# Fourier-Based Automatic Transformation between Mapping Shapes—Cadastral and Land Registry Applications

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- The starting point for each shape is not the same point, see Figure 3, so that they have to be linked manually.
- Each shape consists of a different number of vertices.

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Horizontal displacement error observed between the orthophoto (correct) and the vector cadastral parcel.

**Figure 3.**The corresponding points in the corresponding shapes have different numbers due to two reasons: the start vertices are different and also the direction (clockwise in the blue polygon, and counterclockwise in the red polygon).

**Figure 4.**Transformed shapes using Fourier descriptors. The corresponding points between shapes are shown before (

**a**) and after (

**c**) transformation.

**Figure 5.**Digitized parcels: green is the correct parcel (left) and blue is the parcel to be transformed.

**Figure 6.**Transformed parcels using Fourier descriptors. The corresponding points between shapes are shown before (

**a**) and after (

**b**) transformation.

**Figure 7.**(

**a**) Separation between reference and misregistered polygons, (

**b**) separation histogram from 1000 samples measures between homologous point corresponding to reference and misregistered polygons (20 bin has been created in order to assign every measure to a bin and create the histogram): the mean value for the separation between reference and misregistered polygons is 1.167 m.

**Figure 9.**Buildings: cadastral parcel misregistered (red) and correct ones (blue) and misregistered after Fourier transformation (green).

**Table 1.**Least squares parameter settings: nth harmonic(nth harm) and number of Iterations (N Iter). Mean, Median and Maximum expressed in m. Digitized parcel.

nth Harm/N Iter | Mean | Median | Max |
---|---|---|---|

2 – 2 | 1.169 | 0.953 | 4.499 |

2 – 5 | 1.169 | 0.959 | 4.484 |

2 – 10 | 1.169 | 0.958 | 4.483 |

3 – 2 | 1.170 | 0.949 | 4.517 |

3 – 5 | 1.171 | 0.941 | 4.529 |

3 – 10 | 1.171 | 0.941 | 4.529 |

5 – 2 | 1.169 | 0.966 | 4.461 |

5 – 5 | 1.169 | 0.962 | 4.471 |

5 – 10 | 1.169 | 0.962 | 4.471 |

**Table 2.**Least squares parameter settings: nth harmonic(nth harm) and number of Iterations (N Iter). Mean, Median and Maximum expressed in m. 3 Buildings.

Building 1 | Building 2 | Building 3 | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{n}$th harm/N Iter | Mean | Median | Max | Mean | Median | Max | Mean | Median | Max |

2 – 2 | 0.259 | 0.208 | 0.285 | 1.254 | 1.152 | 3.037 | 1.093 | 0.941 | 3.411 |

2 – 5 | 0.259 | 0.208 | 0.667 | 1.430 | 1.302 | 3.404 | 1.090 | 0.940 | 3.400 |

2 – 10 | 0.259 | 0.208 | 0.667 | 1.439 | 1.318 | 3.420 | 1.090 | 0.940 | 3.400 |

3 – 2 | 0.151 | 0.138 | 0.285 | 0.747 | 0.768 | 1.586 | 0.935 | 0.884 | 1.911 |

3 – 5 | 0.151 | 0.138 | 0.285 | 0.747 | 0.771 | 1.583 | 0.934 | 0.881 | 1.904 |

3 – 10 | 0.151 | 0.138 | 0.285 | 0.747 | 0.771 | 1.583 | 0.934 | 0.881 | 1.904 |

5 – 2 | 0.152 | 0.141 | 0.291 | 0.747 | 0.769 | 1.585 | 0.935 | 0.882 | 1.914 |

5 – 5 | 0.152 | 0.141 | 0.291 | 0.747 | 0.772 | 1.583 | 0.935 | 0.882 | 1.907 |

5 – 10 | 0.152 | 0.141 | 0.291 | 0.747 | 0.772 | 1.583 | 0.935 | 0.882 | 1.907 |

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## Share and Cite

**MDPI and ACS Style**

Reinoso-Gordo, J.F.; Romero-Zaliz, R.; León-Robles, C.; Mataix-SanJuan, J.; Antonio Nero, M.
Fourier-Based Automatic Transformation between Mapping Shapes—Cadastral and Land Registry Applications. *ISPRS Int. J. Geo-Inf.* **2020**, *9*, 482.
https://doi.org/10.3390/ijgi9080482

**AMA Style**

Reinoso-Gordo JF, Romero-Zaliz R, León-Robles C, Mataix-SanJuan J, Antonio Nero M.
Fourier-Based Automatic Transformation between Mapping Shapes—Cadastral and Land Registry Applications. *ISPRS International Journal of Geo-Information*. 2020; 9(8):482.
https://doi.org/10.3390/ijgi9080482

**Chicago/Turabian Style**

Reinoso-Gordo, Juan Francisco, Rocío Romero-Zaliz, Carlos León-Robles, Jesús Mataix-SanJuan, and Marcelo Antonio Nero.
2020. "Fourier-Based Automatic Transformation between Mapping Shapes—Cadastral and Land Registry Applications" *ISPRS International Journal of Geo-Information* 9, no. 8: 482.
https://doi.org/10.3390/ijgi9080482