Data acquisition in array signal processing (ASP) is costly because achieving high angular and range resolutions necessitates large antenna apertures and wide frequency bandwidths, respectively. The data requirements for ASP problems grow multiplicatively with the number of viewpoints and frequencies, significantly increasing the burden of data collection, even for simulation. Implicit Neural Representations (INRs) -- neural network-based models of 3D objects and scenes -- offer compact and continuous representations with minimal radar data. They can interpolate to unseen viewpoints and potentially address the sampling cost in ASP problems. In this work, we select Synthetic Aperture Radar (SAR) as a case from ASP and propose Radon Implicit Field Transform (RIFT). RIFT consists of two components: a classical forward model for radar (Generalized Radon Transform, GRT), and an INR based scene representation learned from radar signals. This method can be extended to other ASP problems by replacing the GRT with appropriate algorithms corresponding to different data modalities. In our experiments, we first synthesize radar data using the GRT. We then train the INR model on this synthetic data by minimizing the reconstruction error of the radar signal. After training, we render the scene using the trained INR and evaluate our scene representation against the ground truth scene. Due to the lack of existing benchmarks, we introduce two main new error metrics: phase-Root Mean Square Error (p-RMSE) for radar signal interpolation, and magnitude-Structural Similarity Index measure(m-SSIM) for scene reconstruction. These metrics adapt traditional error measures to account for the complex nature of radar signals. Compared to traditional scene models in radar signal processing, with only 10% data footprint, our RIFT model achieves up to 188% improvement in scene reconstruction.

We propose new algorithms to efficiently average a collection of points on a Grassmannian manifold in both the centralized and decentralized settings. Grassmannian points are used ubiquitously in machine learning, computer vision, and signal processing to represent data through (often low-dimensional) subspaces. While averaging these points is crucial to many tasks (especially in the decentralized setting), existing methods unfortunately remain computationally expensive due to the non-Euclidean geometry of the manifold. Our proposed algorithms, Rapid Grassmannian Averaging (RGrAv) and Decentralized Rapid Grassmannian Averaging (DRGrAv), overcome this challenge by leveraging the spectral structure of the problem to rapidly compute an average using only small matrix multiplications and QR factorizations. We provide a theoretical guarantee of optimality and present numerical experiments which demonstrate that our algorithms outperform state-of-the-art methods in providing high accuracy solutions in minimal time.