#
Retrieval of Similar Evolution Patterns from Satellite Image Time Series^{ †}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dynamic Time Warping (DTW)

#### 2.2. Determining the Optimal Threshold Using Expectation-Maximization

- Expectation step (E-step). Compute the log-likelihood (i.e., the logarithm of the posterior probability) with respect to the current values of the parameters ${\theta}^{\left(t\right)}$:$$L\left({\theta}^{\left(t\right)}\right)=ln{p}_{{\theta}^{\left(t\right)}}\left(x\right)$$
- Maximization step (M-step). Update the model parameters such that the log-likelihood approaches its maximum, i.e., the convergence of the EM algorithm guarantees that the log-likelihood value is increased with each iteration [19]. It is usually convenient to introduce a mapping between the new model parameters ${\theta}^{(t+1)}$ and the previous ones ${\theta}^{\left(t\right)}$. According to [18] and following the notations in [19], the means, squared standard deviations, and mixture probabilities for each class $k\in \{s,n\}$ can be re-estimated using the following set of relations:$$\begin{array}{ccc}\hfill {\pi}_{k}^{(t+1)}& =& \frac{{\displaystyle \sum _{x}}{\zeta}_{k}^{\left(t\right)}\left(x\right)}{N}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mu}_{k}^{(t+1)}& =& \frac{{\displaystyle \sum _{x}}{\zeta}_{k}^{\left(t\right)}\left(x\right)\xb7x}{{\displaystyle \sum _{x}}{\zeta}_{k}^{\left(t\right)}\left(x\right)}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\sigma}_{k}^{(t+1)}& =& {\left(\frac{{\displaystyle \sum _{x}}{\zeta}_{k}^{\left(t\right)}\left(x\right)\xb7{\left(x-{\mu}_{k}^{\left(t\right)}\right)}^{2}}{{\displaystyle \sum _{x}}{\zeta}_{k}^{\left(t\right)}\left(x\right)}\right)}^{\frac{1}{2}}\hfill \end{array}$$$${\zeta}_{k}^{\left(t\right)}\left(x\right)=\frac{{\pi}_{k}^{\left(t\right)}\mathcal{N}\left(x\right|{\mu}_{k}^{\left(t\right)},{\sigma}_{k}^{\left(t\right)})}{{\sum}_{{k}^{\prime}}{\pi}_{{k}^{\prime}}^{\left(t\right)}\mathcal{N}\left(x\right|{\mu}_{{k}^{\prime}}^{\left(t\right)},{\sigma}_{{k}^{\prime}}^{\left(t\right)})}.$$

## 3. Experiments

## 4. Discussion

#### 4.1. Discovery of Similar Patterns in SITS

#### 4.2. Damage Assessment

#### 4.3. Land Cover and Land Use Mapping over Long Time Series

#### 4.4. Comparison with Other State-of-the-Art SITS Analysis Methods

#### 4.5. Final Remarks

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SITS | Satellite Image Time Series |

USGS | United States Geological Survey |

DTW | Dynamic Time Warping |

LDA | Latent Dirichlet Allocation |

SVM | Support Vector Machine |

ML | Maximum Likelihood |

MLE | Maximum Likelihood Estimation |

EM | Expectation-Maximization |

OA | Overall Accuracy |

MAR | Missed Alarm Rate |

FAR | False Alarm Rate |

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**Figure 3.**Histogram over DTW similarity scores and a mixture of two Gaussian distributions fitted to these scores.

**Figure 4.**Short Landsat SITS comprised of 10 images captured between 1984–1993. Only four representative images (i.e., containing specific changes) of the series are shown: (

**a**) 1984, (

**b**) 1987, (

**c**) 1988, (

**d**) 1992. The distribution of the acquisition moments is shown in (

**e**).

**Figure 5.**Dobrogea Landsat SITS comprised of 13 images captured between 6 May 2000 and 14 September 2001, in the Dobrogea region. Four images of the series are shown, namely: (

**a**) 6 May 2000, (

**b**) 22 May 2000, (

**c**) 9 July 2000, and (

**d**) 29 October 2000. The distribution of the acquisition moments is shown in (

**e**).

**Figure 6.**Long Landsat SITS comprised of 88 images captured in 1984–2011. Only the first and the last images of the series are shown, namely (

**a**) 1984 and (

**b**) 2011. The distribution of the acquisition moments is shown in (

**c**).

**Figure 7.**Spectro-temporal signatures during the construction of the accumulation lakes. The two spectro-temporal signatures are characterized by different temporal evolutions in the period that corresponds to their construction, namely 1984–1993.

**Figure 8.**Pattern discovery in short Landsat SITS. (

**a**) DTW distance image for a query marked inside the Morii & Dridu accumulation lakes. (

**b**) DTW distance image for a query marked inside Mihailesti accumulation lake. (

**c**) Pattern discovery using the proposed method for Morii & Dridu accumulation lakes. (

**d**) Pattern discovery using the proposed method for Mihailesti accumulation lake. In the case of DTW distance images presented in (

**a**,

**b**), dark color represents similar evolutions and bright color represents non-similar evolutions, whereas in (

**c**,

**d**), white pixels correspond to spatio-temporal patterns that are similar to the query, which was selected from the region of interest.

**Figure 9.**Ground truth for short Landsat SITS. White pixels delimit (

**a**) Morii & Dridu accumulation lakes and (

**b**) Mihailesti accumulation lake.

**Figure 10.**Delimitation of areas affected by floods in Dobrogea. (

**a**) DTW distance image for a query in the flooded area. (

**b**) Delimitation of the affected area using the proposed algorithm. (

**c**) Ground truth marking the flooded areas. In the case of DTW distance image presented in (

**a**), dark color represents similar evolutions and bright color represents non-similar evolutions, whereas in (

**b**) white pixels correspond to spatio-temporal patterns that are similar to the query, which was selected from the region of interest.

**Figure 11.**Land cover mapping in long Landsat SITS. (

**a**) DTW distance image for a query made inside the forestry area. (

**b**) Forestry area delimitation using the proposed query-by-example retrieval method. (

**c**) DTW distance image for a query made in an area covered by water. (

**d**) Water delimitation using the proposed query-by-example retrieval method. In the case of DTW distance images presented in (

**a**,

**c**), dark color represents similar evolutions and bright color represents non-similar evolutions, whereas in (

**b**,

**d**), white pixels correspond to spatio-temporal patterns that are similar to the query, which was selected from the region of interest.

**Figure 12.**Land use mapping in long Landsat SITS. (

**a**) DTW distance image for a query made in the urban area. (

**b**) Urban area delimitation using the proposed query-by-example retrieval method. (

**c**) DTW distance image for a query made in the extra-urban area. (

**d**) Extra-urban area delimitation using the proposed query-by-example retrieval method. (

**d**) DTW distance image for a query made inside the agricultural area. (

**e**) Agricultural area delimitation using the proposed query-by-example retrieval method. In the case of DTW distance images presented in (

**a**,

**c**,

**e**), dark color represents similar evolutions and bright color represents non-similar evolutions, whereas in (

**b**,

**d**,

**f**), white pixels correspond to spatio-temporal patterns that are similar to the query, which was selected from the region of interest.

**Table 1.**Example of computation of the Dynamic Time Warping (DTW) distance matrix for two sequences, ’5463545’ and ’0102130’.

0 | 1 | 0 | 2 | 1 | 3 | 0 | |
---|---|---|---|---|---|---|---|

5 | 5 | 9 | 14 | 17 | 21 | 23 | 28 |

4 | 9 | 8 | 12 | 14 | 17 | 18 | 22 |

6 | 15 | 13 | 14 | 16 | 19 | 20 | 24 |

3 | 18 | 15 | 16 | 15 | 16 | 16 | 19 |

5 | 23 | 19 | 20 | 18 | 20 | 18 | 21 |

4 | 27 | 22 | 23 | 20 | 21 | 19 | 22 |

5 | 32 | 26 | 27 | 23 | 24 | 21 | 24 |

Query | Overall Accuracy | Missed Alarm Rate | False Alarm Rate |
---|---|---|---|

Morii and Dridu | 99.68% | 26.84% | 0.23% |

Mihailesti | 99.36% | 30.36% | 0.56% |

Query | Overall Accuracy | Missed Alarm Rate | False Alarm Rate |
---|---|---|---|

Flooded area | 99.96% | 2.36% | 0.13% |

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**MDPI and ACS Style**

Radoi, A.; Burileanu, C. Retrieval of Similar Evolution Patterns from Satellite Image Time Series. *Appl. Sci.* **2018**, *8*, 2435.
https://doi.org/10.3390/app8122435

**AMA Style**

Radoi A, Burileanu C. Retrieval of Similar Evolution Patterns from Satellite Image Time Series. *Applied Sciences*. 2018; 8(12):2435.
https://doi.org/10.3390/app8122435

**Chicago/Turabian Style**

Radoi, Anamaria, and Corneliu Burileanu. 2018. "Retrieval of Similar Evolution Patterns from Satellite Image Time Series" *Applied Sciences* 8, no. 12: 2435.
https://doi.org/10.3390/app8122435