# Typification for Façade Structures Based on User Perception

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## Abstract

**:**

## 1. Introduction

## 2. A User Survey for Typification

#### 2.1. Constraints of Typification

- keeping the area covered by windows,
- keeping the ratio between the height and the width of the windows,
- keeping the distances between windows,
- keeping the distances between windows and the outline of the façade,
- keeping windows distributed in the tendency direction.

#### 2.2. User Survey and the Results

## 3. Verification of the User Survey Using ARG and NEMD Algorithm

_{i}|1 ≤ I ≤ n} and R = {r

_{ij}|1 ≤ I ≤ n, 1 ≤ j ≤ n}. V is the set of n nodes and each v

_{i}represents a window in the façade. R is an $n\times n$ matrix and each r

_{ij}is the relationship between the window v

_{i}and v

_{j}. In this application the node contains attributes about the window such as width and height and the relationship between nodes will represent the spatial and topological relations between the windows on the façade.

_{inner}, in which every element is generated from Equation (1). For example, the D

_{inner}of node v

_{1}in G and v’

_{1}in G’ (v

_{i}and v’

_{i’}) is given in Equation (3), in which the 1st row and 2nd column of inner matrix D

_{inner}, Δ

_{inner}(1, 2) can be calculated with Equation (2).

_{node}(v

_{1}, v’

_{2}) = |0.8 − 0.3| = 0.5, Δ

_{relation}(r

_{11},r’

_{12}) = |0 − 0.2| = 0.2, α = 0.5. Therefore, Δ

_{inner}(1, 2) = 0.35. Similarly, we can calculate all Δ

_{inner}(j, j’) and compose the D

_{inner}for the node pair v

_{1}and v’

_{1}as shown in Equation (3). Based on that, the inner EMD of node v

_{1}in G and v’

_{1}in G’ is 0.1 + 0.3 + 0.35 = 0.75 (the minimum sum of minimum value in each column or row).

_{outer}of G and G’ is also given in Equation (3), in which the first element is 0.75 according to previous calculation. The EMD between G and G’ is 0 based on the D

_{outer}, because G’ is the sub graph of G. But in our application, not only partial but also overall difference between ARGs should be considered. Therefore, the difference between two ARGs is the maximum sum of the minimum value in each column or row of the D

_{outer}, e.g., 0.05 for G and G’ in Figure 4.

#### 3.1. ARG Generation

_{i}= (w

_{i}, h

_{i}) in which is v

_{i}represent the i-th window in the façade; w

_{i}and h

_{i}are the width and height of the window. The relationship between two windows is set to the ratio of the distance between two polygons of the windows and their area sum. However, absolute distance alone is not sufficient to reflect the visual relationship between two windows on the façade since two large windows would look more similar to each other than two smaller ones even if they have same distance.

#### 3.2. Distance Definition

_{node}is composed by two parts: shape distance Δ

_{shape}and area distance Δ

_{area}. Assuming v

_{i}and v

_{j}are two nodes, Δ

_{node}can be calculated as follows:

_{shape}equals the sum area of D

_{1}and D

_{2}(the shaded part in Figure 5c). Since rectangle P

_{1}and P

_{2}are normalized to the rectangle with 1 as their longest edge, Δ

_{shape}is a value between 0 (means exactly the same) and 1 (completely different). In Equation (8), ${W}_{i}$ and ${H}_{i}$ are the total width and length of the façade which contains ${v}_{i}$, so are ${W}_{i}$ and ${H}_{i}$. Δ

_{area}is the normalized area difference. α is a number between 0 and 1 which gives the weight of the shape and area distance in final node distance.

_{ij}and Δ

_{pq}indicate respectively the node distance between ${v}_{i}$ and ${v}_{j}$ and the node distance between v

_{p}and v

_{q}. The combined distance is the same as Equation (1), ${\Delta}_{inner}=\alpha \xb7{\Delta}_{node}+(1-\alpha )\xb7{\Delta}_{relationship}$, where α is a number between 0 and 1 and gives the weight of the node and relationship distance. In our implementation, α is set to 0.5 in Equation (10) because the importance of shape and area are considered to be the same. For the combined distance in Equation (1), α is set to 10/17 according to the value in Table 2, in which the importance value is 10 for the windows and 7 for the relationship between windows. Therefore, the weight for node is set to 10/(10 + 7) and weight for relationship is set to 7/(10 + 7). If there is not apredefined weight, the default weights for the NEMD calculation are identical in the process, e.g., α = 0.5 in Equation (10). Otherwise, the weights are generated to reflect the rational, e.g., α = 10/17 in Equation (1).

#### 3.3. Similarity Values

## 4. The Automatic Approach of Typification

#### 4.1. Typical Distribution of Windows on a Façade

#### 4.2. The Process of Typification

- (1)
- The initial step: let the constraint significances equal one. Then the sides of window can be set initially according to the change of distances among windows: ${a}_{2}={a}_{1}\xb7\frac{\mathrm{min}\left\{{c}_{2},{d}_{2}\right\}}{\mathrm{min}\left\{{c}_{1},{d}_{1}\right\}}$, and ${b}_{2}={b}_{1}\xb7\frac{\mathrm{min}\left\{{c}_{2},{d}_{2}\right\}}{\mathrm{min}\left\{{c}_{1},{d}_{1}\right\}}$.
- (2)
- Put the initial values into Equations (17) and (18), and the number of windows in row and column can be then calculated by:$${M}_{2}=({c}_{2}+{L}_{h2}-2{f}_{2})/({a}_{2}+{c}_{2})$$$${N}_{2}=({d}_{2}+{L}_{v2}-2{e}_{2})/({a}_{2}+{d}_{2})$$
- (3)
- ${M}_{2}$ and ${N}_{2}$ are rounded to the nearest integer. The differences between the calculated values ${M}_{2}$, ${N}_{2}$ and their nearest integer can be utilized to judge whether the process should terminate or not. In our work $\left|round({M}_{2})-{M}_{2}\right|<0.25$ and $\left|round({N}_{2})-{N}_{2}\right|<0.25$ were set as the thresholds below which the process will be terminated.
- (4)
- If the threshold is not yet reached, ${\kappa}_{1}$ will be increased by 0.01, i.e., ${\kappa}_{1}={\kappa}_{1}+0.01$. Then the new ${M}_{2}$ and ${N}_{2}$ will be calculated. If the threshold is reached, the process will terminate; otherwise it will go on to the subsequent step.
- (5)
- ${\kappa}_{2}$ will be increased by 0.01, i.e., ${\kappa}_{2}={\kappa}_{2}+0.01$. Then the new ${M}_{2}$ and ${N}_{2}$ will be calculated. If the threshold is reached, the process will terminate, otherwise it will go on to the subsequent step.
- (6)
- $\tau $ will be increased by 0.01, i.e., $\tau =\tau +0.01$. Then the new ${M}_{2}$ and ${N}_{2}$ will be calculated. If the threshold is reached, the process will terminate, otherwise it will go back to step 4.

## 5. Experiments and Evaluation

## 6. Conclusions and Further Works

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**An example façade and its vector representation: (

**a**) Example façade of the building located in Arnulfstrasse 53, Munich; (

**b**) Windows extracted from the image.

**Figure 2.**Six different options of typification of the example façade in Figure 1: (

**a**) Option 1: constraints 2, 3 and 4; (

**b**) Option 2: constraints 1, 2 and 4; (

**c**) Option 3: constraints 1, 2 and 3; (

**d**) Option 4: constraints 3, 4 and 5; (

**e**) Option 5: constraints 1, 4 and 5; (

**f**) Option 6: constraints 1, 3 and 5.

**Figure 3.**Three different façades with regularly distributed windows. With the (

**a**) is a façade of NH hotel in Munich, (

**b**) and (

**c**) are façades of two normal buildings on Nymphenburg street in Munich.

**Figure 4.**An example of ARG matching ([26]).

**Figure 7.**Typical distribution of windows on a façade: (

**a**) tendency is in horizontal direction; (

**b**) tendency is in vertical direction; (

**c**) no tendency; (

**d**) regularity is disturbed by a door; (

**e**) windows are not equally-sized; (

**f**) the façade is composed of three regular patterns.

**Figure 8.**Distribution of windows: (

**a**) original distribution; (

**b**) possible distribution after the typification.

**Figure 9.**Typification for a regularly distributed façade at two different scales: (

**a**) original distribution; (

**b**) for scale reduced by 2×; (

**c**) manual typification; (

**d**) manual typification.

**Figure 10.**Typifying a façade whose windows are not evenly distributed in both directions: (

**a**) original façade which can be composed of several segments; (

**b**) Typified for scale reduced by 2×.

**Figure 11.**Typifying a façade, in which the windows are distributed irregularly: (

**a**) original façade which can be composed of several segments; (

**b**) for scale reduced by 2×.

Options | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Values | 8.72 | 6.45 | 6.14 | 6.67 | 7.54 | 5.50 |

Value | Constraint | |
---|---|---|

more significant less significant | 10.0 | Keeping ratio of height and width of the windows |

7.0 | Keeping the distances between windows and the outline of the façade, and keeping the distances among windows at the same time | |

5.3 | Keeping the distances between windows and the outline of the façade | |

4.7 | Keeping the distances between windows | |

4.6 | Typification in tendency direction | |

3.9 | Keeping the area covered by windows |

Options | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

NEMD | 43.5 | 47.8 | 56.5 | 55.2 | 47.5 | 50.1 |

User survey | 8.72 | 6.45 | 6.14 | 6.67 | 7.54 | 5.50 |

© 2016 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/).

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**MDPI and ACS Style**

Shen, J.; Fan, H.; Mao, B.; Wang, M.
Typification for Façade Structures Based on User Perception. *ISPRS Int. J. Geo-Inf.* **2016**, *5*, 239.
https://doi.org/10.3390/ijgi5120239

**AMA Style**

Shen J, Fan H, Mao B, Wang M.
Typification for Façade Structures Based on User Perception. *ISPRS International Journal of Geo-Information*. 2016; 5(12):239.
https://doi.org/10.3390/ijgi5120239

**Chicago/Turabian Style**

Shen, Jie, Hongchao Fan, Bo Mao, and Menghe Wang.
2016. "Typification for Façade Structures Based on User Perception" *ISPRS International Journal of Geo-Information* 5, no. 12: 239.
https://doi.org/10.3390/ijgi5120239