We investigate a multiview shape reconstruction problem based on an active surface model whose geometric evolution is driven by radar measurements acquired at sparse locations. Building on our previous work in the context of variational methods for the reconstruction of a scene conceptualized as the graph of a function, we generalize this inversion approach for a general geometry, now described by an active surface, strongly motivated by prior variational computer vision approaches to multiview stereo reconstruction from camera images. While conceptually similar, use of radar echoes within a variational scheme to drive the active surface evolution requires significant changes in regularization strategies compared to prior image based methodologies for the active surface evolution to work effectively. We describe all of these aspects and how we addressed them. While our long term objective is to develop a framework capable of fusing radar as well as other image based information, in which the active surface becomes an explicit shared reference for data fusion. In this paper, we explore the reconstruction using radar as a single modality, demonstrating that the presented approach can provide reconstructions of quality comparable to those from image based methods showing great potential for further development toward data fusion.

Principal Component Analysis (PCA) is a widely used technique for dimensionality reduction in various problem domains, including data compression, image processing, visualization, exploratory data analysis, pattern recognition, time-series prediction, and machine learning. Often, data is presented in a correlated paired manner such that there exist observable and correlated unobservable measurements. Unfortunately, traditional PCA techniques generally fail to optimally capture the leverageable correlations between such paired data as it does not yield a maximally correlated basis between the observable and unobservable counterparts. This instead is the objective of Canonical Correlation Analysis (and the more general Partial Least Squares methods); however, such techniques are still symmetric in maximizing correlation (covariance for PLSR) over all choices of the basis for both datasets without differentiating between observable and unobservable variables (except for the regression phase of PLSR). Further, these methods deviate from PCA's formulation objective to minimize approximation error, seeking instead to maximize correlation or covariance. While these are sensible optimization objectives, they are not equivalent to error minimization. We therefore introduce a new method of leveraging PCA between paired datasets in a dependently coupled manner, which is optimal with respect to approximation error during training. We generate a dependently coupled paired basis for which we relax orthogonality constraints in decomposing unreliable unobservable measurements. In doing so, this allows us to optimally capture the variations of the observable data while conditionally minimizing the expected prediction error for the unobservable component. We show preliminary results that demonstrate improved learning of our proposed method compared to that of traditional techniques.

We propose Directionally Paired Principal Component Analysis (DP-PCA), a novel linear dimension-reduction model for estimating coupled yet partially observable variable sets. Unlike partial least squares methods (e.g., partial least squares regression and canonical correlation analysis) that maximize correlation/covariance between the two datasets, our DP-PCA directly minimizes, either conditionally or unconditionally, the reconstruction and prediction errors for the observable and unobservable part, respectively. We demonstrate the optimality of the proposed DP-PCA approach, we compare and evaluate relevant linear cross-decomposition methods with data reconstruction and prediction experiments on synthetic Gaussian data, multi-target regression datasets, and a single-channel image dataset. Results show that when only a single pair of bases is allowed, the conditional DP-PCA achieves the lowest reconstruction error on the observable part and the total variable sets as a whole; meanwhile, the unconditional DP-PCA reaches the lowest prediction errors on the unobservable part. When an extra budget is allowed for the observable part's PCA basis, one can reach an optimal solution using a combined method: standard PCA for the observable part and unconditional DP-PCA for the unobservable part.