# Improvement of 3D Power Line Extraction from Multiple Low-Cost UAV Imagery Using Wavelet Analysis

## Abstract

**:**

## 1. Introduction

#### Research Problem

## 2. Materials and Methods

#### Description of Data Sets

_{T}ca. 45 m). They were obtained in an open area, locally covered with high vegetation. The diameter of the lines was about 30–40 mm. The data comprised digital photographs and LiDAR data from the Ricopter platform. The photographs were taken with the Sony alpha 6000 camera (with a rigid mount to the VUX-1) with a 24 megapixel 6000 × 4000 matrix and a 16 mm lens. Point clouds from LiDAR were acquired with 380 kHz pulse repetition with the Riegl VUX-1UAV Airborne Laser Scanning system. Power lines were recorded in 3 flight strips at 90 m AGL (Above Ground Level) flight altitude (3 × h

_{T}). The dimensions of the processed section were 670 × 50 m. POSPac MMS 7.1 and RiPROCESS v. 1.6.5 were used for data processing. The point clouds from imaging data were generated in the Pix4D Mapper Pro software with a half-image scale. The data were obtained thanks to the courtesy of the Riegl company. The flight mission took place during the summer.

_{T}). The pylon height was −11 m and the cable diameter was about −6 mm. The point clouds from imaging data were generated in the Pix4D Mapper Pro software with a full-image scale. The data were obtained during summer and winter. Differences in this measurement campaign will be revealed in the experiments.

_{T}) did not exceed the range between 6 and 35 mm. The flight height (linked with the GSD value) was always chosen so that the lines were visible on at least 2–3 pixels. TLS data were also obtained from a distance, so that the footprint (laser diameter) was smaller or equal to the cable cross-section width. The data are presented in Figure 2.

## 3. Methods

#### 3.1. Data Capture

_{FL[AGL]}(2). This value takes into account the distance of the power line from the ground.

_{T}is the approximate pylon height (distance of power lines from ground), f is the focal length, px is the pixel pitch, ϕ

_{T}is the theoretical cable diameter and k is the thickness coefficient. For low-voltage power lines, k = 1, and for high voltage power lines, k = 0.5. It should be noted that the planned flight height should be greater than the highest terrain obstacle in the flight area. It is important to take note of the longitudinal and side overlap between the strips, with the recommended value being 80–85%. This will guarantee that the power lines will show up on the stereogram, especially in the case of side overlap.

#### 3.2. Imagery Pre-Processing

_{IMG}). The conversion scheme is presented in Figure 6.

#### 3.3. Classifying and Filtering Power Lines

#### 3.3.1. Two-Stage Coarse Filtration

_{IMG}) in relation to the neighbouring points falls below the threshold and those in which the value is equal to or higher than the threshold value, t:

_{IMG}< t

_{IMG}≥ t

#### Intensity Condition

_{max}is the dynamic range of standard deviations for the data. The criterion for the level of noise of the data is the relation between the values of the standard deviation and the average.

#### Geometric Conditions

Algorithm 1. Filtering points horizontally | |

Initial: points from data set after coarse filtrationDetermining an approximated line based on power line’s points | |

1: | Look for a linear model for a straight line: Y_t = α + βt, (t,y_t) for t = 1, …, n |

2: | Solve the optimisation problem: |

$\underset{\alpha ,\beta}{min}\left({\displaystyle \sum}_{t=1}^{n}{e}_{t}^{2}\right)=\underset{\alpha ,\beta}{min}({\displaystyle \sum}_{t=1}^{n}{\left({y}_{t}-\alpha -\beta t\right)}^{2})$ | |

where ${\mathrm{e}}_{\mathrm{t}}={\mathrm{y}}_{\mathrm{t}}-{\stackrel{\u02d8}{\mathrm{y}}}_{\mathrm{t}}$ (${\mathrm{y}}_{\mathrm{t}}-$ realisation of variable Y_{t}), ${\stackrel{\u02d8}{y}}_{t}=\alpha +\beta t$ (theoretical value of variable) | |

3: | Determine the coefficients defining the straight line: |

$\begin{array}{c}\stackrel{\u02d8}{\alpha}=\overline{y}-\stackrel{\u02d8}{\beta}\overline{t}\\ \stackrel{\u02d8}{\beta}=\frac{\left({{\displaystyle \sum}}_{t=1}^{n}\left(t-\overline{t}\right)\left(y-\overline{y}\right)\right)}{{{\displaystyle \sum}}_{t=1}^{n}{\left(t-\overline{t}\right)}^{2}}\end{array}$ straight line is constructed | |

4: | Calculate the distance of each point P(P_{y}, P_{x}) from the line d’(P,L) |

5: | Reject the points whose distance exceeds the set d value constituting 65% of the theoretical radius of the cable |

d ≤ 0.65 ⋅ ϕ_{T} output: correct points |

_{T}. The analysis of side trends for the vertical coordinate Z will work in a similar way. Because of the shape of power lines, the analysis of noise “around” the line, considering it vertically, is not a straight line and does not always have to fulfil the condition of a “regular curve”. Furthermore, in dense image matching, the Z coordinate is the most prone to errors at the stage of data acquisition, which later influences the 3D modelling of the object. To analyse the noise distribution around the Z coordinate, Bollinger Bands were used based on the moving average (MA, determined by means of the simple moving average—SMA) and the analysis of side trends. To this end, it was necessary to define the moving average intervals. The number of intervals depends on the course of the power line and may be determined by the quality of the data. For example, we can determine any constant segments of the power line in a point cloud or places where the line’s course or shape changes as intervals. The value can be set at decimetre-long segments of the line. The Bollinger Bands define the natural extremes in the developing trend. The boundaries of the envelopes located below and above the curve in the constant moving average are expressed as a percentage of the distance, the limits being constructed from the Bollinger Band at a distance equal to a certain number of standard deviations. As the standard deviation depends on the variability of data, the bands adopted for the analyses of power lines should differ from the average by a value of 1σ (65%). When constructing the limits, we should use the one sigma rule, according to which the data will be found between the two bands for 65% of points from the interval. Points located outside of the assumed interval should be rejected.

Algorithm 2. Filtering points vertically | |

Initial: points from data set after coarse filtrationDetermining Bollinger Bands for power line approximation | |

1: | Divide the points of the line with the length dL into a set number of intervals n = 0.1 d_{L} |

2: | Determine the moving average: |

${\overline{y}}_{t}=\frac{{y}_{t}+{y}_{t-1}+\dots +{y}_{t-n-1}}{n}$ | |

for y—the variable Z, and t—the current points interval, n—the number of intervals in the average | |

Determine the side trends—upper (UT) and lower (LT): | |

3: | UT = MA + σ |

LT = MA − σ | |

4: | Reject the points outside side trends. output: correct points |

#### 3.4. Denoising Using Wavelet Analysis

_{i}—details, a_Z—approximation of the Z coordinate.

#### 3.4.1. Data Preparation

#### 3.4.2. Wavelet Analysis

_{0}> 1 is the spreading step, on which the shift τ

_{0}(where τ

_{0}= 1) depends. Usually, coefficient s

_{0}= 2. The sampling frequency corresponds to the dyadic sampling.

_{i}is samples of a deterministic function f, $\stackrel{\u02c7}{{v}_{j,k}}$ are estimated coefficients obtained on the basis of the selected threshold t for wavelet coefficients at scale j: $t={\left[{t}_{1},{t}_{2},\dots ,{t}_{J}\right]}^{T}$. If the soft thresholding function is used, nonlinearity is applied to the empirical wavelet coefficients at each scale j, j−1, …, j:

#### 3.5. Quality Check—Point Cloud Quality Index

_{i}) and the estimated values (f(x

_{i})):

_{i}) and the estimated values (f(x

_{i})):

_{o}) parameter of the original signal to the normalised L1 norm (L1

_{dn}) parameter of the signal denoised with a particular threshold method. Equation (10) shows the relation of statistical parameters of coefficients residuals, taking into consideration also the standard deviation of both the original and denoised data (σ

_{o}and σ

_{dn}). The W

_{2dn}index also presents the relation between noisy and smoothened data. Both indexes indicate the relative level of noise in the data.

#### Validation—Comparative Study

## 4. Experiment and Results

#### 4.1. Point Cloud Filtering of Power Lines

#### 4.2. Denoising with Wavelet Transform

_{dn}and w2

_{dn}indexes, the more noiseless the data (bolded in Table 4). This also provides a good comparison of thresholding methods of denoising. In all cases, the RSURE method yields the most accurate data. The analysis was also conducted on the actual parameters of power lines and compared with the theoretical values and those obtained via laser scanning. Table 5 shows the values of point cloud resolution, line span and diameter before (ϕ

_{UAVo}) and after correction (ϕ

_{UAVc}), and sag before (

**S**) and after correction (

_{UAVc}**S**).

_{UAVc}**S**—original,

_{UAVo}**S**—corrected) and laser scanning (LS) data. The largest differences are noticeable for resolution, cable diameter and point cloud resolution. There was less improvement for the line sag value. It is also interesting that sag values for the same line could be different in different seasons because of temperature. The sag value was usually smaller in winter than in summer. This is related to the properties of the power line material. These properties can be noticed for Line S2_L1_UAV in Table 5 and Table 7.

_{UAVc}## 5. Discussion and Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The view for the three data sets: (

**a**) forest area and high-voltage power lines: platform, imagery, point clouds (image), point clouds ultra-light light detection and ranging (LiDAR); (

**b**) rural area and mean-voltage power lines: platform, imagery, point clouds (image), point clouds (TLS); (

**c**) urban area and low-voltage power lines: platform, imagery, point clouds (image), point clouds (TLS).

**Figure 5.**Pattern for generating power lines point clouds from unmanned aerial vehicle (UAV) imaging data (dotted lines indicate optional stages).

**Figure 8.**Data denoising workflow, Z—vertical coordinate of UAV imagery point clouds, D

_{i}—details; a_Z—approximation of the Z coordinate.

**Figure 9.**Diagram of wavelet decomposition [45].

**Figure 11.**Results of different stages of coarse filtration: (

**a**) original points, (

**b**) intensity filtration—horizontal view, (

**c**) intensity filtration—horizontal view, (

**d**) result of filtering with intensity and geometric condition—horizontal view, (

**e**), (

**f**)—LiDAR point clouds—horizontal and vertical view.

**Figure 12.**The fragment of decomposition of the Z coordinate with the DB6 L6 wavelet, s—original signal, a—approximation, d1–d6—decomposition details.

**Figure 14.**The fragment of decomposition of the Z denoised coordinate with the DB6 L6 wavelet, s—original signal, a—approximation, d1–d6—decomposition details.

**Figure 16.**Power line (data set III) at different stages of the proposed algorithm: (

**a**) the original point cloud, (

**b**) close-up, (

**c**) the results of coarse filtration with intensity condition, (

**d**) the results of coarse filtration with geometric condition, (

**e**) points denoised with the use of wavelet analysis (

**f**) close-up of wavelet analysis results.

**Figure 17.**High-voltage power lines from UAV multi-images (

**a**) lines and towers before and after intensity filtering, (

**b**) lines before and after filtering.

Feature | Set I | Set II | Set III | |
---|---|---|---|---|

Objects and area | Power line type Cable diameter (cm) Scene type | High-voltage (ϕ _{T} = 30 mm)Open and forest area | Low-voltage (ϕ _{T} = 6 mm)Rural area | Low-voltage (ϕ _{T} = 28 mm)Urban area |

Image and point clouds | Camera model GSD (cm) Flight height (m) Point cloud resolution (cm) | Sony Alpha 6000 4 cm 90 AGL 4 | Sony RX100 0.3 23 m, 30 m 0.5–1.0 | Sony RX100 0.5 28 m 0.6–1.2 |

Set IVReference data type | Laser scanning model and type | Riegl VUX-1UAV (UAV-ALS) | Leica P40 (TLS) | Leica P40 (TLS) |

SET I (Summer) | SET II (Summer) | SET II (Winter) | SET III (Autumn) | |
---|---|---|---|---|

Minimum | 0.499 | 0.499 | 0.499 | 0.499 |

Maximum | 0.560 | 0.560 | 0.560 | 0.560 |

Mean | 0.509 | 0.523 | 0.529 | 0.512 |

SD | 0.019 | 0.013 | 0.015 | 0.009 |

Max. SD | 0.031 | 0.031 | 0.031 | 0.031 |

K coefficient | 0.20 | 0.15 | 0.20 | 0.07 |

Threshold value t | 0.522 | 0.523 | 0.525 | 0.513 |

Filtration | 68% | 67% | 75% | 84% |

**Table 3.**Statistics for the coefficients and reconstruction of the original and denoised signals for a power line (data set III).

Original Data | Fixed from Threshold | RSURE | HSURE | |||||
---|---|---|---|---|---|---|---|---|

Coeff. | Rec. | Coeff. | Rec. | Coeff. | Rec. | Coeff. | Rec. | |

SD | 0.024 | 0.020 | 0.018 | 0.012 | 4.769 × 10^{−07} | 1.972 × 10^{−07} | 0.018 | 0.013 |

MAD | 0.005 | 0.009 | 0.002 | 0.002 | 1.045 × 10^{−08} | 6.24 × 10^{−09} | 0.002 | 0.002 |

L1 norm | 129.4 | 182.6 | 19.55 | 33.93 | 0.0001 | 0.0001 | 22.09 | 38.16 |

L2 norm | 2.879 | 2.879 | 1.808 | 1.808 | 4.837 × 10^{−05} | 0.0001 | 1.877 | 1.877 |

Max norm | 0.624 | 0.459 | 0.567 | 0.389 | 3.645 × 10^{−05} | 0.0001 | 0.574 | 0.398 |

w1_{dn} | 6.0 | 4.6 | 82.7 | 105.2 | 5.4 | 4.2 | ||

w2_{dn} | 8.8 | 7.5 | 50,201 | 101,935 | 8.1 | 7.0 |

**Table 4.**Statistics for the coefficients and reconstruction of the original and denoised signal for a power line (data set II).

Original Data | Fixed from Threshold | RSURE | HSURE | |||||
---|---|---|---|---|---|---|---|---|

Coeff. | Rec. | Coeff. | Rec. | Coeff. | Rec. | Coeff. | Rec. | |

SD | 0.004 | 0.003 | 0.002 | 0.001566 | 0.001991 | 0.0014 | 0.0022 | 0.0015 |

MAD | 0.002 | 0.002 | 0.0003 | 0.000225 | 0.0002494 | 0.0002 | 0.0003 | 0.0002 |

L1 norm | 25.3 | 35.3 | 2.77 | 4.69 | 2.44 | 4.21 | 2.77 | 4.69 |

L2 norm | 0.447 | 0.447 | 0.226 | 0.226 | 0.203 | 0.203 | 0.226 | 0.226 |

Max norm | 0.086 | 0.077 | 0.078 | 0.066 | 0.077 | 0.066 | 0.077 | 0.066 |

w1_{dn} | 5.8 | 5.6 | 6.6 | 6.3 | 5.8 | 5.6 | ||

w2_{dn} | 18.2 | 20.3 | 20.4 | 22.6 | 18.2 | 20.3 |

Line | Point Cloud Resolution (mm) | Span (m) | Theor. Cable Diameter ϕ_{T} (mm) | Cable Diameter ϕ_{UAVo} (mm) | Cable Diameter Corrected ϕ_{UAVc} (mm) | Sag S_{UAVo} (m) | Sag Corrected S_{UAVc} (m) | |
---|---|---|---|---|---|---|---|---|

SET I | S1_L1_ALS | 38 | 292.69 | 30 | 46 | - | 5.52 | - |

S1_L1_UAV | 43 | 291.88 | 96 | 64 | 5.71 | 5.61 | ||

S1_L2_ALS | 38 | 291.19 | 65 | - | 8.10 | - | ||

S1_L2_UAV | 43 | 291.25 | 88 | 58 | 8.24 | 8.31 | ||

SET II | S2_L3_TLS | 6 | 48.94 | 6 | 10 | - | 0.26 | - |

S2_L3_UAV | 4 | 48.41 | 10 | 8 | 0.22 | 0.25 | ||

S2_L1_TLS (winter) | 4 | 44.52 | 13 | - | 0.47 | - | ||

S2_L1_UAV (summer) | 10 | 44.86 | 26 | 15 | 0.76 | 0.69 | ||

S2_L1_UAV (winter) | 5 | 43.78 | 12 | 10 | 0.51 | 0.46 | ||

SET III | S3_L1_TLS | 7 | 40.62 | 28 | 30 | - | 1.61 | - |

S3_L1_UAV | 11 | 40.62 | 42 | 29 | 1.58 | 1.63 |

SET I | SET II | SET III | ||||
---|---|---|---|---|---|---|

Sag Delta [m] | S1_L1 | S1_L2 | S2_L3 | S2_L1 (Winter) | S2_L1 (Summer) | S3_L1 |

S–_{UAVo}S_{LS} | 0.19 | 0.14 | 0.03 | 0.04 | 0.03 | |

S_{UAVc–}S_{LS} | 0.09 | 0.20 | 0.01 | 0.01 | 0.02 | |

S–_{UAVo}S_{UAVc} | 0.09 | 0.07 | 0.02 | 0.04 | 0.07 | 0.05 |

Correction level | 2% | 1% | 10% | 10% | 9% | 3% |

**Table 7.**Comparison of diameter before and after correction for UAV imagery and laser scanning data.

Diameter Delta (mm) | S1_L1 | S1_L2 | S2_L3 | S2_L1 (Winter) | S2_L1 (Summer) | S3_L1 |
---|---|---|---|---|---|---|

ϕ_{UAVo}–ϕ_{T} | 66 | 58 | 4 | 6 | 20 | 14 |

ϕ_{UAVc}–ϕ_{T} | 34 | 28 | 2 | 4 | 9 | 1 |

ϕ_{LS}–ϕ_{T} | 16 | 35 | 4 | 7 | 7 | 2 |

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Fryskowska, A.
Improvement of 3D Power Line Extraction from Multiple Low-Cost UAV Imagery Using Wavelet Analysis. *Sensors* **2019**, *19*, 700.
https://doi.org/10.3390/s19030700

**AMA Style**

Fryskowska A.
Improvement of 3D Power Line Extraction from Multiple Low-Cost UAV Imagery Using Wavelet Analysis. *Sensors*. 2019; 19(3):700.
https://doi.org/10.3390/s19030700

**Chicago/Turabian Style**

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2019. "Improvement of 3D Power Line Extraction from Multiple Low-Cost UAV Imagery Using Wavelet Analysis" *Sensors* 19, no. 3: 700.
https://doi.org/10.3390/s19030700