Fast 3D Modeling with Approximated Convolutional Kernels

This paper introduces a novel regression methodology for 3D reconstruction, with applications in robotics tasks such as terrain modeling and implicit surface calculation. The proposed methodology is based on projections into a high-dimensional space, that is able to fit arbitrarily complex data as a continuous function using a series of kernel evaluations within a linear regression model. We avoid direct kernel calculation by employing a novel sparse random Fourier feature vector, that approximates any shift-invariant kernel as a series of dot products relative to a set of inducing points placed throughout the input space. The varying properties of these inducing points produce non-stationarity in the resulting model, and can be jointly learned alongside linear regression weights. Furthermore, we show how convolution with arbitrary kernels can be performed directly in this high-dimensional continuous space, by training a neural network to learn the Fourier transform of the convolutional output based on information from the input kernels. Experimental results in terrain modeling and implicit surface calculation show that the proposed framework is able to outperform similar techniques in terms of computational speed without sacrificing accuracy, while enabling efficient convolution with arbitrary kernels for tasks such as global localization and template matching within these applications.