Iterative Continuous Convolution for 3D Template Matching and Global Localization

This paper introduces a novel methodology for 3D template matching that is scalable to higher-dimensional spaces and larger kernel sizes. It uses the Hilbert Maps framework to model raw pointcloud information as a continuous occupancy function, and we derive a closed-form solution to the convolution operation that takes place directly in the Reproducing Kernel Hilbert Space defining these functions. The result is a third function modeling activation values, that can be queried at arbitrary resolutions with logarithmic complexity, and by iteratively searching for high similarity areas we can determine matching candidates. Experimental results show substantial speed gains over standard discrete convolution techniques, such as sliding window and fast Fourier transform, along with a significant decrease in memory requirements, without accuracy loss. This efficiency allows the proposed methodology to be used in areas where discrete convolution is currently infeasible. As a practical example we explore the key problem in robotics of global localization, in which a vehicle must be positioned on a map using only its current sensor information, and provide comparisons with other state-of-the-art techniques in terms of computational speed and accuracy.