# Camera Calibration for Coastal Monitoring Using Available Snapshot Images

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Camera Mathematical Models

- radial and tangential distortions: ${k}_{1\U0001f7c9}$, ${k}_{2\U0001f7c9}$, ${p}_{1\U0001f7c9}$, and ${p}_{2\U0001f7c9}$ (dimensionless);
- pixel size: ${s}_{c\U0001f7c9}$ and ${s}_{r\U0001f7c9}$ (dimensionless); and
- decentering: ${o}_{c}$ and ${o}_{r}$ (in pixels),

- real world co-ordinates of the center of vision: ${x}_{c}$, ${y}_{c}$, and ${z}_{c}$ (in units of length); and
- Eulerian angles: $\varphi $, $\sigma $, and $\tau $ (in radians).

#### 2.2. Error Definition and Calibration Procedure

#### 2.3. Experimental Setup

## 3. Results

#### 3.1. Error Analysis

#### 3.2. Influence of the Obliquity of the Number of Gcps

#### 3.3. Calibration Parameters

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CPU | Central Processing Unit |

GCP(s) | Ground Control Point(s) |

RMS | Root Mean Square |

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**Figure 1.**Images from Castelldefels (

**A**, at ${41}^{\circ}{15}^{\prime}54.{9}^{\prime \prime}$ N, ${1}^{\circ}{59}^{\prime}29.{1}^{\prime \prime}$ E) and Barcelona (

**B**, at ${41}^{\circ}{23}^{\prime}16.{5}^{\prime \prime}$ N, ${2}^{\circ}{11}^{\prime}50.{9}^{\prime \prime}$ E) video monitoring stations (coo.icm.csic.es).

**Figure 2.**Real-world to pixel transformation: camera position (${x}_{c}$, ${y}_{c}$, ${z}_{c}$) and Eulerian angles ($\varphi $, $\sigma $ and $\tau $).

**Figure 3.**Angles ${A}_{1}$ ($\tau \sim {55}^{\circ}$), ${A}_{2}$ ($\tau \sim {40}^{\circ}$), and ${A}_{3}$ ($\tau \sim {15}^{\circ}$) to analyze the influence of obliquity.

**Figure 4.**Subsets ${S}_{0}$ to ${S}_{7}$, for angle ${A}_{1}$, considered to analyze the influence of the GCPs distribution.

**Figure 5.**Errors ${\u03f5}_{P}$ for angle ${A}_{1}$ and models ${M}_{1}$ (

**A**) and ${M}_{2}$ (

**B**) as a function of the GCP calibration subset (${S}_{0}$ to ${S}_{7}$).

**Figure 6.**Errors $\tilde{\u03f5}$ for ${M}_{1}$ and angle ${A}_{1}$ at all the available points for the different sets ${S}_{k}$. The GCPs for each set are here highlighted with white circles in the center and correspond to the points in Figure 4.

**Figure 7.**Errors $\tilde{\u03f5}$ for ${M}_{2}$ and angle ${A}_{1}$ at all the available points for the different sets ${S}_{k}$. The GCPs for each set are here highlighted with white circles in the center and correspond to the points in Figure 4.

**Figure 8.**Errors ${\u03f5}_{Q}$ and ${\u03f5}_{G}$ for angle ${A}_{1}$ and models ${M}_{1}$ (

**A**) and ${M}_{2}$ (

**B**) as a function of the GCP calibration subset (${S}_{0}$ to ${S}_{7}$).

**Figure 9.**Errors ${\u03f5}_{Q}$ and ${\u03f5}_{G}$ for angle ${A}_{1}$ with $\tau \sim {55}^{\circ}$ (

**A**,

**B**); ${A}_{2}$ with $\tau \sim {40}^{\circ}$ (

**C**,

**D**); and ${A}_{3}$ with $\tau \sim {15}^{\circ}$ (

**E**,

**F**); and for models ${M}_{1}$ (

**A**,

**C**,

**E**) and ${M}_{2}$ (

**B**,

**D**,

**F**) as a function of the GCP calibration set (${S}_{0}$ to ${S}_{7}$).

**Figure 10.**Errors ${\u03f5}_{Q}$ and ${\u03f5}_{G}$ for angle ${A}_{1}$ for different numbers of GCPs (for sets ${S}_{1}$ to ${S}_{7}$): 6 GCPs (

**A**,

**B**); 8 GCPs (

**C**,

**D**); and 12 GCPs (

**E**,

**F**) and for models ${M}_{1}$ (

**A**,

**C**,

**E**) and ${M}_{2}$ (

**B**,

**D**,

**F**).

**Figure 11.**Radial distortion ${k}_{1\U0001f7c9}$ (

**A**,

**B**) and pixel size ${s}_{c\U0001f7c9}$ (

**C**,

**D**) for models ${M}_{1}$ (

**A**,

**C**) and ${M}_{2}$ (

**B**,

**D**) for angle ${A}_{1}$.

**Figure 12.**Demeaned camera position co-ordinates ${x}_{c}$, ${y}_{c}$, and ${z}_{c}$ for angles ${A}_{1}$ (

**A**,

**D**,

**G**), ${A}_{2}$ (

**B**,

**E**,

**H**), and ${A}_{3}$ (

**C**,

**F**,

**I**) for model ${M}_{2}$. The unit length “u” corresponds to the side of the squares of the grid.

**Figure 13.**Demeaned camera Eulerian angles $\varphi $, $\sigma $, and $\tau $ for angles ${A}_{1}$ (

**A**,

**D**,

**G**), ${A}_{2}$ (

**B**,

**E**,

**H**), and ${A}_{3}$ (

**C**,

**F**,

**I**) for model ${M}_{2}$.

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## Share and Cite

**MDPI and ACS Style**

Simarro, G.; Calvete, D.; Souto, P.; Guillén, J.
Camera Calibration for Coastal Monitoring Using Available Snapshot Images. *Remote Sens.* **2020**, *12*, 1840.
https://doi.org/10.3390/rs12111840

**AMA Style**

Simarro G, Calvete D, Souto P, Guillén J.
Camera Calibration for Coastal Monitoring Using Available Snapshot Images. *Remote Sensing*. 2020; 12(11):1840.
https://doi.org/10.3390/rs12111840

**Chicago/Turabian Style**

Simarro, Gonzalo, Daniel Calvete, Paola Souto, and Jorge Guillén.
2020. "Camera Calibration for Coastal Monitoring Using Available Snapshot Images" *Remote Sensing* 12, no. 11: 1840.
https://doi.org/10.3390/rs12111840