Efficient Multidimensional Diracs Estimation With Linear Sample Complexity

Abstract

Estimating Diracs in continuous two or higher dimensions is a fundamental problem in imaging. Previous approaches extended one-dimensional (1-D) methods, like the ones based on finite rate of innovation (FRI) sampling, in a separable manner, e.g., along the horizontal and vertical dimensions separately in 2-D. The separate estimation leads to a sample complexity of $\mathcal {O}\left(K^D\right)$ for $K$ Diracs in $D$ dimensions, despite that the total degrees of freedom only increase linearly with respect to $D$ . We propose a new method that enforces the continuous-domain sparsity constraints simultaneously along all dimensions, leading to a reconstruction algorithm with linear sample complexity $\mathcal {O}(K)$ , or a gain of $\mathcal {O}\left(K^{D-1}\right)$ over previous FRI-based methods. The multi-dimensional Dirac locations are subsequently determined by the intersections of hypersurfaces (e.g., curves in 2-D), which can be computed algebraically from the common roots of polynomials. We first demonstrate the performance of the new multidimensional algorithm on simulated data: multidimensional Dirac location retrieval under noisy measurements. Then, we show results on real data: radio astronomy point source reconstruction (from LOFAR telescope measurements) and the direction of arrival estimation of acoustic signals (using Pyramic microphone arrays).