Low Rank Properties for Estimating Microphones Start Time and Sources Emission Time

The absence of unknown timing information about the microphones recording start time and the sources emission time presents a challenge in several applications, including joint microphones and sources localization. Compared with traditional optimization methods that try to estimate unknown timing information directly, low rank property (LRP) contains an additional low rank structure that facilitates a linear constraint of unknown timing information for formulating corresponding low rank structure information, enabling the achievement of global optimal solutions of unknown timing information with suitable initialization. However, the initialization of unknown timing information is random, resulting in local minimal values for estimation of the unknown timing information. In this paper, we propose a combined low rank approximation method to alleviate the effect of random initialization on the estimation of unknown timing information. We define three new variants of LRP supported by proof that allows unknown timing information to benefit from more low rank structure information. Then, by utilizing the low rank structure information from both LRP and proposed variants of LRP, four linear constraints of unknown timing information are presented. Finally, we use the proposed combined low rank approximation algorithm to obtain global optimal solutions of unknown timing information through the four available linear constraints. Experimental results demonstrate superior performance of our method compared to state-of-the-art approaches in terms of recovery rate (the number of successful initialization for any configuration), convergency rate (the number of successfully recovered configurations), and estimation errors of unknown timing information.