Harmonic and Phase-Modulated Activation Functions for Implicit Neural Representations: A Comprehensive Benchmark Study

It is well-known that activation functions are crucial in determining spectral expressiveness, training dynamics, and reconstruction accuracy in implicit neural representations (INRs), which employ coordinate-based multilayer perceptrons to represent continuous signals. Despite showing excellent performance, sinusoidal activations, for example SIREN, are limited in their adaptability to diverse signal types due to their fixed harmonic structure. In this paper, we propose two novel periodic activation functions for INRs. (1) Harmonic generalizes sinusoidal activations by combining the fundamental frequency with learned second and third harmonics through per-neuron trainable amplitude coefficients, resulting in a richer spectral basis within the SIREN initialization framework. (2) PM-FINER (Phase-Modulated FINER) extends the variable-periodic FINER activation by embedding frequency modulation synthesis directly into the instantaneous phase, enabling data-driven phase distortion via a learnable modulation index and carrier ratio. We conducted comprehensive experiments spanning nine architectural configurations (including SIREN, WIRE, FINER, Gaussian, Harmonic, PM-FINER, and an additional direct comparison against the Subtractive Modulative Network (SMN)), using six natural images, three learning rate schedulers, and three random seeds, totaling 486 main training runs (534 runs total including an ω0 sensitivity sweep). Our evaluation combined peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and rigorous statistical analysis, such as paired t-tests, Wilcoxon signed-rank tests, Cohen’s d effect sizes, and Friedman rank tests. Under cosine annealing, Harmonic achieves a mean PSNR gain of 6.08 dB over SIREN and 2.57 dB over FINER (both p<0.001, Cohen’s d>3.7), while PM-FINER ranks statistically on par with Harmonic (mean difference 0.17 dB, p=0.36), outperforming all of the other baselines. Compared with SMN, Harmonic outperforms it by +7.94 dB under cosine annealing (Bonferroni-adjusted p<10−5, Cohen’s d=12.3), winning on all six images. Additionally, the Friedman ranking across the six images confirmed Harmonic (with mean rank =1.33) and PM-FINER (with mean rank =1.67), being the top two methods under cosine annealing. Our results establish interpretable multi-harmonic and phase-modulated activations as real alternatives to the existing INR activation functions.