Neural distance fields (NDF) have emerged as a powerful tool for addressing challenges in 3D computer vision and graphics downstream problems. While significant progress has been made to learn NDF from various kind of sensor data, a crucial aspect that demands attention is the supervision of neural fields during training as the ground-truth NDFs are not available for large-scale outdoor scenes. Previous works have utilized various forms of expected signed distance to guide model learning. Yet, these approaches often need to pay more attention to critical considerations of surface geometry and are limited to small-scale implementations. To this end, we propose a novel methodology leveraging second-order derivatives of the signed distance field for improved neural field learning. Our approach addresses limitations by accurately estimating signed distance, offering a more comprehensive understanding of underlying geometry. To assess the efficacy of our methodology, we conducted comparative evaluations against prevalent methods for mapping and localization tasks, which are primary application areas of NDF. Our results demonstrate the superiority of the proposed approach, highlighting its potential for advancing the capabilities of neural distance fields in computer vision and graphics applications.

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