# Joint SAR Image Time Series and PSInSAR Data Analytics: An LDA Based Approach
## 2 Evolving Structures in SAR Imagery  
When looking at a scene, the user is able to perceive well-defined structures based on visual characteristics. However, the landscape is the result of set of transformations generated by the vegetation lifecycle, geomorphological phenomena, or human activities. It is important to understand the shape of structures around us in the context of Earth surface dynamics. Temporal evolution is an important feature that must be integrated in the scene analysis together with structural features in order to define an advanced contextual meaning. While the process becomes complex, a successful data exploitation demands for scene decomposition into atoms representing primitive features. They are considered basic elements, a collection that seems rather orderless and preventing of a coherent analysis with respect to the information that can be extracted. The challenge is to discover those rules that generate meaningful feature combination, define semantics, and create knowledge and understanding for the user. To this end, the Latent Dirichlet Allocation (LDA) model addresses the latency of scene content and uses a Bag of Words approach to discover hidden information inside the data. LDA is a generative probabilistic model able to shape semantic associations within a collection of discrete data [21]. Although the method is designed for the peculiarities of text analysis, a collection of EO images time series, seen as a random discrete process, will respond to the algorithm's requirements as long as relevant equivalences can be made for the parameters of the LDA model [21]: word, document, and corpus. For this aim, the SITS will be hierarchically decomposed into basic elements representing visual words such that documents and corpus can derive without difficulty. Land surfaces similarly reflecting the echoes over the analyzed period of time will be included in the same class, with the belief that the results will picture a symbolic representation linked to the scene's nature. The proposed approach is presented in Figure 1. A SITS consist of a set of EO images acquired at different moments of time over the same area, entirely covering the area, without exceeding it. From the LDA algorithm's perspective, the first action to be completed refers to the definition of the baseline element of a SITS. We propose the temporal signature (the values of a pixel location in all the images in the time series) as the main element composing the SITS. Therefore, the SITS is decomposed in a set of R \(\times\) C temporal signatures, where (R, C) is the size of the scene. By performing an unsupervised k-means [22] clustering over the entire collection of temporal signatures, we define elements with basic understanding. As the goal is to group those pixels that share a comparable evolution over time, the similarity threshold will be forced to stay close to its maximum value. The process lies on the Euclidean distance, ensuring that the comparison between two signatures will consider measurements performed at the same moment of time (i.e., pixels (x\({}_{i}\)y\({}_{i}\)) and (x\({}_{i}\)y\({}_{i}\)) are first compared at moment t\({}_{1}\), and then at moments t\({}_{2}\), t\({}_{3}\) and so on). The obtained k-means classes will represent visual words. In order to avoid limitations in content representations (due to a high similarity threshold), tests and previous results recommends a number of 150-250 visual words [17]. At the next step, we associate a patch to the document. We experimentally verified that smaller patches (i.e., 5 \(\times\) 5 or 10 \(\times\) 10 pixels) enable better and more coherent content separation. The sequence of words defining the document is given by a Bag of Words representation of the visual words included in the patch. The corpus, a collection of documents, is associated with the analyzed SITS. Given the reduced amount of structures observable in the SAR imagery, we chose to work with 150 visual words and patches of 10 \(\times\) 10 pixels.