# Classifying the Built-Up Structure of Urban Blocks with Probabilistic Graphical Models and TerraSAR-X Spotlight Imagery

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## Abstract

**:**

## 1. Introduction

#### 1.1. Descriptive Attributes for Classifying Urban Structure Types

#### 1.2. Main Assumptions and Contributions of This Paper

## 2. Methods

#### 2.1. Context-Based Classification with Probabilistic Graphical Models

_{i}is the unknown class label of block i and s

_{i}is a vector of attributes extracted from block i. The letter j denotes a block belonging to the set of neighbors N

_{i}from block i. Z(S) is the partition function of the distribution, which scales the probabilities to the interval [0,1]. The so-called association factors $logp({c}_{i}|{s}_{i})$ represent the likelihood terms or the prior probabilities of the classes for each block i, whereas the interaction factors ${\varphi}_{ij}({c}_{i},{c}_{j})$ map each possible joint class assignment of c

_{i}and c

_{j}to a real positive number. Since the goal is to maximize Equation (1), in other words, to find the joint class assignment for all c

_{i}for which the conditional probability P(C|S) is maximum (this is commonly referred to as the estimation of the maximum a posteriori probability, or MAP solution), we may disregard the partition function Z(S) and reformulate Equation (1) as the following energy function:

#### 2.2. Model Parameterization and Neighborhood Definition Criteria

#### 2.3. Urban Blocks’ Attributes

^{2}were discarded from further analysis. This threshold value was chosen given the fact that the footprint of a house or building is hardly smaller than that. The remaining dark and bright areas had their size, compactness, perimeter-area ratio and length-to-width ratio computed. Table 1 shows which descriptive attributes were computed from the dark and bright areas and then submitted for the classification with the Random Forest algorithm. The parameters of some of these attributes were set empirically based on the image resolution and the typical area range and shape of different types of buildings in the test-site, namely single-family houses, apartment buildings and warehouses. When transferring the model to other cities, adjustments in the parameter values may be necessary. Alternatively, additional attributes with different parameter values may be submitted to the Random Forest algorithm, which is able to perform attribute relevance analysis and is robust to correlated variables [42].

## 3. Experiments

#### 3.1. Image and Auxiliary Data

#### 3.2. Study Area and Urban Structure Type Classes

#### 3.3. Classification Experiments

## 4. Results

#### 4.1. Model Comparison

#### 4.2. Accuracy Analysis

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Construction of a simple undirected probabilistic graphical model for the classification of urban structure types. (

**a**) Blocks are considered neighbors if the Euclidean distance between their centers of mass is not larger than a threshold (th). (

**b**) Each block’s unobserved class ${c}_{i}$ is represented in the graph by a node. (

**c**) An edge is created between each pair of neighboring blocks, thus forming an interaction factor. (

**d**) If it is defined that, besides its neighbors unobserved variable ${c}_{j}$, ${c}_{i}$ also depends on its observed attributes ${s}_{i}$. The number of association factors in the graph equals the number of blocks in the study site. The number of pairwise interaction factors in the graph equals the number of pairs of neighboring blocks.

**Figure 2.**Outcome of the main steps involved in the computation of block attributes from extracted lines and polygons. (

**a**) The input to the procedure is the intensity SAR image of each block. (

**b**) Lines extracted from the input image. (

**c**) Polygons extracted from the input image. (

**d**) Lines and the block’s boundaries projected to the local coordinate system. (

**e**) Polygons and the block’s boundaries projected to the local coordinate system. (

**f**) Network created from the extracted polygons.

**Figure 3.**The image data footprints and the study area. Each footprint covers an area of approximately 10 × 5 km. Their intersection area was defined as our study area. The central point coincides with the historical center of Munich (Germany).

**Figure 4.**Plot-graphs of the overall accuracies obtained with the four different models explored in this work and with $\lambda $ parameters varying from 0.01–1.0. The colors of the lines indicate the neighborhood definition criterion.

**Figure 5.**Ground-truth map and the standard and context-based classifications. (

**a**) Standard classification performed with the Random Forest algorithm, i.e., without considering context. (

**b**) context-based classification performed with model CRF 1 and the neighborhood definition criterion of a fixed radius of 240 m. (

**c**) The official urban structure type (UST) map of Munich generalized for five classes and considered as the ground-truth.

**Table 1.**Urban blocks attributes computed from the extracted bright and dark areas. Bright and dark areas were extracted by applying threshold operations on the Synthetic Aperture Radar intensity and coherence images.

Dark and bright areas’ attributes: |

Number of bright areas |

Number of dark areas |

Number of bright areas larger than 200 pixels |

Number of dark areas larger than 200 pixels |

Number of bright areas larger than 500 pixels |

Number of dark areas larger than 500 pixels |

Number of bright areas larger than 1000 pixels |

Number of dark areas larger than 1000 pixels |

Sum of bright areas larger than 200 pixels/block’s area |

Sum of dark areas larger than 200 pixels/block’s area |

Sum of bright areas larger than 500 pixels/block’s area |

Sum of dark areas larger than 500 pixels/block’s area |

Sum of bright areas larger than 1000 pixels/block’s area |

Sum of dark areas larger than 1000 pixels/block’s area |

Sum of bright areas/block’s area |

Sum of dark areas/block’s area |

Total area of all bright areas |

Total area of all dark areas |

Mean area of all bright areas |

Mean area of all dark areas |

Maximum area among bright areas |

Maximum area among dark areas |

Standard deviation of the area of all bright areas |

Standard deviation of the area of all dark areas |

Maximum compactness among bright areas |

Maximum compactness among dark areas |

Compactness of the largest bright area |

Compactness of the largest dark area |

Maximum length-to-width ratio among bright areas |

Maximum length-to-width ratio among dark areas |

Minimum perimeter-area ratio among bright areas |

Minimum perimeter-area ratio among dark areas |

Number of bright areas with length-to-width ratio higher than 5 |

Number of dark areas with length-to-width ratio higher than 5 |

Number of bright areas with length-to-width ratio higher than 10 |

Number of dark areas with length-to-width ratio higher than 10 |

**Table 2.**Attributes computed from the lines extracted from the Synthetic Aperture Radar intensity images of each block. The line extraction procedure is described in detail in [49].

Line attributes: |

Maximum line length |

Mean angle difference between a line and its closest line |

Mean angle diff. between a line and the block boundary closest to it |

Mean angle diff. between a line and the block boundary most parallel to it |

Mean angle diff. between a line and the line most parallel to it |

Mean angle diff. between a line and the line most perpendicular to it |

Mean distance between a line and its closest line |

Mean distance between a line and the line most parallel to it |

Mean distance between a line and the line most perpendicular to it |

Mean distance between a line and the block boundary closest to it |

Mean distance between a line and the block boundary most parallel to it |

Mean distance between a line and the block boundary most perpendicular to it |

Mean orientation of the lines |

Mean length of the lines |

Min. angle diff. between a line and the block boundary closest to it |

Min. distance between a line and the line most parallel to it |

Min. distance between a line and the line most perpendicular to it |

Min. distance between a line and the block boundary most parallel to it |

Min. distance between a line and the block boundary most perpendicular to it |

Number of lines |

Number of lines longer than 50 m |

Number of lines longer than 100 m |

Std. dev. of angle difference between a line and its closest line |

Std. dev. of distance between a line and its closest block boundary |

Std. dev. of distance between a line and its closest line |

Std. dev. of distance between a line and the line most parallel to it |

Std. dev. of distance between a line and the line most perpendicular to it |

Std. dev. of the lines length |

Std. dev. of the orientation of all lines |

**Table 3.**Attributes computed from the polygons extracted from the SAR intensity image of each block. The polygon extraction procedure is described in detail in Novack [49]. * The pertinence of the polygon to an ideal building polygon shape. The computation of this pertinence is demonstrated in Novack and Stilla [45].

Polygon attributes: |

Maximum pertinence * |

Mean area |

Mean angle difference between a polygon and its closest block boundary |

Mean distance between a polygon and its closest block boundary |

Mean distance between a polygon and the polygon most parallel to it |

Mean distance between a polygon and the polygon most perpendicular to it |

Mean, max. and std. dev. of the polygons area |

Mean, max. and std. dev. of the polygons length-to-width |

Mean, max. and std. dev. of the polygons compactness |

Mean, max. and std. dev. of the polygons orientation |

Number of pairs of polygons parallel to each other |

Number of pairs of polygons perpendicular to each other |

Number of polygons |

Number of polygons with area > 3rd, 4th and 5th 5-quantiles of the area values |

Number of polygons with pertinence > [0.05, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5] |

Number of pol. with pertinence > [3rd, 4th, 5th] of the 5-quantiles of the pertinence values |

Std. dev. of the angle diff. between a polygon and its closest block boundary |

Std. dev. of the distance between a polygon and its closest block boundary |

Std. dev. of the distance between a polygon and the polygon most parallel to it |

Std. dev. of the distance between a polygon and the polygon most perpendicular to it |

Std. dev. of the polygons orientation |

3rd, 4th and 5th 5-quantiles of the polygons area |

3rd, 4th and 5th 5-quantiles of the pertinence values |

Network structure attributes: |

Density of the graph [51] |

Highest mean membership of two connected nodes |

Mean node pertinence |

Membership value of the node with highest membership |

Number of nodes |

Number of edges |

Number of edges/number of nodes |

Number of edges connecting parallel nodes |

Number of edges connecting parallel nodes/number of edges |

Number of edges connecting parallel nodes/Number of edges connecting perpendicular nodes |

Number of edges connecting perpendicular nodes |

Number of edges connecting perpendicular nodes/number of edges |

Std. dev. of the membership from the two connected nodes with highest membership mean |

Network Moran’sIof attributes [52]: |

Area |

Distance to closest block boundary |

Length to width ratio |

Orientation |

Orientation difference to closest block boundary |

Rectangular fit |

For each of the six attributes above, the following measures were computed: |

Expected I based on random permutation of the values |

Expected I under normality assumption |

Difference between I and expected I based on random permutations |

Difference between I and expected I under normality assumption |

p-value of I based on random permutation of the values |

p-value of I under the normality assumption (one-sided) |

**Table 5.**The confusion matrices of the standard and best context-based classifications. The best contextual classification was obtained with model CRF 1 and the neighborhood definition criterion of a fixed radius of 240 m. The overall accuracies achieved by the standard and contextual classifications are 68.91% and 75.95%, respectively. The respective Kappa indices are 0.57 and 0.65. The considered classes are Parks and Vegetated Areas (PVA), Detached and Semi-Detached Housing (DSDH), Large Buildings and Industrial Areas (LBIA), Dense Block Development (DBD) and Regular Block Development (RBD).

PVA | DSDH | LBIA | DBD | RBD | Total | User’s Acc. (%) | Prod.’s Acc. (%) | |
---|---|---|---|---|---|---|---|---|

Std. class.: | ||||||||

PVA | 184 | 10 | 3 | 3 | 15 | 215 | 85.58 | 84.40 |

DSDH | 22 | 144 | 6 | 12 | 35 | 219 | 65.75 | 57.60 |

LBIA | 2 | 5 | 42 | 6 | 7 | 62 | 67.77 | 30.00 |

DBD | 3 | 25 | 61 | 491 | 48 | 628 | 78.18 | 85.68 |

RBD | 7 | 66 | 28 | 64 | 91 | 256 | 35.54 | 46.42 |

Total | 218 | 250 | 140 | 573 | 196 | |||

Ctxt. class.: | ||||||||

PVA | 176 | 8 | 5 | 12 | 14 | 215 | 81.86 | 85.85 |

DSDH | 15 | 155 | 2 | 6 | 41 | 219 | 70.77 | 65.12 |

LBIA | 4 | 4 | 45 | 4 | 5 | 62 | 72.25 | 41.66 |

DBD | 5 | 10 | 33 | 546 | 34 | 628 | 82.94 | 88.63 |

RBD | 5 | 61 | 23 | 48 | 119 | 256 | 46.48 | 55.86 |

Total | 205 | 238 | 108 | 616 | 213 |

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**MDPI and ACS Style**

Novack, T.; Stilla, U.
Classifying the Built-Up Structure of Urban Blocks with Probabilistic Graphical Models and TerraSAR-X Spotlight Imagery. *Remote Sens.* **2018**, *10*, 842.
https://doi.org/10.3390/rs10060842

**AMA Style**

Novack T, Stilla U.
Classifying the Built-Up Structure of Urban Blocks with Probabilistic Graphical Models and TerraSAR-X Spotlight Imagery. *Remote Sensing*. 2018; 10(6):842.
https://doi.org/10.3390/rs10060842

**Chicago/Turabian Style**

Novack, Tessio, and Uwe Stilla.
2018. "Classifying the Built-Up Structure of Urban Blocks with Probabilistic Graphical Models and TerraSAR-X Spotlight Imagery" *Remote Sensing* 10, no. 6: 842.
https://doi.org/10.3390/rs10060842