Efficient recursive implementation and evaluation of the analytical change of basis

Develop and evaluate an efficient recursive implementation of the analytically derived change of basis between the exponential kernels and logarithmic B-spline temporal functions.

Background

The paper approximates logarithmic B-spline temporal functions using polynomial combinations of recursively updated exponential kernels, allowing event-by-event processing. It notes that the change of basis can instead be derived analytically using Gram matrices formed from dot products of the basis functions. However, an efficient recursive implementation and its evaluation are not developed in the paper and are explicitly deferred to future work.

References

It has not escaped our notice that the change of basis can be derived analytically using Gram-matrices formed by dot-products of the involved basis functions. We leave this, with efficient recursive implementation and evaluation, for future work.

— Efficient Multi-Timescale Event Representations for Feed-Forward Object Detection  (2609.05049 - Lundell et al., 4 Sep 2026) in Section 3, subsection “Polynomial Approximation of Log-Time B-Spline functions” (after Eq. defining the polynomial approximation)