Laser-scanned as-built 3D modeling of air-conditioning ducts

based on Manhattan world assumption

Masaya Narumi1, Satoshi Kanai2, Hiroaki Date3, Eisuke Wakisaka4, and Shintaro Sakamoto 5

1) M.S., Student, Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan. Email:

m_narumi@sdm.ssi.ist.hokudai.ac.jp

2) Dr. Eng., Prof., Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan. Email:

kanai@ssi.sit.hokudai.ac.jp

3) Dr. Eng., Assoc. Prof., Graduate School of Information Science and Technology, Hokkaido University, Sapporo, Japan.

Email: hdate@ssi.ist.hokudai.ac.jp

4) Researcher., Research & Development Center, Shinryo Corporation, Tsukuba, Japan. Email: wakisaka.ei@shinryo.com

5) Dr. Eng., Senior Researcher., Research & Development Center, Shinryo Corporation, Tsukuba, Japan.

Email: sakamoto.sh@shinryo.com

Abstract:

We propose a fully automatic recognition algorithm for air-conditioning ducts with rectangular section (rectangular ducts) from massive laser-scanned point clouds for 3D as-built modeling of ducts. Almost all air-conditioning ducts are installed along three orthogonal dominant axes in building spaces, and the Manhattan world assumption is employed for efficient recognition. In the proposed algorithm, the orthogonal axes are first recognized from the point clouds, and then the rectangular sections of the ducts are detected on the plane perpendicular to one of the axes using the combinational Hough transform. Second, the extracted sections are mutually connected to estimate the 3D straight portions of the ducts. Finally, the multiple straight portions of the ducts are connected and the duct portions which lack scanned points are filled via region-growing-based interpolation. The algorithm was evaluated, achieving around 89% precision and recall for recognizing the straight portions of the rectangular ducts and 83% recall for connections among ducts. Keywords: object recognition, air-conditioning ducts, laser scanning, Manhattan-world assumption

1. INTRODUCTION

The frequency of updating air-conditioning systems in manufacturing facilities and buildings has been increasing along with, the need for as-built modeling of air-conditioning ducts for more efficient renovation works. A duct with a rectangular section (rectangular duct) is a major component of the piping of air-conditioning systems (Figure 1); therefore, as-built modeling of rectangular ducts is urgently needed. Although 3D point clouds obtained from terrestrial laser scanners have been generally used for this purpose, as-built 3D modeling of rectangular ducts from massive 3D laser-scanned point clouds is a manual and tedious work.

Several studies have investigated the automatic recognition of piping systems from massive laser-scanned point clouds [1, 2, 3, 4]. However, they only focused on recognizing and creating as-built models of cylindrical pipes in plants, and they cannot be directly applied to the recognition and modeling of rectangular ducts. While,

Figure 1. A process of as-built 3D modeling of air-conditioning ducts

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commercial software is available for creating as-built 3D models of rectangular ducts [5] based on laser-scanned point clouds, it still requires manual intervention and has not achieved complete automatic recognition and modeling.

Therefore, to address this need, this study has proposed a fully automatic algorithm for 3D as-built modeling of air-conditioning rectangular ducts with connection parts from massive laser-scanned point clouds. Since almost all air-conditioning ducts are arranged along the three orthogonal dominant axes in facility space, we employed the Manhattan world assumption for efficient recognition and modeling. The algorithm is described in detail and its recognition performance is verified for laser-scanned point clouds.

2. RELATED WORKS

For object recognition from laser-scanned point clouds in industrial scene, various methods have been proposed. Cabaleo [6] proposes a recognition method for metal frames using projection of the scanned points on the floor plane and Hough transform based line detection. And Sara [7] also proposed a recognition method for primitive shape objects using sharp feature detection and object detection based on predefined library. However, these methods do not work for the scanned points including a lot of defects, and it’s difficult to apply it to the point cloud of complex duct piping system with many occlusions.

Moreover, Hyojoo [8] and Jing [9] respectively propose object recognition method for complex shaped objects such as indicators or instrumentation by making database from target objects shape and matching the point clouds with the object databases. However, these methods can only work for specific or standardized shaped objects, and cannot work for the recognition the shapes of rectangular ducts whose shapes are unstandardized and length are not uniquely defined.

On the other hand, Mizoguchi [10] and Mahmond [11] propose a recognition method for cylindrical pipes based on the fact known as the Manhattan world assumption [12] that most of pipes are installed along with the major orthogonal axes of the building, and detecting circular sections of the planes perpendicular to the axes. However in [10], detail shape of the pipes and their connection are not be detected. While in [11], the connectivity of the pipes is not recognized, and occlusion robustness previously discussed is not considered.

3. RECOGNITION ALGORITHM FOR AIR-CONDITIONING DUCTS WITH RECTANGULAR SECTIONS

3.1 Overview of the algorithm

As shown in Figure 1, most air-conditioning ducts are placed almost parallel to one of the dominant orthogonal axes, defined by the intersections among ceilings or wall surfaces. Therefore, we assume that the air-conditioning ducts are installed according to the Manhattan world assumption. Thus, the recognition algorithm of ducts from laser-scanned point clouds is simplified.

The proposed algorithm involves four steps (Figure 2). First, the three dominant orthogonal axes are estimated by extracting the floors, ceilings, and wall surfaces from the input point clouds (A1). Second, the rectangular sections of the ducts are detected from the cross-sectionals point clouds perpendicular to one of the estimated axes (A2). Third, straight dusts are recognized by connecting the rectangular sections of the ducts (A3). Fourth, the connected parts of the straight ducts are recognized and missing parts of the ducts are then filled via region-growing-based interpolation (A4). Finally, the recognition results are exported to a dedicated CAD system for air-conditioning system design (A5). In the following sections, algorithms steps A1 to A5 are described.

3.2 Estimation of the three dominant orthogonal axes from point clouds (A1)

3.2.1 Detecting point clouds on the floor

First, the distribution histogram of the vertical points is generated from the input point clouds 𝑃𝑂 = {𝑝𝑖𝑜},

followed by a search for the interval in which a majority of the points reside. Next, a subset of point clouds within, the interval is extracted as the candidate floor surface.

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Subsequently, all planar regions are extracted from the candidate though region growing, and the region with the largest number of points is considered the floor plane.

3.2.2 Estimation of the three dominant axes

Next, the floor plane is fitted using least-squares methods to the floor plane, and the normal vector, which is treated as the dominant z axis, is calculated from the floor plane. Subsequently, the input set of point clouds 𝑃𝑂 is transformed to 𝑃𝑧 = {𝑝𝑖

𝑧} and the dominant z axis is aligned with the z axis of the global coordinate system. Then, as shown in Figure 3(a), at all transformed point clouds ∀𝑝𝑖

𝑧 ∈ 𝑃𝑧 , the normal vector 𝒏𝒊 is

Figure 2. Overview of the recognition and as-built modeling algorithm

Figure 3. Estimation and alignment of dominant axes

(b) 𝒏𝒊′ , which is the

projection of 𝒏𝒊 (c) Calculation of 𝜃𝑖

(e) Generated point cloud set 𝑃𝑧𝑥

(a) Transformed point clouds 𝑃𝑧

with 𝒏𝒊

(d) Histgram with respect to 𝜃𝑖

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estimated using the k-nearest neighbor search and the local least-square plane fit. Then as shown in Figure 3(b), the vector 𝒏𝒊′, which is the projection of 𝒏𝒊 to the xy plane is calculated. Then as shown in Figure 3(c), the reference vector 𝒏𝒓

′ , which is parallel to the xy plane is arbitrarily specified, and the angle 𝜃𝑖 between 𝒏𝒊′ and 𝒏𝒓

′ is calculated. In addition, as shown in Figure 3(d), the direction of the dominant x axis is determined as the one that fits the median direction of the most frequent interval in the histogram with respect to 𝜃𝑖. Finally as shown in Figure 3(e), the point cloud set 𝑃𝑧 is transformed to 𝑃𝑧𝑥 to align the dominant x axis with the x axis of the global coordinate system. These steps generate the point cloud set 𝑃𝑧𝑥 where the dominant three axes of the object alignment are aligned with the xyz axes.

After obtaining the dominant axes, the point clouds of the floor plane form the background and are removed to improve the recognition efficiency. The point clouds of the ceilings and wall planes are similarly removed.

3.3 Detection of the sectional shapes of ducts with rectangular shapes (A2)

First, the point cloud sets 𝑃𝑧𝑥 is divided by a set of planes {𝑆𝑖} that are positioned at equal intervals and perpendicular to each dominant axis. Next, the partial point cloud 𝑃𝑑 bounded by two consecutive parallel planes 𝑆𝑖 𝑎𝑛𝑑𝑆𝑖+1 is obtained, and the sectional point cloud 𝑃𝑠 is obtained by projecting 𝑃𝑑 onto 𝑆𝑖 . Thereafter, the point cloud 𝑃𝑐 that is located on the longitudinal edges of straight ducts is extracted as follows.

(1) As shown in Figure 4(a), the corner point 𝑝𝑣𝑖 that is located at one of the right-angle corners of a rectangle is identified in 𝑃𝑠 using the combinational Hough transform [13]:

(2) the point cloud 𝑃𝑠𝑣 , which is the union of 𝑃𝑠 and 𝑃𝑣 = {𝑝𝑣𝑖}, is partitioned to {𝐶𝑖} using Euclidian clustering (Figure 4(b)) and, as shown in Figure 4(c), the axis-aligned bounding box 𝐵𝑖 is fitted to 𝐶𝑖, which contains only the corner point 𝑝𝑣𝑖; and,

(3) about each 𝐵𝑖 and 𝐶𝑖 (Figure 4(d)), the appropriateness of 𝐵𝑖 as the sectional rectangle of the duct is examined by equation (1)

where 𝑁𝑖𝑛𝑛𝑒𝑟 is the number of points near 𝐵𝑖’s centroid and 𝐴𝑖 the aspect ratio of 𝐵𝑖 . Figure 4(e) shows that if 𝐵𝑖 satisfies equation (1), it is identified as the cross-section of a rectangular duct. Finally, the four vertices

of 𝐵𝑖 are recorded as the vertices of the duct.

𝑁𝑖𝑛𝑛𝑒𝑟 ≤ 𝑁𝑡ℎ (𝑁𝑡ℎ: points threshold) 𝐴𝑡ℎ1 ≤ 𝐴𝑖 ≤ 𝐴𝑡ℎ2 (𝐴𝑡ℎ1, 𝐴𝑡ℎ2: aspect ratio threshold)

(1)

Figure 4. Extraction of the sectional shapes of ducts

(a) 𝑃𝑠𝑣 ,sum set of 𝑃𝑠 and 𝑃𝑣 (b) Euclidian clustering (c) Fitting bounding box

(d) Judging adequacy of duct’s sectional shapes (e) Estimated vertices of duct’s sectional shapes

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3.4 Recognition of straight ducts (A3)

3.4.1 Recognition of longitudinal edges in a straight duct

From the all points of 𝑃𝑐 obtained in chapter 2.3, a seed point 𝑝𝑠𝑒𝑒𝑑 is chosen from the vertices at the corners of the rectangle 𝐵𝑖 with the highest point density relative to its perimeter. Then, the other points located nearly on a virtual edge line which passes through 𝑝𝑠𝑒𝑒𝑑 and is oriented to the longitudinal direction of the duct are collected. The collected point clouds are further divided into small groups using Euclidian clustering. Among the points in a group, two end points 𝑝𝑠 and 𝑝𝑒 are found along the virtual edge, and a line segment 𝑙(𝑝𝑠, 𝑝𝑒) is generated. Finally as shown in Figure 5(a), a set of line segments located on four longitudinal edges of every duct are identified.

3.4.2 Recognition of straight ducts

After the disjointed line segments on the edges of ducts are found, the connectivity of the duct is finally recognized based on the overlap relation among the segments. As shown in Figure 5(a), at any point 𝑝𝑖 on line segment 𝑙, votes are cast for the other three line segments on the same duct. Then, as shown in Figure 5(b), the three segments 𝑙′, 𝑙′′, 𝑎𝑛𝑑 𝑙′′′ with the highest votes are collected from all points in 𝑙, and segment group 𝐿 ={𝑙, 𝑙′, 𝑙′′, 𝑙′′′} is constructed on the duct. As shown in Figure 5(c), the endpoints of the four segments 𝑙, 𝑙′, 𝑙′′, 𝑎𝑛𝑑 𝑙′′′ are relocated to the longitudinal position limit where at least three out of four segments appear on the cross-section. Therefore, the geometry of both end faces of the duct are beautified to form congruent rectangles, and a straight duct shape with relatively short length is generated as cuboid 𝑣. Finally, a complete straight duct 𝐷 is recognized by taking a union of multiple cuboids 𝑣 that intersect along the longitudinal direction.

After the straight ducts are recognized, the ducts located very close to any ceiling plane are recognized as beams and are removed.

3.5 Recognition of connections among straight ducts and interpolation of absent parts (A4)

After the recognition of straight ducts, the connectivity among them is estimated in the following manner. First, as shown in Figure 6(a), the Manhattan distance 𝑑𝑚 = 𝑑𝑥 + 𝑑𝑦 + 𝑑𝑧 is calculated between the centroids of the two end faces 𝑐1 and 𝑐2 on any different duct pair 𝐷1 and 𝐷2, where 𝑑𝑥 , 𝑑𝑦 , and 𝑑𝑧 are the x, y, and z components of vector 𝑐1 𝑐2⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗. If 𝑑𝑚 ≤ 𝑑𝑡ℎ1 , where 𝑑𝑡ℎ1 is the distance threshold, ducts 𝐷1 and 𝐷2 are considered as being connected to each other.

Then, if there are any connection parts between two ducts, the type of the connection parts (elbow, S-curve, or T-junction) is classified considering 𝑑𝑥 , 𝑑𝑦, and 𝑑𝑧 and the normal vectors of the two end faces 𝒆1 and 𝒆2. As shown in Figure 6(b), if one of 𝑑𝑥 , 𝑑𝑦 , or 𝑑𝑧 is close to zero and 𝒆1 ⊥ 𝒆2, the connection is classified to an elbow. If one of 𝑑𝑥, 𝑑𝑦 𝑜𝑟 𝑑𝑧 is close to zero and 𝒆1 ∥ 𝒆2, the connection parts is classified as S-curve. If two ducts are connected to the same duct, the connection is classified as T-junction.

Because ducts are mainly near ceilings and on top of each other, the point clouds of the two side surfaces of the

Figure 5. Recognition of straight ducts

(a) Segmented group on

longitudinal edge of duct

(b) Recognized segment

group 𝐿(𝑙, 𝑙′, 𝑙′′, 𝑙′′′)

(c) Forming process for each

segments 𝐿 contains

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ducts may be missing in the scanned data; moreover, the point clouds of their bottom surfaces can be captured only in some cases. Even using the algorithm described above, a rectangular duct model 𝐷 cannot be generated in the space where only the bottom surfaces of ducts are scanned. Therefore, to interpolate an additional duct model in space, the following algorithm is introduced. If the space between ducts 𝐷1 and 𝐷2 satisfies equation (2) and the number of scanned points on the bottom portion of the duct’s space is adequate, a new duct 𝐷12, which has the same sectional shape as 𝐷1 and 𝐷2 and includes both 𝐷1 and 𝐷2, is inserted and it replaces 𝐷1 and 𝐷2 (Figure 6(c)).

However, if one of the two straight ducts is missing in the above case, the missing space between the ducts cannot be interpolated. To solve this, the following region-growing-based algorithm is implemented. As shown in Figure 6(d), four boundary edges on the end face of duct 𝐷 are first identified, and the number of scanned points close to either of the edges is examined. Then, if the number of points exceeds the threshold 𝑑𝑡ℎ3, the position of the end face moves outward by 𝑑𝑡ℎ3. This process is repeated until a small number of scanned points exist near the boundary edges of the duct’s end faces.

By performing this process for all duct combinations, the rectangular duct piping system that consists of straight ducts and connecting elements is recognized.

4. RECOGNITION PERFORMANCE

The indoor space of a building where air-conditioning ducts were installed and ceiling boards were removed during renovation was scanned by a terrestrial laser scanner(Leica HDS 7000). The original scanned points included around 94 million points generated from nine scanner positions. The points were down-sampled to 2,836,148 points using the voxel grid method of point cloud library(PCL) [14] with a 20 mm voxel width. Then the down-sampled points were used as the input point cloud (Figure 7).

𝑑𝑚 ≤ 𝑑𝑡ℎ2 (𝑑𝑡ℎ2: threshold of Manhattan distance, 𝑑𝑡ℎ1 ≤ 𝑑𝑡ℎ2)

𝑑𝛼 ≅ 0, 𝑑𝛽 ≅ 0 (α, β ∈ {𝑥, 𝑦, 𝑧}, and directions different from the longitudinal one) (2)

Figure 6. Recognition of connection between each straight ducts and interpolation of absent part

(b) Recognition type of connection

(c) Interpolation using Manhattan distance (d) Interpolation using region-growing

(a) Calculation Manhattan distance

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We developed C++ prototype software for the as-built modeling of rectangular ducts with the aid of PCL. A set of rectangular ducts were recognized with the software using the parameters shown in Table 1. The point clouds that included the floors, ceilings and wall planes were automatically removed from the input point cloud as shown in Figure 8, and the rectangular ducts in the remaining points were finally recognized. Figure 9 shows the recognized ducts as colored points and the connections as colored lines. Though many beams with rectangular sections resembling rectangular ducts exist on the ceiling, the result shows that only the ducts and their connections were appropriately extracted and recognized without any confused. As listed in Table 2, the total processing time for the duct recognition was about 6 min for 2.8 M points, and sectional analysis has a dominant computational cost.

Table 2. Processing time for recognition

Time to processing [sec]

Elimination of background 75.0

Sectional analysis 179.3

Recognition of straight duct 0.4

Recognition of connection and

interpolation of absent part95.4

Total 350.0

Parameter Value

[points]

1.0

3.0

1.5[m]

3.5[m]

50[mm]

Table 1. Parameter setting

Figure 7. Input point clouds (2,836,148 points) Figure 8. Background removed point clouds

(1,144,460 points)

Figure 9. Recognition result of rectangular ducts

(colored points show straight ducts and colored lines shows connections among them)

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To evaluate the recognition success rate, we first visually identified the actual straight duct portions in the input

point clouds (true duct) and compared them with the recognition results. The omission or commission duct

portions in the recognized ducts were examined, and their lengths were visually estimated. The accuracy

evaluation was based on the duct length, where recall is defined as the length (ducts successfully

recognized)/length (all true ducts), and precision as length (ducts successfully recognized)/length (all

recognized ducts). The recognition accuracy results are summarized in Table 3. About 89% precision and recall

were achieved in the case of straight ducts. Moreover, the recall for connections between the recognized straight

ducts was 83% in terms of the number of connections. The over-recognized portions mainly consisted of small

straight portions close to the end faces of the elbow, S-curve, and T-junction (Table 3). On the other hand, under-

recognized portions did not have enough scanned point density for region growing on the side or bottom surface

of the ducts.

5. DISCUSSION

As listed in Table 3, some amount of omission (11%) and commission (10%) errors have been observed in the recognition result of straight ducts. The main reason of them are examined in this section.

The point clouds included in the red and yellow rounded-rectangles in Figure 9 were not recognized as duct portions and were classified as omissions. These omissions are fundamentally caused by very sparse scanned point density, but they give the following different results of under-recognition.

In yellow rectangle portions, region growing processes could not continue due to the very sparse point density. On the other hand, in red rectangle portions, the point clouds in upper portion have an enough point density to interpolate the duct shape by region growing. However, a narrow portion between the upper portion and lower green duct does not include any point clouds due to occlusions, and the portion blocked region growing. This issue could be solved by the right selection of the threshold of an extension range in region growing. However, this extension might increase a commission error, and a right threshold has to be carefully selected. The right threshold selection still remains as an open problem.

Meanwhile, the over-recognized ducts as commission appear near the end of the recognized rectangular ducts, because, as shown in Figure 10, a part of the point clouds sampled in the entry or exit of a curved ducts (shown in two red rectangles in the figure) connected to rectangular ducts tend to be falsely joined to the part of a rectangular duct sections. However, these commissions can be corrected by post-processing in 3D CAD systems of duct design, and do not become an important issue.

Recall 89%

Precision 90%

F-measure 0.89

Table 3. Accuracy of recognition of straight ducts

Recognized ducts Under recognized All true ducts

Ground truth 51.78[m] 6.67[m] 58.45[m]

Over recognized 5.79[m]

All recognized ducts 57.57[m]

Figure 10. Example of over recognition

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There is still another potential limitation in the proposed algorithm. For the correct recognition, a certain number of points must be scanned on at least two faces among four in every rectangular duct. However, as shown in Figure 11, even though a sufficient number of the points are sampled on two faces, the dimension of recognized ducts could be smaller than actual ducts when the points are not distributed on the whole surfaces due to the occlusion.

6. CONCLUSION

We proposed a recognition algorithm of air-conditioning ducts with rectangular sectional shapes from massive laser-scanned point clouds for 3D as-built modeling of ducts. Using Manhattan world assumption, efficient recognition and modeling were achieved, and the duct recognition problem was reduced to the combination of detected 2D rectangular sections of ducts on planes perpendicular to one the of three dominant axes, bridging the detected duct sections and recognizing the duct connection parts as elbows, S-curves, and T-junctions. A region-growing-based interpolation was also introduced to recover duct portions with insufficient number of scanned points. The total processing time for duct recognition was about 6 min for 2.8 M points, and we obtained around 90% precision and recall from the scanned point clouds for the straight portions of the rectangular ducts, and 83% recall for the duct connections. The omission and commission errors in the recognition and their reasons were also examined.

In future, we will investigate the applicability of the proposed algorithm to a larger scale and more complex point clouds, and include a recognition function for circular pipes by introducing a robust sectional shape detection method [15].

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