Zero-delay Consistent and Smooth Trainable Interpolation

The question of how to produce a smooth interpolating curve from a stream of uncertainty regions, which become available sequentially, is addressed in this paper. To this end, we formalize the concept of real-time interpolator (RTI): a trainable recurrent unit that reconstructs smooth signals that are consistent with the received uncertainty regions in an online manner. More specifically, an RTI works under the requirement of reconstructing a section of the signal immediately after an uncertainty region is revealed (zero delay), without changing the reconstructed signal in the previous sections. Particularly, this work formulates the design of spline-based RTIs and proposes a data-driven training procedure, which minimizes the average curvature of the interpolated signals over a set of example sequences. These sequences are representative of the nature of the data sequence to be interpolated, allowing to tailor the RTI to any specific signal source. Our overall design allows for different possible schemes due to its modular structure, but in this work, we present two approaches, namely, the parametrized RTI and the recurrent neural network (RNN)-based RTI, including their architectures and properties. Experimental results show that the two proposed RTIs can be trained to achieve improved performance (in terms of the curvature loss metric) with respect to a myopic-type RTI that only exploits the local information at each time step while maintaining smooth, zero-delay, and consistency requirements.