Efficiency multiples
Growth Contraction
A 18,104 28.83 19.04 66 1.51
B(a) 24,160 56.64 22.98 40.6 2.46
B(a)-1 33,510 89.19 33.57 37.6 2.66
B(a)-2 41,130 128.39 41.5 32.3 3.09
B(b) 53,765 182.6 53.66 29.4 3.4
B(b)-1 67,566 289.12 79.99 27.7 3.61
B(b)-2 75,206 375.37 91.12 24.3 4.12
Fig. 22. Comparison of running time.
4.3. Parameter settings
Based on the analysis in Section 3, the optimal parameter settings
were selected as ;
<
= 2.5L
0
, U
<
= 65°, U = 0.98, D = 17.5L
0
, ;
= 5L
0
and U
= 30°. The optimal parameter settings were suggested as default
parameters, because for all point cloud data tested, the trace recognition
results based on the optimal parameter settings were more consistent with
that observed from the pictures or the point clouds of the corresponding
rock masses than the results of the growth method (Li et al., 2016).
In addition, the parameter settings were influenced by point clouds
quality, because parameter settings were dependent on the recognition
results and the recognition results were influenced by the point clouds
quality. The point clouds quality was interfered by many factors. Plants,
shelters of natural rock mass and trivial grains and fractures caused by
blasting disturbance of tunnel excavation faces were the disturbance
factors when point clouds were obtained directly from laser scanning.
When point clouds were obtained from 3D reconstruction based on
photogrammetry, light intensity, shadows, dust and even lens distortion of
cameras were considered as disturbance factors.
Although there were many factors influencing point clouds quality
and the parameter settings, the proposed method based on the optimal or
default parameters was more robust than the growth method (Li et al.,
2016). The reasons are: (1) For the proposed method, point cloud skeleton
extraction (Section 2.2) reduces the interference of noisy points; and (2)
Trace segments connection (Section 2.5) considers more about the real
extending trend of traces than the growth method.
In conclusion, there is no need for the proposed method to select new
parameters when every time encountered a new rock mass, for the optimal
or default parameters have served well for trace recognition based on
point cloud data we have had. We suggest reselecting the parameters only
when the results exceed the users’ expectation or specific results that the
users need.
5. Conclusions
This paper proposed a new method for trace mapping based on 3D
point clouds of rock mass surfaces. Features points were generated by
NTV first, and then the proposed method was performed to detect traces.
Compared with the growth method, our method (trace feature points
contraction, trace feature points connection, trace segments linearization,
and trace segments connection) performed two principal advantages: (1)
Trace mapping result was more accurate because the detected traces were
smoother, more linearly outstretched and more robust to noisy points, and
the detected traces could better match the principal trends of the real
traces; and (2) The proposed method was more efficient and the
enhancement of efficiency was more remarkable as the number of feature
points increased.
A sensitivity analysis was conducted to identify the optimal
parameters of the proposed method. In our cases, the optimal threshold
radius ;
<
in contraction algorithm was 2-3 times the average edge length
of triangular mesh. The optimal threshold angle U
<
in feature point
connection algorithm was 60°-70°. The optimal linearization threshold G
in trace segment linearization algorithm was 0.96-0.98. In trace segment
connection algorithm, the optimal threshold distance P was 15-20 times
the average edge length of triangular mesh. The threshold distance ;
was 6-8 times the average edge length of triangular mesh, and threshold
angle U
was 30°-45°.
The case study indicated that the proposed method provided more
efficient and accurate measurements of discontinuity geometric
parameters. As a supplement to traditional measurement of discontinuity
traces, the proposed method could achieve quick and accurate trace
mapping in engineering fields. The results of the proposed method can be
used to (1) calculate the tunnel surrounding rock quality indices such as
rock mass rating (RMR) (Bieniawski, 1988) and Q value (Barton, et al.,
1974), (2) evaluate the rock mass blasting disturbance during tunnel
excavations, (3) divide units of tunnel surrounding rock block units, (4)
construct models of geological bodies (ISRM, 1978), and (5) serve for
mechanism analysis of tunnel surrounding rock (Zhu et al., 2016).
Declaration of Competing Interest
The authors wish to confirm that there are no known conflicts of
interests associated with this publication and there has been no significant
financial support for this work that could have influenced its outcome.
Acknowledgments
This work was supported by the Special Fund for Basic Research on
Scientific Instruments of the National Natural Science Foundation of
China (Grant No. 4182780021), Emeishan-Hanyuan Highway Program,
and Taihang Mountain Highway Program.
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Keshen Zhang obtained his BSc degree in Civil Engineering from Shandong Jianzhu University, China, in 2011, and his MSc degree in
Architecture
and Civil Engineering from Tongji University, Shanghai, China, in 2019. He is now a PhD candidate majoring in Civil Engineering at
Tongji
University. His research interest is elaborate collection and analysis of rock mass information based on photogrammetry and LIDAR.
