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Comparing Implicit Neural Representations and B-Splines for Continuous Function Fitting from Sparse Samples

Hongze Yu, Yun Jiang, Jeffrey A. Fessler

TL;DR

This preliminary empirical study compares a positional-encoded INR with a cubic B-spline model for continuous function fitting from sparse random samples, isolating the representation capacity difference by only using coefficient-domain Tikhonov regularization, and empirically support the superior representation capacity of INRs for sparse data fitting.

Abstract

Continuous signal representations are naturally suited for inverse problems, such as magnetic resonance imaging (MRI) and computed tomography, because the measurements depend on an underlying physically continuous signal. While classical methods rely on predefined analytical bases like B-splines, implicit neural representations (INRs) have emerged as a powerful alternative that use coordinate-based networks to parameterize continuous functions with implicitly defined bases. Despite their empirical success, direct comparisons of their intrinsic representation capabilities with conventional models remain limited. This preliminary empirical study compares a positional-encoded INR with a cubic B-spline model for continuous function fitting from sparse random samples, isolating the representation capacity difference by only using coefficient-domain Tikhonov regularization. Results demonstrate that, under oracle hyperparameter selection, the INR achieves a lower normalized root-mean-squared error, yielding sharper edge transitions and fewer oscillatory artifacts than the oracle-tuned B-spline model. Additionally, we show that a practical bilevel optimization framework for INR hyperparameter selection based on measurement data split effectively approximates oracle performance. These findings empirically support the superior representation capacity of INRs for sparse data fitting.

Comparing Implicit Neural Representations and B-Splines for Continuous Function Fitting from Sparse Samples

TL;DR

This preliminary empirical study compares a positional-encoded INR with a cubic B-spline model for continuous function fitting from sparse random samples, isolating the representation capacity difference by only using coefficient-domain Tikhonov regularization, and empirically support the superior representation capacity of INRs for sparse data fitting.

Abstract

Continuous signal representations are naturally suited for inverse problems, such as magnetic resonance imaging (MRI) and computed tomography, because the measurements depend on an underlying physically continuous signal. While classical methods rely on predefined analytical bases like B-splines, implicit neural representations (INRs) have emerged as a powerful alternative that use coordinate-based networks to parameterize continuous functions with implicitly defined bases. Despite their empirical success, direct comparisons of their intrinsic representation capabilities with conventional models remain limited. This preliminary empirical study compares a positional-encoded INR with a cubic B-spline model for continuous function fitting from sparse random samples, isolating the representation capacity difference by only using coefficient-domain Tikhonov regularization. Results demonstrate that, under oracle hyperparameter selection, the INR achieves a lower normalized root-mean-squared error, yielding sharper edge transitions and fewer oscillatory artifacts than the oracle-tuned B-spline model. Additionally, we show that a practical bilevel optimization framework for INR hyperparameter selection based on measurement data split effectively approximates oracle performance. These findings empirically support the superior representation capacity of INRs for sparse data fitting.
Paper Structure (18 sections, 12 equations, 4 figures)

This paper contains 18 sections, 12 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Problem Formulation
  4. B-Spline Formulation
  5. INR Formulation
  6. Oracle Hyperparameter Search for B-Spline
  7. Bilevel Optimization for INR
  8. Oracle Analysis of INR Regularization
  9. Evaluation Metric
  10. Experiments
  11. Results
  12. B-Spline Results
  13. INR Results
  14. Discussion and Conclusions
  15. Comparison of Representation Capacity
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Cubic B-spline fitting with oracle grid search over regularization strength $\lambda$ and number of knots $M$. Top row: NRMSE heatmap showing the optimal at $M=75$, $\lambda=2.51\times10^{-2}$ (red star); NRMSE versus $\lambda$ for different $M$; NRMSE versus $M$ for different $\lambda$. Bottom row: optimal reconstruction, absolute error map, and cross-section at $y=1.50$. The optimal NRMSE is 0.1540.
  • Figure 2: Effect of Tikhonov regularization on cubic B-spline fitting with $M=75$ knots. (a) True function with random samples. (b) Unregularized fit showing oscillatory artifacts. (c) Optimally regularized fit ($\lambda=0.0251$) with reduced artifacts but increased blur. (d) NRMSE versus $\lambda$, illustrating the bias-variance tradeoff.
  • Figure 3: INR hyperparameter sensitivity analysis over encoder and MLP weight decay ($\lambda_{\mathrm{Enc}}$, $\lambda_{\mathrm{MLP}}$). (a) Oracle NRMSE with 100% training samples. (b) Oracle NRMSE with 80% training samples. (c) Validation loss with 80%/20% train/validation split. Markers indicate optima: cyan triangle (100% oracle), magenta square (80% oracle), red star (validation grid search), yellow diamond (BayesOpt).
  • Figure 4: INR reconstruction results under different hyperparameter selection criteria. Each row shows the reconstruction, absolute error map, and cross-section at $y=1.50$. (a) Oracle grid search with 100% samples (NRMSE $= 0.129$). (b) Oracle grid search with 80% samples (NRMSE $= 0.132$). (c) Validation loss grid search with 80%/20% split (NRMSE $= 0.137$). (d) Bilevel optimization with 80%/20% split (NRMSE $= 0.131$). (e) Unregularized fitting (NRMSE $= 0.382$).