Constrained Tensor Decompositions for SAR Data: Agricultural Polarimetric Time Series Analysis

Tensor decompositions are a powerful tool for multidimensional data analysis, interpretation, and signal processing. This work develops a constrained tensor decomposition framework for complex multidimensional Synthetic Aperture Radar (SAR) data. The framework generalizes the Canonical Polyadic (CP) decomposition by formulating it as an optimization problem and allows precise control over the shape and properties of the output factors. The implementation supports complex tensors, automatic differentiation, different loss functions, and optimizers. We discuss the importance of constraints for physical validity, interpretability, and uniqueness of the decomposition results. To illustrate the framework, we formulate a polarimetric time series decomposition and apply it to data acquired over agricultural areas to analyze the development of four crop types at X, C, and L bands over the period of twelve weeks. The obtained temporal factors describe the changes in the crops in a compact way and show a correlation to certain crop parameters. We extend the existing polarimetric time series change analysis with the decomposition to show the changes in more detail and provide an interpretation through the polarimetric factors. The decomposition framework is extensible and promising for joint information extraction from multidimensional SAR data

Constrained Tensor Decompositions for Polarimetric Time Series Change Analysis

Synthetic Aperture Radar (SAR) sensors provide data in polarimetric, interferometric, temporal, and spatial dimensions depending on the acquisition setup. With the increasing availability of multi-dimensional SAR data the joint processing and information extraction from several data dimensions grows in relevance. Some existing works already combine two data dimensions for different tasks. For example, forest height inversion methods [2] jointly use polarimetry and interferometry, the Sum of Kronecker Products (SKP) decomposition [3] exploits polarimetry and tomography, polarimetric change detection [1] combines the polarimetric and temporal dimensions, and convolutional neural networks [4] for land cover classification can use spatial and polarimetric features. We propose a decomposition framework for multi-dimensional SAR data that allows an arbitrary number of data dimensions and is based on the Canonical Polyadic (CP) tensor decomposition. The decomposition is formulated as an optimization problem allowing precise control over the shape and properties of the output factors. We take into account the specifics of SAR data by adding constraints for physical validity and interpretability. In order to demonstrate our approach, we formulate a decomposition for polarimetric time series using the proposed framework. The algorithm decomposes a stack of polarimetric coherency matrices into R components, each defined by a polarimetric and a temporal factor. We then use the factors to evaluate F-SAR data in X, C, and L bands obtained over agricultural areas during the CROPEX 2014 campaign. We analyze the evolution of four different crop types and the changes in the signal related to the growth, drying, fruit maturation, and harvest. The obtained factors describe the changes in the crops in a compact way and show correlations to certain crop parameters. The results are visualized using the polarimetric change matrices proposed in [1] and show additional fine-grained changes in comparison to the original method. The decomposition framework is an extensible and promising tool for joint information extraction from multi-dimensional SAR data. It can be used to improve existing methods or extend them with new data dimensions. For example, implementing the SKP decomposition using the framework allows to obtain more than two components, or enables to integrate an additional data dimension such as time. Furthermore, it is possible to integrate physical models into the data-driven tensor decomposition approach. The framework implementation builds on top of PyTorch and supports automatic differentiation and optimization in the complex domain. This simplifies the decomposition design, facilitates experiments with different data dimensions, and allows to concentrate on the choice of the constraints or interpretation of the factors. [1] Alberto Alonso-González et al. "Polarimetric SAR Time Series Change Analysis over Agricultural Areas". In: IEEE Transactions on Geoscience and Remote Sensing 58.10 (2020), pp. 7317-7330. [2] SR Cloude and KP Papathanassiou. "Three-stage inversion process for polarimetric SAR interferometry". In: IEE Proceedings-Radar, Sonar and Navigation 150.3 (2003), pp. 125-134. [3] Stefano Tebaldini. "Algebraic Synthesis of Forest Scenarios from Multibaseline PolInSAR Data". In: IEEE Transactions on Geoscience and Remote Sensing 47.12 (2009), pp. 4132-4142. [4] Xiao Xiang Zhu et al. "Deep learning meets SAR: Concepts, models, pitfalls, and perspectives". In: IEEE Geoscience and Remote Sensing Magazine 9.4 (2021), pp. 143-172

Model-based Tensor Decompositions for Bio- and Geophysical Parameter Retrieval

The increasing availability of multidimensional SAR data motivates the development of new techniques for analysis, decomposition, and joint information extraction. In this work, we explore the integration of physical polarimetric models into tensor decompositions to estimate geophysical parameters like soil moisture. In vegetated areas, SAR backscatter signal contains both the information about the ground and the vegetation due to the penetration into the media. Therefore, to accurately estimate the soil parameters, the methods should take into account the vegetation backscatter. Polarimetric physical models approximate the signal with a sum of different components that describe both the ground and the vegetation contributions [1]. In order to avoid ambiguous model inversion, the total number of model parameters is limited by the number of observables. Current approaches typically use physical models with a small number of parameters. This can result in cases where the model is not able to accurately describe the data or has a small validity range. Moving towards the inversion of more complex models, we propose to enlarge the observation space by integrating and jointly processing additional data dimensions such as spatial, temporal or polarimetric information. We introduce model-based tensor decompositions that directly operate on data tensors representing them as a sum of model-based tensor components. The larger observation space combined with sharing of certain parameters along the new data dimensions allows to use more complex and accurate models. To illustrate the approach, we present a method to estimate soil moisture from a combination of polarimetric and spatial data, represented as a three-dimensional tensor. Given a small spatial image patch with several independent polarimetric coherency matrices in every pixel, we assume a constant soil moisture across the patch, while letting other parameters like vegetation backscatter intensity vary from pixel to pixel. The model inversion is formulated as an optimization problem. We use the physical model to reconstruct an approximation tensor from the physical parameters and iteratively minimize the distance between the approximation and the measured data. After convergence, the algorithm provides the physical parameters that fit the data best. Since the model is a differentiable mathematical function, minimization is performed with an optimizer based on gradient descent. The algorithm is implemented in PyTorch taking advantage of automatic differentiation and advanced optimizers. We evaluate the proposed method on high-resolution airborne F-SAR data obtained by DLR over agricultural areas during the HTERRA 2022 campaign in the province of Foggia, Italy. The decomposition provides a characterization of the dominant scattering mechanism for each resolution cell. In addition, soil moisture estimation at a 12 meter resolution is obtained. The larger observation space and the use of the more accurate model allow the inversion in more regions than compared to a simpler X-Bragg model. References [1] I. Hajnsek, T. Jagdhuber, H. Schon, and K. P. Papathanassiou, "Potential of estimating soil moisture under vegetation cover by means of PolSAR," IEEE Transactions on Geoscience and Remote Sensing, vol. 47, no. 2, pp. 442-454, 2009

Soil Moisture Estimation from Multi-dimensional SAR Data

Synthetic Aperture Radar (SAR) sensors provide high-resolution and weather-independent images of the Earth surface offering sensitivity to geometrical and dielectric properties. Current and planned satellite missions with large coverage and short revisit times make SAR a promising tool for continuous soil moisture monitoring. SAR signal over vegetated areas includes the backscatter of the ground and the vegetation. To accurately estimate the soil moisture, the methods should take into account the vegetation backscatter. Physical models address this by having different components that model both the ground and the vegetation contributions [1]. The model components take physical parameters as an input and predict the SAR signal. In order to avoid ambiguous model inversion, the total number of parameters is limited by the number of observables. Current approaches typically use physical models with a small number of parameters unable to fully capture the complexity of real scenarios. To allow inversion of more complex models with a larger number of parameters, we propose to increase the observation space by jointly processing several data dimensions such as spatial, temporal or polarimetric information. The key idea is to share certain model parameters across the data dimensions keeping the number of parameters smaller than the number of observables. To illustrate the approach, we present a method to estimate soil moisture from a combination of polarimetric and spatial data. Given a small spatial image patch with polarimetric data for each pixel, we assume a constant soil moisture across the patch, while letting other parameters like vegetation backscatter intensity vary from pixel to pixel. The model inversion is formulated as an optimization problem where the physical model is represented by a differentiable function that maps parameters to the predicted backscatter. Parameters can then be estimated by minimizing the difference between observed and predicted data in an iterative fashion by gradient descent. The minimization is performed with PyTorch taking advantage of automatic differentiation and advanced optimizers. We evaluate the proposed method on high-resolution airborne F-SAR data obtained by DLR over vegetated agricultural areas during the CROPEX campaign. With a larger observation space more input parameters can be uniquely inverted allowing to use more complex and accurate physical models. [1] Irena Hajnsek et al. "Potential of Estimating Soil Moisture under Vegetation Cover by Means of PolSAR". In: IEEE Transactions on Geoscience and Remote Sensing 47.2 (2009), pp. 442-454