# Wall Structure Geometry Verification Using TLS Data and BIM Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Building Structures’ Geometry Verification

#### 2.1. Geometry Identification from BIM Model

_{ref}, Y

_{ref}are the coordinates of the reference point, h is thickness of the wall, and X

_{dir}, Y

_{dir}are the elements of the unit vector defining the direction of the wall. The plus or minus sign in the above equations is chosen according to whether we determine the inner or outer planes of the wall.

**u**and

**v**, the normal vector n of the plane is calculated using Equation (4). Then, the general equation of the plane (5) can be defined from the calculated elements of the normal vector.

#### 2.2. Plane Segmentation from Point Clouds

- Edge-based methods,
- Model-based methods,
- Surface-based methods, region-based methods,
- Clustering-based methods,
- Graph-based methods.

**A**

^{n}

^{×3}is the matrix of reduced coordinates (reduced to the subsets centroid), n is the number of the selected number of neighboring points, the column vectors of

**U**

^{n}

^{×n}are normalized eigenvectors of the matrix

**AA**

^{T}, the column vectors of

**V**

^{3}

^{×3}are normalized eigenvectors of the matrix

**A**

^{T}

**A**. The matrix

**Σ**

^{n}

^{×3}is a diagonal matrix with the first three singular numbers of the matrix

**A**

^{T}

**A**on the main diagonal. Therefore, the normal vector of the regression plane is the column vector of

**V**, which belongs to the smallest singular number of the matrix

**A**

^{T}

**A**[38]. The components of the normal vector are the coefficients a, b, and c of the general equation of the estimated plane. The coefficient d is calculated by fitting the coordinates of the centroid point (X

_{mean}, Y

_{mean}, Z

_{mean}), and the components of the normal vector to the Equation (8):

- Normal vector-based filtration,
- Curve segmentation.

#### 2.2.1. Normal Vector-Based Filtration

_{point}is the normal vector of the local plane calculated from k-nearest neighbors, n

_{PoC}is the normal vector of the corresponding regression plane of the point cloud and α is the angle that the two vectors make at each other.

#### 2.2.2. Curve Segmentation

- Preprocessing: Creation of the input images from the planar point cloud.
- Evolution: Curve segmentation of the plane using images.
- Postprocessing: Creation of point cloud segments and selection of wall segment.

#### Preprocessing

**x**= (x

_{1},x

_{2},x

_{3}), the coordinates x

_{1}and x

_{2}describe the position of the point in the plane and the x

_{3}axis is orthogonal to the regression plane. First, we create a regular square mesh in the x

_{1}, x

_{2}plane. Then, we represent the properties of the point cloud by a set of pixels (bitmap) images, one image for each property. For example, we take the R (red) channel of color and define the value of each pixel (square of the mesh) as the mean value of the R channel of all points which lie in the pixel. Finally, we rescale the values in each image to the interval [0,1]. Figure 8 shows the image for the intensity channel.

#### Evolution

**x**= (x

_{1},x

_{2}). The evolution is driven by a suitably designed velocity field

**v**, therefore, the basic evolution model is:

**x**/∂t denotes the time derivative of the position vector, i.e., the velocity of the point

**x**. The equation is coupled with an initial condition—the initial curve is a small circle (or multiple circles) placed inside the segmented region.

**v**is considered in the form:

**N**(

**x**) denotes the positively oriented normal vector at point

**x**of the curve, k(

**x**) is the signed curvature at

**x**, and ∇ is the gradient operator. The role of the first term B

**N**is to expand the segmentation curve in the normal direction from its initial shape through the segmented region towards its border, the “blowing” function B controls the speed of the expansion and is defined using the bitmap images. The idea behind computing the value of B(

**x**) for a point x on the curve is to compare the properties at point x to the average of the properties inside the evolving curve. If the properties are similar, the value of B(

**x**) is large (the point

**x**should move fast), if they differ a lot, B(

**x**) is small (i.e.,

**x**should move slowly).

_{0}≪ 1 (i.e., the edge attraction does not dominate), keep it unchanged until the curve is close to the border (moving very slowly) and then switch λ(t) to 1, which turns off the expansion and attracts the curve towards the edges.

**N**is called curvature regularization and has a smoothing effect. We use it to deal with the noise and to smooth sharp edges of the segmentation curve, mainly during the expansion phase. The parameter δ(t) weighs the influence of the term.

#### Postprocessing

#### 2.3. Creation of Deviation Maps

_{P}, Y

_{P}, and Z

_{P}are the coordinates of a given point of the subset.

_{IFC}is the normal vector of the wall plane from the BIM model and n

_{PoC}is the normal vector of the estimated regression plane from the segmented point cloud. The deviation between the two models’ coefficient d is calculated using Equation (14):

_{IFC}is the distance of the IFC plane from the origin of the coordinate system and d

_{PoC}is the distance of the estimated regression plane from the origin of the coordinate system.

## 3. Case Study

## 4. Conclusions

^{®}software. In the future, approaches for the verification of other structural parts (with cylindrical or spherical geometry) will be proposed and added. We also plan to incorporate the identification of holes in the walls, which need to be identified in the BIM model, while finding the same holes in the point cloud. The approach will be programmed and fully automated.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Identification of geometry from BIM model—(

**1**) BIM model, (

**2**) IFC file (

**3**) derivation of the wall’s geometry, (

**4**) estimation of the parameters of the wall’s surface.

**Figure 4.**Plane segmentation procedure: (

**a**) imput point cloud, (

**b**) seed point and the 102,400 nearest neighbors, (

**c**) 1,638,400 nearest neighbors, (

**d**) inlier points.

n | a_{IFC} | a_{PoC} | b_{IFC} | b_{PoC} | c_{IFC} | c_{PoC} | d_{IFC} [m] | d_{PoC} [m] |

1 | 1.0000 | 0.9999 | 0.0000 | 0.0023 | 0.0000 | 0.0006 | −13.400 | −13.444 |

2 | −1.0000 | −0.9999 | 0.0000 | 0.0022 | 0.0000 | 0.0005 | 16.235 | 16.252 |

3 | 1.0000 | 0.9999 | 0.0000 | 0.0013 | 0.0000 | 0.0004 | −16.350 | −16.378 |

4 | −1.0000 | −0.9999 | 0.0000 | 0.0005 | 0.0000 | 0.0014 | 19.250 | 19.240 |

5 | 0.0000 | 0.0021 | −1.0000 | −0.9999 | 0.0000 | 0.0010 | −11.575 | −11.552 |

6 | 0.0000 | 0.0030 | −1.0000 | −0.9999 | 0.0000 | 0.0019 | −11.575 | −11.524 |

7 | 0.0000 | 0.0006 | 1.0000 | 0.9999 | 0.0000 | −0.0013 | 11.460 | 11.444 |

8 | 0.0000 | −0.0022 | −1.0000 | −0.9999 | 0.0000 | −0.0028 | −9.760 | −9.730 |

9 | 0.0000 | −0.0013 | −1.0000 | −0.9999 | 0.0000 | −0.0016 | −10.060 | −10.042 |

10 | −1.0000 | −0.9999 | 0.0000 | 0.0011 | 0.0000 | 0.0015 | 16.800 | 16.801 |

11 | 0.0000 | 0.0088 | 1.0000 | 0.9999 | 0.0000 | 0.0002 | 7.375 | 7.209 |

12 | −1.0000 | −0.9999 | 0.0000 | 0.0012 | 0.0000 | 0.0006 | 19.250 | 19.248 |

13 | 1.0000 | 0.9999 | 0.0000 | 0.0023 | 0.0000 | 0.0003 | −15.700 | −15.699 |

_{IFC}, b

_{IFC}, c

_{IFC}, and d

_{IFC}are the parameters of the plane from IFC, a

_{PoC}, b

_{IPoC}, c

_{PoC}, and d

_{PoC}are the parameters of the plane from IFC.

n | $\mathsf{\alpha}$ [°] | $\mathsf{\delta}$ [mm] | max [mm] | min [mm] | avg [mm] | absmax [mm] |

1 | 0.1346 | −44 | 24 | 6 | 16 | 24 |

2 | 0.1279 | −17 | 18 | −9 | 11 | 18 |

3 | 0.0793 | −28 | 20 | 4 | 11 | 20 |

4 | 0.0865 | 9 | 21 | 5 | 10 | 21 |

5 | 0.1338 | 23 | 23 | 6 | 14 | 23 |

6 | 0.2045 | 51 | 22 | 7 | 14 | 22 |

7 | 0.0815 | 16 | 20 | 2 | 12 | 20 |

8 | 0.2042 | 30 | 23 | 9 | 16 | 23 |

9 | 0.1192 | 18 | 21 | 7 | 13 | 21 |

10 | 0.1045 | −1 | 26 | 7 | 16 | 26 |

11 | 0.5026 | 166 | 22 | −4 | 9 | 22 |

12 | 0.0761 | 2 | 19 | −1 | 9 | 19 |

13 | 0.1316 | 1 | 41 | −3 | 14 | 41 |

_{max}is the absolute maximum deviation between the IFC plane and the segmented point cloud.

n | max [mm] | min [mm] | abs_{max} [mm] | std [mm] |

1 | 5 | −9 | 9 | 2 |

2 | 4 | −16 | 16 | 2 |

3 | 7 | −9 | 9 | 1 |

4 | 10 | −15 | 15 | 2 |

5 | 7 | −6 | 7 | 2 |

6 | 7 | −5 | 7 | 2 |

7 | 10 | −9 | 10 | 2 |

8 | 11 | −5 | 11 | 2 |

9 | 5 | −6 | 6 | 1 |

10 | 11 | −9 | 11 | 1 |

11 | 10 | −4 | 10 | 1 |

12 | 14 | −5 | 14 | 2 |

13 | 19 | −24 | 24 | 2 |

_{max}is the absolute maximum deviation between the regression plane and the segmented point cloud, std is the standard deviation calculated from the orthogonal distances of the segmented points from the regression plane.

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**MDPI and ACS Style**

Bariczová, G.; Erdélyi, J.; Honti, R.; Tomek, L.
Wall Structure Geometry Verification Using TLS Data and BIM Model. *Appl. Sci.* **2021**, *11*, 11804.
https://doi.org/10.3390/app112411804

**AMA Style**

Bariczová G, Erdélyi J, Honti R, Tomek L.
Wall Structure Geometry Verification Using TLS Data and BIM Model. *Applied Sciences*. 2021; 11(24):11804.
https://doi.org/10.3390/app112411804

**Chicago/Turabian Style**

Bariczová, Gabriela, Ján Erdélyi, Richard Honti, and Lukáš Tomek.
2021. "Wall Structure Geometry Verification Using TLS Data and BIM Model" *Applied Sciences* 11, no. 24: 11804.
https://doi.org/10.3390/app112411804