# Object-Based Time-Constrained Dynamic Time Warping Classification of Crops Using Sentinel-2
## 1 Introduction  
Humankind has impacted almost every place on Earth causing important environmental changes [1, 2]. One of the most widespread landscape changes is related to the cultivation of crops, which has been spatially extended and intensified to face the food supply needs of a constantly growing global population [3, 4]. Efficient monitoring of crops is important in providing food security, understanding productivity, and supporting agricultural policies. In this regard, satellite image analysis can provide timely and relevant spatial and temporal information products and tools [5, 6]. In recent decades, satellite images with global coverage and different spectral and temporal characteristics have become freely available, such as MODIS [7] and Landsat [8]. The recent Sentinel-2 satellite, a high resolution, multi-spectral spaceborne Earth imaging mission, is providing imagery with high temporal resolution (five days), spectral (13 bands), and spatial resolution (10-60 m). These assets are making the Sentinel-2 product suitable for satellite image time series (SITS) analysis [6]. SITS offers advantages over single-image approaches, particularly by allowing the incorporation of the phenological characteristics of vegetation throughout a year or multiple years in the classification task [9]. Given the large amounts of SITS acquired by diverse spaceborne platforms, we need efficient methods to turn this data into information [10]. Numerous methods and software packages for analyzing SITS exist, each with advantages and disadvantages [11], including the popular BFAST (Breaks For Additive Seasonal and Trend) [12], TIMESAT [13], SPIRITS (Software for the Processing and Interpretation of Remotely Sensed Image Time Series) [14], DTW (Dynamic Time Warping) [11], or WELD (Web Enabled Landsat Data) [9]. These methods are used in a wide range of applications including those focused on seasonality information extraction [15], phenological change detection [16], reconstructing cloud free time series [17], land cover mapping [11; 18], automated mapping of crops [19; 20], agricultural dynamics mapping during different cropping years [21], and detecting trends in forest disturbance and recovery [22]. SITS analysis aims at extracting meaningful information from pixels for a certain goal (e.g., crop mapping), but this comes with several issues that need to be addressed: (1) the shortcoming of sample availability, which can be overcome by using reference data from the past; (2) the irregular temporal sampling of images and the missing information in the datasets (e.g., cloud contamination) that can obscure the analysis; and (3) the variability of pseudo-periodic phenomena (e.g., vegetation cycles, which are influenced by weather conditions) [11]. All of these challenges require the development of methods capable of coping with the distortions of the temporal profile of physical variables and artifacts of SITS datasets [11]. One method that addresses these issues is Dynamic Time Warping (DTW), initially developed for speech recognition [23] and recently applied in the analysis of SITS [11; 24; 25; 26]. DTW compares the similarity between two temporal sequences by finding their optimal alignment and providing a dissimilarity measure [23; 27]. Previous research demonstrated that DTW performs successfully in various data mining scenarios [28] such as data mining of sequential data [29; 30] or spoken word recognition [23]. DTW has recently become a focus for the remote sensing community as well. [PERSON] et al. [11] introduced DTW from a theoretical point of view and demonstrated the utility of DTW in analyzing SITS. [PERSON] et al. [31] used DTW to re-align multi-year Landsat time series to a common phenology in order to eliminate the yearly phenological discrepancies. The authors demonstrated their approach for different forest types in eight different locations distributed across the United States. [PERSON] et al.