# The Role of Global Appearance of Omnidirectional Images in Relative Distance and Orientation Retrieval

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Global Appearance Descriptors

#### 2.1. Descriptors Based on the Discrete Fourier Transform

#### 2.2. Descriptors Based on Histograms of Oriented Gradients

#### 2.3. Descriptors Based on Gist

#### 2.4. Descriptors Based on Wi-SURF

#### 2.5. Descriptors Based on BRIEF-Gist

#### 2.6. Descriptors Based on Radon Transform

## 3. Solving the Absolute Localization Problem

#### 3.1. Descriptors Based on the Discrete Fourier Transform

- A set of artificial rotations is applied to the test image. The shift theorem of the unidimensional DFT can be used to generate the argument matrices of the test image rotated siblings. The step between consecutive rotations is $\Delta \varphi $. This is equivalent to making a shift of the columns of the panoramic image with a magnitude of d pixels, where $\Delta \varphi =d\xb72\pi /{N}_{2}$. In the experiments, we consider $d=\{1,2,\cdots ,{N}_{2}-1\}$. This means that the angular step between consecutive artificial rotations is $\Delta \varphi =2\pi /{N}_{2}$. This is the resolution of the method.
- After this process, a set of ${n}_{rot}=2\pi /\Delta \varphi $ arguments matrices are available at time instant t.$${\{{\Phi}_{0},{\Phi}_{1},\cdots ,{\Phi}_{{n}_{rot}}\}}_{t}={\left\{{\Phi}_{\alpha}\right\}}_{t},\alpha =0,\dots ,{n}_{rot}$$
- The Hadamard product of the matrix ${\Phi}_{t}$ and every matrix ${\Phi}_{\alpha}$ is calculated. The sum of the components of each resulting matrix is obtained, and the result is an array of data:$${\{{m}_{0},{m}_{1},\cdots ,{m}_{{n}_{rot}}\}}_{t}={\left\{{m}_{\alpha}\right\}}_{t},\alpha =0,\dots ,{n}_{rot}$$
- The estimated relative rotation is the $\alpha $ value whose coefficient ${m}_{\alpha}$ presents the maximum value.$$\alpha ={arg\phantom{\rule{0.166667em}{0ex}}max}_{\alpha}\left\{{m}_{\alpha}\right\}$$$${\theta}_{ti}=\frac{2\pi \alpha}{{n}_{rot}}$$$${\theta}_{t}={\theta}_{i}+{\theta}_{ti}$$In this equation, ${\theta}_{i}$ is the orientation that the robot had when the map image $i{m}_{i}$ was captured, with respect to the global reference system.

#### 3.2. Descriptors Based on Histograms of Oriented Gradients

#### 3.3. Descriptors Based on Gist

#### 3.4. Descriptors Based on Wi-SURF

#### 3.5. Descriptors Based on BRIEF-Gist

#### 3.6. Descriptors Based on the Radon Transform

#### 3.6.1. Radon–Fourier Method

#### 3.6.2. Radon–POC Method

## 4. Experimental Setup

#### 4.1. Sets of Images

#### 4.2. Addition of Noise and Occlusions

## 5. Results and Discussion

#### 5.1. Image Retrieval Problem

- Weighted metric distance:$$dis{t}_{p}(\overrightarrow{r},\overrightarrow{s})={\left(\sum _{i=1}^{l}{\omega}_{i}\xb7{|{r}_{i}-{s}_{i}|}^{p}\right)}^{\frac{1}{p}}$$If we consider ${\omega}_{i}=1$, $i=1,\cdots ,l$, the Minkowski distance is obtained. Two particular cases will be considered: $dis{t}_{1}$ (Manhattan distance), which is defined from the Minkowski distance with $p=1$, and $dis{t}_{2}$ (Euclidean distance), doing $p=2$.
- Pearson correlation coefficient. It is a similitude coefficient that can be obtained as:$$si{m}_{Pea}(\overrightarrow{r},\overrightarrow{s})=\frac{{\overrightarrow{r}}_{d}^{T}\xb7{\overrightarrow{s}}_{d}}{|{\overrightarrow{r}}_{d}\left|\right|{\overrightarrow{s}}_{d}|}$$$$dis{t}_{3}(\overrightarrow{r},\overrightarrow{s})=1-si{m}_{Pea}(\overrightarrow{r},\overrightarrow{s})$$
- Inner product: It is also a similitude coefficient that can be calculated as the scalar product between the two vectors to compare.$$si{m}_{cos}(\overrightarrow{r},\overrightarrow{s})=\frac{{\overrightarrow{r}}^{T}\xb7\overrightarrow{s}}{|\overrightarrow{r}\left|\right|\overrightarrow{s}|}$$As shown in the equation, $\overrightarrow{r}$ and $\overrightarrow{s}$ are usually normalized. In this case, this measure is known as cosine similitude and takes values in the range $[-1,+1]$. The corresponding distance value is:$$dis{t}_{4}(\overrightarrow{r},\overrightarrow{s})=1-si{m}_{in}(\overrightarrow{r},\overrightarrow{s})$$

^{®}at 3 GHz and using the mathematical tool Matlab

^{®}. These time results are not absolute, they depend of the computer which runs the process. They are comparable because all the calculations have been done with the same machine.

#### 5.2. Estimation of the Position

#### 5.3. Estimation of the Orientation

#### 5.4. Evaluation with a Trajectory Dataset

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Bird’s eye view of the capture points of the training set of images. The size of the grid is $40\times 40$ cm.

**Figure 3.**Library. Bird’s eye view of the capture points of the training set of images. The size of the grid is $40\times 40$ cm.

**Figure 4.**Corridor. Bird’s eye view of the capture points of the training set of images. The size of the grid is $40\times 40$ cm.

**Figure 5.**Sample image from the training test with (

**a**) different levels of added Gaussian noise (${\sigma}^{2}=\{0,0.0025,0.05,0.01,0.02,0.05\}$) and (

**b**) sequence of occlusions considered ($\{0,5,10,20,40\}\%$).

**Figure 6.**FS image retrieval problem. Success rate of the method. ${k}_{1}$ and ${k}_{2}$ are, respectively, the number of rows and columns of the descriptor (Table 1).

**Figure 7.**HOG image retrieval problem. Success rate of the method. ${k}_{5}$ is the number of horizontal cells and ${b}_{1}$ the number of bins per histogram (Table 1).

**Figure 8.**Gist image retrieval problem. Success rate of the method. ${k}_{7}$ is the number of horizontal blocks and ${m}_{1}$ the number of Gabor filters to build the descriptor (Table 1).

**Figure 9.**WS image retrieval problem. Success rate of the method. ${k}_{9}$ is the number of horizontal cells and ${w}_{1}$ the number of windows per cell (Table 1).

**Figure 10.**BG image retrieval problem. Success rate of the method. ${k}_{10}$ is the number of horizontal cells and ${w}_{2}$ the number of windows per cell (Table 1).

**Figure 11.**RT–F image retrieval problem. Success rate of the method. ${k}_{11}$ is the number of blocks and ${p}_{1}$ the relative angle (deg) between the lines in each set (Table 1).

**Figure 12.**RT–POC image retrieval problem. Success rate of the method. ${p}_{1}$ is the relative angle (deg) between the lines en each set (Table 1).

**Figure 20.**FS average localization error with noise: (

**a**) no filter and (

**b**) homomorphic filtering. Legend: — Original, — Noise 1, — Noise 2, — Noise 3, — Noise 4, — Noise 5.

**Figure 21.**FS average localization error with occlusions: (

**a**) no filter and (

**b**) homomorphic filtering. Legend: — Original, — Occlusion 1, — Occlusion 2, — Occlusion 3, — Occlusion 4.

**Figure 22.**HOG average localization error with noise: (

**a**) no filter and (

**b**) homomorphic filter. Legend: — Original, — Noise 1, — Noise 2, — Noise 3, — Noise 4, — Noise 5.

**Figure 23.**HOG average localization error with occlusions: (

**a**) no filter and (

**b**) homomorphic filter. Legend: — Original, — Occlusion 1, — Occlusion 2, — Occlusion 3, — Occlusion 4.

**Figure 24.**Gist average localization error with noise: (

**a**) no filter and (

**b**) homomorphic filtering. Legend: — Original, — Noise 1, — Noise 2, — Noise 3, — Noise 4, — Noise 5.

**Figure 25.**Gist average localization error with occlusions: (

**a**) no filter and (

**b**) homomorphic filtering. Legend: — Original, — Occlusion 1, — Occlusion 2, — Occlusion 3, — Occlusion 4.

**Figure 26.**WS average localization error with noise: (

**a**) no filter and (

**b**) homomorphic filtering. Legend: — Original, — Noise 1, — Noise 2, — Noise 3, — Noise 4, — Noise 5.

**Figure 27.**WS average localization error with occlusions: (

**a**) no filter and (

**b**) homomorphic filtering. Legend: — Original, — Occlusion 1, — Occlusion 2, — Occlusion 3, — Occlusion 4.

**Figure 28.**BG average localization error with noise: (

**a**) no filter and (

**b**) homomorphic filtering. Legend: — Original, — Noise 1, — Noise 2, — Noise 3, — Noise 4, — Noise 5.

**Figure 29.**BG average localization error with occlusions: (

**a**) no filter and (

**b**) homomorphic filtering. Legend: — Original, — Occlusion 1, — Occlusion 2, — Occlusion 3, — Occlusion 4.

**Figure 31.**FS orientation estimation in the presence of occlusions. Average orientation error (deg).

**Figure 34.**HOG orientation estimation in the presence of occlusion. Average orientation error (deg).

**Figure 36.**Gist orientation estimation in the presence of occlusion. Average orientation error (deg).

**Figure 44.**Average errors with the COLD dataset in the presence of noise. (

**a**) Average position error (cm) and (

**b**) average orientation error (deg). Legend: — Original, — Noise 1, — Noise 2, — Noise 3, — Noise 4.

**Figure 45.**Average errors with the COLD dataset in the presence of occlusions. (

**a**) Average position error (cm) and (

**b**) average orientation error (deg). Legend: — Original, — Occlusion 1, — Occlusion 2, — Occlusion 3, — Occlusion 4.

Descriptor | Parameters |
---|---|

FS | ${k}_{1}\Rightarrow $ number of rows, position descriptor ${\mathbf{A}}_{j}$ ${k}_{2}\Rightarrow $ number of columns, position descriptor ${\mathbf{A}}_{j}$ ${k}_{3}\Rightarrow $ number of rows, orientation descriptor ${\Phi}_{j}$ ${k}_{4}\Rightarrow $ number of columns, orientation descriptor ${\Phi}_{j}$ |

HOG | ${b}_{1}\Rightarrow $ number of bins per histogram, position descriptor ${\overrightarrow{h}}_{1j}$ ${k}_{5}\Rightarrow $ number of horizontal cells, position descriptor ${\overrightarrow{h}}_{1j}$ ${b}_{2}\Rightarrow $ number of bins per histogram, orientation descriptor ${\overrightarrow{h}}_{2j}$ ${l}_{1}\Rightarrow $ width of vertical cells, orientation descriptor ${\overrightarrow{h}}_{2j}$ ${d}_{1}\Rightarrow $ distance between vertical cells, orientation descriptor ${\overrightarrow{h}}_{2j}$ ${k}_{6}=\frac{{N}_{2}}{{d}_{1}}\Rightarrow $ number of vertical cells, orientation descriptor ${\overrightarrow{h}}_{2j}$ |

Gist | ${m}_{1}\Rightarrow $ number of orientations (Gabor filters), position descriptor ${\overrightarrow{g}}_{1j}$ ${k}_{7}\Rightarrow $ number of horizontal blocks, position descriptor ${\overrightarrow{g}}_{1j}$ ${m}_{2}\Rightarrow $ number of orientations (Gabor filters), orientation descriptor ${\overrightarrow{g}}_{2j}$ ${l}_{2}\Rightarrow $ width of vertical blocks, orientation descriptor ${\overrightarrow{g}}_{2j}$ ${d}_{2}\Rightarrow $ distance between vertical blocks, orientation descriptor ${\overrightarrow{g}}_{2}$ ${k}_{8}=\frac{{N}_{2}}{{d}_{2}}\Rightarrow $ number of vertical blocks, orientation descriptor ${\overrightarrow{g}}_{2j}$ |

WS | ${w}_{1}\Rightarrow $ number of windows per cell, descriptor ${\overrightarrow{ws}}_{j}$ ${k}_{9}\Rightarrow $ number of horizontal blocks, descriptor ${\overrightarrow{ws}}_{j}$ $s{p}_{1}\Rightarrow $ horizontal space between windows, descriptor ${\overrightarrow{ws}}_{j}$ |

BG | ${w}_{2}\Rightarrow $ number of windows per cell, descriptor ${\overrightarrow{bg}}_{j}$ ${k}_{10}\Rightarrow $ number of horizontal blocks, descriptor ${\overrightarrow{bg}}_{j}$ |

RT | ${p}_{1}\Rightarrow $ degrees between lines where Radon is calculated, matrix r ${k}_{11}\Rightarrow $ number of columns, position descriptor ${{\mathbf{A}}_{\mathbf{RT}}}_{j}$ ${k}_{12}\Rightarrow $ number of columns, orientation descriptor ${{\Phi}_{RT}}_{j}$ ${N}_{x}\Rightarrow $ omnidirectional images’ size is ${N}_{x}\times {N}_{x}$ |

**Table 2.**Contents of the map, for localization and orientation estimation, per image included in the model $i{m}_{j},\phantom{\rule{0.277778em}{0ex}}j=1,\cdots ,n$.

Descriptor | Localization | Orientation |
---|---|---|

FS | ${\mathbf{A}}_{j}\in {\mathbb{R}}^{{k}_{1}\times {k}_{2}}$ | ${\Phi}_{j}\in {\mathbb{R}}^{{k}_{3}\times {k}_{4}}$ |

HOG | ${\overrightarrow{h}}_{1j}\in {\mathbb{R}}^{{k}_{5}\xb7{b}_{1}\times 1}$ | ${\overrightarrow{h}}_{2j}\in {\mathbb{R}}^{{k}_{6}\xb7{b}_{2}\times 1}$ |

Gist | ${\overrightarrow{g}}_{1j}\in {\mathbb{R}}^{2\xb7{k}_{7}\xb7{m}_{1}\times 1}$ | ${\overrightarrow{g}}_{2j}\in {\mathbb{R}}^{{k}_{8}\xb7{m}_{2}\times 1}$ |

WS | ${\overrightarrow{ws}}_{j}\in {\mathbb{R}}^{{k}_{9}\xb7{w}_{1}\xb764\times 1}$ | |

BG | ${\overrightarrow{bg}}_{j}\in {\mathbb{R}}^{{k}_{10}\xb7{w}_{2}\times 1}$ | |

RT–F | ${{\mathbf{A}}_{\mathbf{RT}}}_{j}\in {\mathbb{R}}^{\frac{360}{{p}_{1}}\times {k}_{11}}$ | ${{\Phi}_{RT}}_{j}\in {\mathbb{R}}^{\frac{360}{{p}_{1}}\times {k}_{12}}$ |

RT–POC | $r\in {\mathbb{R}}^{\frac{360}{{p}_{1}}\times 0.5\xb7{N}_{x}}$ |

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**MDPI and ACS Style**

Román, V.; Payá, L.; Peidró, A.; Ballesta, M.; Reinoso, O.
The Role of Global Appearance of Omnidirectional Images in Relative Distance and Orientation Retrieval. *Sensors* **2021**, *21*, 3327.
https://doi.org/10.3390/s21103327

**AMA Style**

Román V, Payá L, Peidró A, Ballesta M, Reinoso O.
The Role of Global Appearance of Omnidirectional Images in Relative Distance and Orientation Retrieval. *Sensors*. 2021; 21(10):3327.
https://doi.org/10.3390/s21103327

**Chicago/Turabian Style**

Román, Vicente, Luis Payá, Adrián Peidró, Mónica Ballesta, and Oscar Reinoso.
2021. "The Role of Global Appearance of Omnidirectional Images in Relative Distance and Orientation Retrieval" *Sensors* 21, no. 10: 3327.
https://doi.org/10.3390/s21103327