Implicit Neural Representation (IDIR)

Updated 26 December 2025

IDIR is a coordinate-based method that uses neural networks to represent signals continuously, bypassing traditional discrete structures.
It facilitates mesh-free inverse scattering and deformable registration by encoding shapes and fields via implicit representations.
The approach provides resolution independence, full differentiability, and memory efficiency, enhancing modeling in imaging and physics.

Implicit neural representations (INRs) are coordinate-based neural networks that model signals—such as images, audio, 3D shapes, or physical fields—as continuous functions parameterized by neural network weights rather than explicit discrete structures. The term “IDIR” initially referred specifically to the INR-based shape-inverse scattering framework introduced in "Implicit Neural Representation for Mesh-Free Inverse Obstacle Scattering" (Vlašić et al., 2022), but the methodology and terminology have evolved and now encompass a broader family of approaches where the function of interest is represented by the level-set or direct output of an implicit MLP. IDIR frameworks are central to contemporary research in mesh-free inverse problems, deformable registration, physics-based modeling, and high-fidelity geometric inference.

1. Core Formulation: Coordinate-Based Implicit Representation

In the IDIR framework for inverse obstacle scattering (Vlašić et al., 2022), the obstacle boundary is implicitly defined by the zero-level set of a neural network

$f_\theta : \mathbb{R}^d \to \mathbb{R},\qquad S = \partial\widehat\Omega = \{x\mid f_\theta(x) = 0\},$

where $f_\theta(x)$ approximates the signed distance function (SDF) to the obstacle boundary. The neural network is typically a multilayer perceptron (MLP), often with periodic (SIREN-style) activations $\sigma(u)=\sin(\omega_0 u)$ to enhance high-frequency expressivity. For 3D shape IDIR, the SIREN configuration (2–5 hidden layers of width 128–512, $\omega_0 \ge 30$ in input layers) yields accurate implicit geometry recovery.

The general IDIR paradigm applies equally to fields beyond shapes: for deformable image registration, the implicit representation models the displacement vector field (DVF) as a neural network $f: \mathbb{R}^d \to \mathbb{R}^d$ (Hadramy et al., 17 Jul 2025, Drozdov et al., 26 Sep 2025), or to physical quantities in inverse PDE problems.

2. Mesh-Free Inverse Problem Solving with IDIR

The inversion strategy in mesh-free inverse obstacle scattering employs the IBIM (Implicit Boundary Integral Method) (Vlašić et al., 2022):

The SDF network $f_\theta$ defines the obstacle, enabling direct computation of geometric quantities (normals, curvature) via automatic differentiation.
The forward PDE (e.g., sound-hard Helmholtz equation) is solved by replacing explicit boundary integrals over $\partial\Omega$ with smooth volume integrals over a tubular neighborhood $T_\epsilon$ :

$\int_{\partial\Omega} f(y)\,dS(y) = \int_{T_\epsilon} f(P(x)) J(x) \delta_\epsilon(f_\theta(x))\,dx,$

where $J(x)$ is the Jacobian correction and $P(x)$ projects $x$ onto the implicit surface.

Measured data (e.g., scattered acoustic fields) are compared against the forward simulation, and the loss is differentiated w.r.t. $\theta$ (the INR parameters) using autograd, efficiently updating the shape through a continuous and differentiable process.

This framework is generically extendable to other PDE-constrained inverse problems where the domain (geometry) or inhomogeneity is represented implicitly.

3. Generative Regularization in IDIR

IDIR incorporates a generative prior via a latent-variable hypernetwork (Vlašić et al., 2022). Denote $g_\phi: \mathbb{R}^k \to \mathbb{R}^{|\theta|}$ as a hypernetwork mapping a low-dimensional latent vector $z$ to INR weights $\theta$ :

Encoder $E$ maps discrete SDFs to $z$ .
RealNVP-style normalizing flows $f_\psi$ impose a gaussian prior on the $z$ -space.
During inverse fitting, only the latent vector $z$ is optimized (not the full $\theta$ ), enforcing that $f_{g_\phi(z)}$ lies near the generative SDF manifold. This provides strong regularization and accelerates and robustifies convergence—especially critical when observations are sparse or noisy.

4. Advances in Deformable Image Registration: cIDIR and KAN-IDIR

Implicit neural representation frameworks for deformable image registration constitute a major application of the IDIR approach.

4.1 Conditioned INR for Regularization (cIDIR)

In cIDIR (Hadramy et al., 17 Jul 2025), the INR $f(x;\lambda)$ outputs a DVF, where $\lambda$ denotes regularization hyperparameters (e.g., smoothness weights). By conditioning the MLP via a harmonizer network $h(\lambda)$ (predicting sinusoidal activation parameters), a single trained network encapsulates the entire Pareto frontier of regularization-weighted registration solutions. The objective integrates similarity and regularization terms over a uniform prior $p(\lambda)$ : $\mathcal{L}(\theta) = \mathbb{E}_{\lambda\sim U[0,1]} \mathbb{E}_{x\in \Omega} \left[ (1-\lambda)\mathcal{L}_{sim} + \lambda \mathcal{L}_{reg} \right].$ At inference, the optimal $\lambda$ is selected via a segmentation-driven Dice maximization, circumventing retraining for each user-supplied or data-specific regularizer.

4.2 Kolmogorov–Arnold Networks in IDIR (KAN-IDIR)

KAN-IDIR and RandKAN-IDIR (Drozdov et al., 26 Sep 2025) substitute the conventional MLP with a Kolmogorov–Arnold Network employing learnable univariate Chebyshev polynomials. This enhances functional expressivity and registration accuracy, while a randomized basis selection scheme in RandKAN-IDIR reduces computational and memory overhead without loss in accuracy. Both methods maintain resolution independence, avoid dense displacement grids, and achieve state-of-the-art target registration errors across multiple modalities and anatomical regions.

5. Computational and Representational Benefits

IDIR frameworks inherit the principal advantages of continuous coordinate-based neural parameterizations:

Resolution-independence: Evaluation and optimization are not tied to a fixed spatial discretization.
Full differentiability: Calculus on shape or DVF is analytic via autodiff, enabling higher-order regularization penalties (e.g., Jacobian determinant, bending energy).
Memory efficiency: Only the weights of the network are stored, sidestepping large voxel or mesh grids.
Regularization flexibility: Physics-based and generative priors can be imposed in the latent or weight space.
Speed and generalizability: Data-driven priors and hypernetwork-based architectures (as in latent-IDIR) accelerate optimization and improve robustness to noise or limited data.

6. Quantitative Performance and Empirical Findings

Empirical assessments across core IDIR applications report:

Physics-based SIREN-MLP IDIR achieves low Chamfer distances (down to $0.03$ for ETH-80 shapes) under realistic measurement noise and limited-view.
cIDIR attains mean Target Registration Error (TRE) of $1.33$ mm (DIR-LAB CT, bending regularization), outperforming both prior INR (IDIR) and CNN-based models, and requiring a single training pass for all regularization weights.
KAN-IDIR and RandKAN-IDIR yield competitive or superior Dice/h95 segmentation scores, lower runtime, reduced seed-dependent variability, and near-zero non-invertibility rates compared to MLP-based IDIR and classical iterative solvers.

Representative summary table (DIR-LAB; mean TRE, runtime, memory):

Method	TRE [mm] ↓	Runtime [s] ↓	VRAM [GB] ↓
pTV (classic)	0.95	442	–
IDIR (MLP)	1.07	261	4.1
KAN-IDIR	0.98	63	2.2
RandKAN-IDIR	0.99	43	1.4

7. Limitations and Future Directions

IDIR methods face several open challenges:

Instance-specific optimization cost: IDIR registration is not real-time, though accelerated architectures (RandKAN-IDIR, low-dimensional latent regularization) reduce per-instance compute.
Hyperparameter sensitivity: Optimal architecture, regularization strength, or basis complexity must be tuned per task.
Continuous domain generalization: Representations built on discrete signals (e.g., hash-mapping in DINER) do not smoothly generalize to new continuous coordinates unless coupled with continuous mapping modules.
Extendibility to multi-modal or very high-dimensional signals: Future IDIR work may combine spectral dictionary design (Yüce et al., 2021), band-limited activations (Ko et al., 19 Aug 2025), or adaptive kernel transformations (Zheng et al., 7 Apr 2025) for further gains in scalability, expressivity, and efficiency.

In conclusion, the IDIR (Implicit Neural Representation) paradigm enables flexible, compact, and fully differentiable modeling of signals, shapes, and physical mappings across a wide range of scientific, imaging, and inverse problems. Frameworks such as mesh-free inverse scattering IDIR (Vlašić et al., 2022), conditioned INR registration (cIDIR) (Hadramy et al., 17 Jul 2025), and KAN-IDIR (Drozdov et al., 26 Sep 2025) exemplify high-fidelity data fitting, seamless regularization, and analytic geometric reasoning, marking IDIR as a cornerstone in modern representation learning and computational imaging.