# Estimation of 3D Indoor Models with Constraint Propagation and Stochastic Reasoning in the Absence of Indoor Measurements

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## Abstract

**:**

## 1. Introduction

- Whereas [17] describes the overall approach, this article discusses the relevant methods and algorithms in more depth.
- Whereas [17] represents position of rooms and windows in a continuous domain and performs reasoning with inequalities in these domains, in this paper, we represent geometry in discrete domains during the combinatorial part and apply a method of constraint propagation in finite domains that is considerably more efficient.
- This article deals with doors and estimates their sizes and positions and thus provides an access for indoor navigation.
- It generalizes from 2D floor plan layouts to 3D indoor models.

## 2. Related Work

## 3. Modeling Floor Plans with Constraints

#### 3.1. Hard and Soft Constraints

#### 3.2. Probability Density Functions

## 4. Constraint Propagation for Topological Floor Plan Derivation

## 5. Conditional Linear Gaussian Models for Stochastic Floor Plan Prediction

## 6. Results and Discussion

#### Derivation of a 3D Indoor Model from Predicted Floor Plans

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Our approach derives floor plans automatically (

**bottom right**) from sparse observations like window locations from possibly LoD3 exterior models, footprint and room information such as room areas (

**top**). No additional indoor measurements are needed. For the comparison, a reference floor plan is depicted (

**bottom left**).

**Figure 2.**(

**a**) reasoning process: the combination of constraint propagation and a maximum a posteriori probability inference reduces a huge search space with continuous and discrete parameters to a small set of solutions with the most likely hypothesis; (

**b**) distribution of the width of office rooms: a Gaussian mixture is a good approximation to a skew symmetric or multimodal distribution.

**Figure 3.**(

**a**) illustration of location and shape parameters for a floor f and an $ith$ room with a single window used during the reasoning process; (

**b**) example for adding auxiliary fix rooms (green) in order to model floor plans with a non-rectangular footprint.

**Figure 4.**Excerpt from the relational database schema. Location and shape parameters of rooms are a.o. used for prior knowledge and evaluation.

**Figure 5.**(

**a**–

**c**) Floor plan prediction demonstrated on three examples with regard to different requirements.

**Figure 6.**The incorporation of room numbers in (

**b**) contributed to better results than without room number information in (

**a**).

**Figure 7.**Histograms of the location of doors depending on the width of rooms. This information is used for the prediction of door locations in derived floor plans for 3D indoor modeling.

**Figure 8.**3D indoor model of the resulted floor plan from Figure 5b. Door locations and story heights are derived and classified from background knowledge.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Loch-Dehbi, S.; Dehbi, Y.; Plümer, L.
Estimation of 3D Indoor Models with Constraint Propagation and Stochastic Reasoning in the Absence of Indoor Measurements. *ISPRS Int. J. Geo-Inf.* **2017**, *6*, 90.
https://doi.org/10.3390/ijgi6030090

**AMA Style**

Loch-Dehbi S, Dehbi Y, Plümer L.
Estimation of 3D Indoor Models with Constraint Propagation and Stochastic Reasoning in the Absence of Indoor Measurements. *ISPRS International Journal of Geo-Information*. 2017; 6(3):90.
https://doi.org/10.3390/ijgi6030090

**Chicago/Turabian Style**

Loch-Dehbi, Sandra, Youness Dehbi, and Lutz Plümer.
2017. "Estimation of 3D Indoor Models with Constraint Propagation and Stochastic Reasoning in the Absence of Indoor Measurements" *ISPRS International Journal of Geo-Information* 6, no. 3: 90.
https://doi.org/10.3390/ijgi6030090