Entropy-Based Registration of Point Clouds Using Terrestrial Laser Scanning and Smartphone GPS
<p>Relation between P and Q: (<b>a</b>,<b>b</b>) two different postures of P and Q with the same distance r, and the distance r is not enough to recover the transformation between them; and (<b>c</b>) the uncertainty of Q’s location.</p> "> Figure 2
<p>The relation between γ and its corresponding entropy: points in (<b>a</b>,<b>b</b>) are previously registered, and the entropy is calculated when points in (<b>b</b>) are being rotated γ degrees around the z axis. The entropy change with γ is recorded in (<b>c</b>).</p> "> Figure 3
<p>Transformation from 3D to 2D: (<b>a</b>) original points; (<b>b</b>) ground filtering; (<b>c</b>) clustering result with small clusters colored by green; (<b>d</b>) filtering result and the number of points is 3,407,526; and (<b>e</b>) projecting result and the number of points in each block is rendered by different colors of the block center. The number of points is 27,706.</p> "> Figure 3 Cont.
<p>Transformation from 3D to 2D: (<b>a</b>) original points; (<b>b</b>) ground filtering; (<b>c</b>) clustering result with small clusters colored by green; (<b>d</b>) filtering result and the number of points is 3,407,526; and (<b>e</b>) projecting result and the number of points in each block is rendered by different colors of the block center. The number of points is 27,706.</p> "> Figure 4
<p>Search for the minimum entropy: (<b>a</b>,<b>b</b>) the original point clouds; (<b>c</b>,<b>d</b>) searching process; and (<b>e</b>) the postures corresponding to the minimum entropy.</p> "> Figure 5
<p>Search for the initial parameters: (<b>a</b>) the searching process for rotation angles; and (<b>b</b>) the searching process for initial scan distance.</p> "> Figure 6
<p>Two criteria for initial distance and rotation angles selection: (<b>a</b>) The entropy change when <math display="inline"> <semantics> <mrow> <msub> <mi>r</mi> <mi>k</mi> </msub> </mrow> </semantics> </math> = 14.2272 m and <math display="inline"> <semantics> <mrow> <msub> <mi>κ</mi> <mi>p</mi> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> = 93°, <math display="inline"> <semantics> <mrow> <msub> <mi>κ</mi> <mi>q</mi> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math> = 108°, in S1 and S2 of the first data set in <a href="#sec3dot1-sensors-17-00197" class="html-sec">Section 3.1</a>. The calculation of <math display="inline"> <semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <msub> <mi>r</mi> <mi>k</mi> </msub> </mrow> <mrow> <mi>A</mi> <mo>−</mo> <mi>M</mi> </mrow> </msubsup> </mrow> </semantics> </math> corresponding to 12.2272 m is also shown. (<b>b</b>) How <math display="inline"> <semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <msub> <mi>r</mi> <mi>k</mi> </msub> </mrow> <mrow> <mi>R</mi> <mo>−</mo> <mi>M</mi> </mrow> </msubsup> </mrow> </semantics> </math> works on S3 and S4 in the second data set in <a href="#sec3dot2-sensors-17-00197" class="html-sec">Section 3.2</a>.</p> "> Figure 7
<p>View of data set 1 after registration and three scan positions.</p> "> Figure 8
<p>IME results with different d<sub>G</sub>, when t<sub>G</sub> is fixed to 10 m: (<b>a</b>) mean angular error (MAE); (<b>b</b>) translation error (MTE); (<b>c</b>) scan distance error (∆d<sub>S</sub>); and (<b>d</b>) runtime (T).</p> "> Figure 8 Cont.
<p>IME results with different d<sub>G</sub>, when t<sub>G</sub> is fixed to 10 m: (<b>a</b>) mean angular error (MAE); (<b>b</b>) translation error (MTE); (<b>c</b>) scan distance error (∆d<sub>S</sub>); and (<b>d</b>) runtime (T).</p> "> Figure 9
<p>Nadir view of the spatial distributions for S1 and S2 after IME when: (<b>a</b>) d<sub>G</sub> = 1 m; (<b>b</b>) d<sub>G</sub> = 2.5 m; (<b>c</b>) d<sub>G</sub> = 5 m; and (<b>d</b>) d<sub>G</sub> = 10 m, and t<sub>G</sub> is fixed to 10 m.</p> "> Figure 10
<p>IME results with different t<sub>G</sub>, when d<sub>G</sub> is fixed to 2.5 m: (<b>a</b>) mean angular error (MAE); (<b>b</b>) mean translation error (MTE); (<b>c</b>) scan distance error (∆d<sub>S</sub>); and (<b>d</b>) runtime (T).</p> "> Figure 11
<p>Nadir view of the spatial distributions after IME when: (<b>a</b>) t<sub>G</sub> = 2.5 m; (<b>b</b>) t<sub>G</sub> = 5 m; (<b>c</b>) t<sub>G</sub> = 10 m; and (<b>d</b>) t<sub>G</sub> = 20 m and d<sub>G</sub> is fixed to 2.5 m.</p> "> Figure 12
<p>Distribution of the 6 scans.</p> "> Figure 13
<p>The IME result of S1–S6: (<b>a</b>) the Nadir view corresponding to the minimum entropy, which is a local convergence for IME; and (<b>b</b>) the theoretically correct posture for S1 and S2, which corresponds to the fifth small entropy.</p> "> Figure 14
<p>IME results with different d<sub>G</sub>, when t<sub>G</sub> is fixed to 2.5 m: (<b>a</b>) mean angular error (MAE); (<b>b</b>) mean translation error (MTE); (<b>c</b>) scan distance error (∆d<sub>S</sub>); and (<b>d</b>) runtime (T).</p> "> Figure 15
<p>IME results with different t<sub>G</sub> when d<sub>G</sub> is fixed to 0.5 m: (<b>a</b>) mean angular error (MAE); (<b>b</b>) mean translation error (MTE); (<b>c</b>) scan distance error (∆d<sub>S</sub>); and (<b>d</b>) runtime (T).</p> "> Figure 16
<p>Comparison of two initial value selection criteria, using minimum entropy and using deviation distance: (<b>a</b>) MAE; (<b>b</b>) MTE; and (<b>c</b>) Scan distance; and (<b>d</b>) the selecting process in detail. Although the distance 24.5 m corresponds to the smallest entropy, 36.3 m, which is much closer to reference distance of 38.302 m, is selected based on deviation distance.</p> "> Figure 17
<p>Matching results using SIFT: (<b>a</b>) the result of SIFT feature matching on S1–S2 in data set 2 and the false matching is mainly because of viewpoint changes and self-similarity; (<b>b</b>) the false match from holes; and (<b>c</b>) the correct match on the background with no correspondence in the space of point cloud.</p> ">
Abstract
:1. Introduction
- combining the terrestrial laser scanner with smartphone for coarse registration;
- using 2D projection entropy to measure the distribution coherence between two scans; and
- presenting the Iterative Minimum Entropy (IME) algorithm to correct initial transformation parameters and reduce the effect of positioning error from the smartphone GPS.
2. Methodology
2.1. Combining the Terrestrial Laser Scanner with Smartphone
2.2. Registration with 2D Projection Entropy
2.2.1. Searching for the Minimum Entropy in 3D Space
2.2.2. Transformation from 3D to 2D Space
- Ground filtering: The ground is removed using progressive densification of a triangle mesh.
- Clustering above-ground points: The clustering process is based on distance. If the distance between p and its closest point q is within the distance threshold, then p and q will be identified as the same cluster.
- Removing small clusters: In this paper, a cluster is removed if the number of points in the cluster is less than 500.
2.2.3. Correcting Initial Transformation Parameters Using Iterative Minimum Entropy
3. Experiments and Discussion
3.1. Data Set 1
3.2. Data Set 2
3.3. Comparison
4. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Scan | Number of Points | ||
---|---|---|---|
Original | After Preprocessing | After Projecting | |
S1 | 50,164,568 | 21,225,851 | 15,017 |
S2 | 53,101,629 | 21,256,225 | 15,749 |
S3 | 54,158,850 | 21,858,623 | 15,631 |
Scan pair | Scan Distance (m) | Reference (m) |
---|---|---|
S1–S2 | 34.2272 | 24.5040 |
S1–S3 | 25.8609 | 28.264 |
S2–S3 | 40.4741 | 31.6358 |
Scans | Stage | ∆φ (°) | Δω (°) | Δκ (°) | ∆x (m) | ∆y (m) | ∆z (m) | RMSD (m) | ∆dS (m) |
---|---|---|---|---|---|---|---|---|---|
S1–S2 | IME | 0.317 | 2.501 | −0.639 | 3.784 | 0.644 | 0.433 | 0.643 | 0.220 |
ICP | 0.034 | 0.173 | 0.020 | 0.162 | 0.081 | −0.006 | 0.011 | −0.063 | |
S1–S3 | IME | −1.011 | 0.316 | 0.171 | −1.065 | 1.482 | −0.358 | 0.517 | −1.292 |
ICP | −0.008 | 0.010 | −0.003 | −0.006 | 0.004 | −0.006 | 0.007 | −0.007 | |
S2–S3 | IME | −1.986 | −1.613 | −1.128 | −2.665 | −0.608 | −0.281 | 0.853 | −2.27 |
ICP | −0.143 | −0.059 | 0.035 | −0.07 | −0.031 | −0.044 | 0.007 | −0.076 |
Scan | Number of Points | ||
---|---|---|---|
Original | After Preprocessing | After Projecting | |
S1 | 16,941,512 | 14,168,335 | 31,652 |
S2 | 20,841,940 | 12,642,348 | 11,736 |
S3 | 10,091,225 | 7,832,464 | 15,022 |
S4 | 14,232,022 | 10,745,991 | 7119 |
S5 | 22,715,235 | 14,635,858 | 9299 |
S6 | 16,474,111 | 10,901,182 | 4819 |
Distance (m) | S1 | S2 | S3 | S4 | S5 | S6 | |
---|---|---|---|---|---|---|---|
Reference (m) | |||||||
S1 | 0 | 43.4 | 84.4 | 98.5 | 53.1 | 59.5 | |
S2 | 28.712 | 0 | 43 | 69.6 | 37.9 | 72.7 | |
S3 | 74.967 | 46.529 | 0 | 44.5 | 48.5 | 90.6 | |
S4 | 88.723 | 65.984 | 38.302 | 0 | 54.4 | 76 | |
S5 | 63.112 | 56.282 | 63.939 | 47.843 | 0 | 48.7 | |
S6 | 58.443 | 65.705 | 90.439 | 77.565 | 28.896 | 0 |
Scan Pair | Overlap Rate | Stage | ∆φ (°) | Δω (°) | Δκ (°) | ∆x (m) | ∆y (m) | ∆z (m) | RMSD (m) | ∆dS (m) |
---|---|---|---|---|---|---|---|---|---|---|
S1–S2 | 17.3% | IME | 2.23 | 1.396 | 0.230 | 0.133 | 0.798 | 0.609 | 0.25 | 0.795 |
ICP | 0.205 | 0.402 | −0.029 | 0.024 | 0.039 | 0.005 | 0.018 | 0.045 | ||
S1–S3 | 10.84% | IME | 1.192 | 1.691 | −2.865 | 0.749 | −0.998 | 0.186 | 0.480 | 0.567 |
ICP | 0.422 | 0.545 | 0.329 | 0.302 | −0.244 | 0.075 | 0.039 | −0.078 | ||
S1–S6 | 7.67% | IME | - | - | - | - | - | - | - | - |
ICP | - | - | - | - | - | - | - | - | ||
S2–S3 | 23.5% | IME | −1.038 | 0.295 | −5.895 | 5.236 | 1.404 | −0.423 | 0.58 | 1.963 |
ICP | −0.246 | 0.145 | −0.131 | 0.028 | 0.109 | 0.0797 | 0.028 | 0.112 | ||
S3–S4 | 17.8% | IME | −0.466 | 1.330 | 6.090 | −2.170 | −0.922 | 0.111 | 0.103 | 0.983 |
ICP | 0.304 | −0.382 | −0.047 | 0.126 | −0.109 | 0.108 | 0.012 | −0.167 | ||
S4–S5 | 18.1% | IME | −0.313 | −1.940 | −2.115 | −0.371 | −1.641 | 0.671 | 0.523 | −0.733 |
ICP | 0.087 | 0.085 | −0.144 | 0.0379 | −0.114 | 0.0004 | 0.021 | 0.013 | ||
S4–S6 | 10.52% | IME | 0.140 | −1.10 | 1.091 | −0.182 | 0.433 | −0.059 | 0.140 | −0.096 |
ICP | −0.091 | 0.011 | −0.004 | 0.079 | −0.002 | −0.029 | 0.037 | 0.077 | ||
S5–S6 | 30.1% | IME | −0.551 | −0.779 | −4.011 | −1.148 | 1.134 | −0.730 | 0.231 | 0.820 |
ICP | 0.027 | 0.0054 | 0.0005 | 0.017 | −0.002 | −0.022 | 0.011 | 0.102 |
SIFT Based Method | The Proposed Method | ||||||||
---|---|---|---|---|---|---|---|---|---|
Key Points | Average Match | Successful Match (%) | Average MAE (°) | Average MTE (m) | Average RMSD (m) | Average MAE (°) | Average MTE (m) | Average RMSD (m) | |
Data set 1 | 3678 | 72 | 69.4 | 1.741 | 1.09 | 0.715 | 1.076 | 1.258 | 0.671 |
Data set 2 | 7608 | 5 | - | - | - | - | 1.532 | 0.958 | 0.330 |
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Chen, M.; Wang, S.; Wang, M.; Wan, Y.; He, P. Entropy-Based Registration of Point Clouds Using Terrestrial Laser Scanning and Smartphone GPS. Sensors 2017, 17, 197. https://doi.org/10.3390/s17010197
Chen M, Wang S, Wang M, Wan Y, He P. Entropy-Based Registration of Point Clouds Using Terrestrial Laser Scanning and Smartphone GPS. Sensors. 2017; 17(1):197. https://doi.org/10.3390/s17010197
Chicago/Turabian StyleChen, Maolin, Siying Wang, Mingwei Wang, Youchuan Wan, and Peipei He. 2017. "Entropy-Based Registration of Point Clouds Using Terrestrial Laser Scanning and Smartphone GPS" Sensors 17, no. 1: 197. https://doi.org/10.3390/s17010197