Learning KAN-based Implicit Neural Representations for Deformable Image Registration

## Abstract

Abstract Deformable image registration (DIR) is a cornerstone of medical image analysis, enabling spatial alignment for tasks like comparative studies and multi-modal fusion. While learning-based methods (e.g., CNNs, transformers) offer fast inference, they often require large training datasets and struggle to match the precision of classical iterative approaches on some organ types and imaging modalities. Implicit neural representations (INRs) have emerged as a promising alternative, parameterizing deformations as continuous mappings from coordinates to displacement vectors.
However, this comes at the cost of requiring instance-specific optimization, making computational efficiency and seed-dependent learning stability critical factors for these methods.
In this work, we propose KAN-IDIR and RandKAN-IDIR, the first integration of Kolmogorov-Arnold Networks (KANs) into deformable image registration with implicit neural representations (INRs). The learnable activation functions of KANs and their inherent suitability for approximating physical systems make them ideal for modeling deformation fields. Our proposed randomized basis sampling strategy reduces the required number of basis functions in KAN while maintaining registration quality, thereby significantly lowering computational costs. We evaluated our approach on three diverse datasets (lung CT, brain MRI, cardiac MRI) and compared it with competing instance-specific learning-based approaches, dataset-trained deep learning models, and classical registration approaches. KAN-IDIR and RandKAN-IDIR achieved the highest accuracy among INR-based methods across all evaluated modalities and anatomies, with minimal computational overhead and superior learning stability across multiple random seeds. Additionally, we discovered that our RandKAN-IDIR model with randomized basis sampling slightly outperforms the model with learnable basis function indices, while eliminating its additional training-time complexity. Source code is available at https://github.com/anac0der/KAN-IDIR .

## 1 Introduction

Deformable image registration (DIR) is a fundamental task in the field of medical image processing. Its primary goal is to determine a spatial transformation that aligns one image with another, ensuring that corresponding regions appear in the same locations. This process enables accurate comparison, fusion, or further analysis of imaging data.

Figure: Figure 1: Target registration error (TRE) distributions for IDIR and proposed KAN-IDIR across all DIR-Lab cases, comparing outlier characteristics between optimal (lowest mean error; first two histograms) and suboptimal (highest mean error; last two histograms) seed initializations from a total of 10 seeds. The dashed red line indicates the 3 mm acceptability threshold, while the dotted purple line marks the maximum outlier value per method. Our proposed KAN-IDIR demonstrates: (1) fewer outliers overall in optimal seed conditions – 54 vs. 64 for IDIR; and (2) consistent performance across seed selections – $>3$mm outlier count difference for best/worst seed is 4 for KAN-IDIR vs. 56 for IDIR.
Refer to caption: img/overall_histogram_collage_idir_kanidir_min.png

With recent advances in artificial intelligence technologies, image-to-image architectures such as convolutional neural networks (CNNs) and transformers have become increasingly popular in medical image registration (Balakrishnan et al., 2019a; Chen et al., 2022). These approaches predict displacement vector fields for image pairs, offering inference speeds several orders of magnitude faster than classical iterative methods – an essential advantage for real-time clinical applications.
However, such learning-based methods face three primary limitations: (1) their performance relies on large training datasets, which are often unavailable for specific organs or imaging modalities; (2) they do not consistently outperform classical methods for certain anatomical structures; and (3) they exhibit a degradation of registration quality when input data resolution varies.
An alternative approach to dataset-trained registration methods is to learn the deformation field as a mapping from continuous voxel coordinates to the corresponding displacement vector. This mapping is parametrized by a neural network, referred to as an implicit neural representation (INR). INRs have demonstrated remarkable performance in tasks like novel view synthesis (NeRFs) (Mildenhall et al., 2020) and have been leveraged in several methods for DIR (Wolterink et al., 2022; Harten et al., 2023, 2024).

INR-based solutions address the main limitations of dataset-trained deep learning models: their reliance on large training datasets, sensitivity to domain shifts, and difficulties with handling substantially different resolutions. However, these methods require instance-specific optimization, which makes computational efficiency and seed-dependent learning stability crucial for the practical use of these methods. Most existing INR architectures are based on multi-layer perceptrons (MLPs) with various activation functions (Sitzmann et al., 2020; Saragadam et al., 2023; Liu et al., 2024a). Recently, Kolmogorov-Arnold Networks (KANs), with their learnable activations, have emerged as a promising alternative to traditional MLPs in regression tasks (Liu et al., 2024c). Several recent studies have examined the INRs based on KANs across various tasks (Mehrabian et al., 2024; Li et al., 2025). To date, however, no prior work has investigated KAN-based INRs for deformable image registration, leaving their potential to advance registration techniques unexplored.

In this paper, we propose a novel approach to DIR using Implicit Neural Representations enhanced with Kolmogorov-Arnold Networks.
Our key contributions are:

1. We introduce the first integration of Kolmogorov-Arnold Networks with randomized basis sampling for INR-based deformable image registration. Our approach enhances registration accuracy while maintaining low computational and memory costs compared to classical or neural network-based methods. Additionally, it demonstrates improved seed-dependent learning stability compared to other INRs (see Figure [1](https://arxiv.org/html/2509.22874v1#S1.F1)).

2. Our models, namely KAN-IDIR and RandKAN-IDIR, are the only INR-based solutions that achieve the best or best-in-class performance across all three evaluated datasets spanning different organs and imaging modalities.
Specifically, they deliver superior accuracy and runtime/memory efficiency on DIR-Lab (lung CT), surpass other instance-specific solutions on OASIS-1 (brain MRI), and achieve the leading perfomance and best deformation regularity on ACDC (cardiac MRI). Besides that, the proposed models result in fewer outliers compared to the other INR-based baseline method for the DIR-Lab dataset (see Figure [1](https://arxiv.org/html/2509.22874v1#S1.F1)).

3. Surprisingly, our RandKAN-IDIR model with randomized basis sampling achieved similar performance to the adaptive KAN variant that learns function indices, while eliminating their additional computational overhead during training.

## 2 Related Works

Implicit Neural Representations. Implicit neural representations model signals as continuous, differentiable functions parameterized by neural networks. Following (Sitzmann et al., 2020), an INR is defined as the solution to an equation:

$$ $\mathcal{C}(\mathbf{x},\Phi,\nabla_{\mathbf{x}}\Phi,\nabla_{\mathbf{x}}^{2}\Phi,\ldots)=0,\quad\Phi:\mathbf{x}\mapsto\Phi(\mathbf{x}).$ (1) $$

Here, a network is trained to parameterize the function $\Phi$ that maps coordinates $\mathbf{x}\in\mathbb{R}^{n}$ to target values while satisfying the implicit constraint $\mathcal{C}$ in Eq. [1](https://arxiv.org/html/2509.22874v1#S2.E1).

The choice of INR architecture is critical, as it determines the class of functions that the network can effectively represent. While ReLU networks primarily capture low-frequency signals (Mildenhall et al., 2020), research has focused on alternative activations, e.g. SIREN’s sine functions (Sitzmann et al., 2020) or wavelet activations (Saragadam et al., 2023). Notably, SIREN remains a strong baseline in most applications.

### 2.1 Medical Image Registration

Medical Image Registration. Recent advances in deep learning (DL) for medical image registration have been heavily CNN-based methods. The VoxelMorph framework (Balakrishnan et al., 2019a) was a major breakthrough, enabling fast deformable registration using CNNs to learn deformation fields in unsupervised manner. It was further extended with probabilistic modeling with stationary velocity fields (SVFs) (Dalca et al., 2019), diffeomorphic networks in LapIRN approach (Mok & Chung, 2020b), and methods ensuring symmetry and topology preservation (Mok & Chung, 2020a; Greer et al., 2021; Tian et al., 2023). The rapid development of vision transformers introduced several strong approaches for DIR that utilized the ability of transformers to extract more complex spatial patterns from data that resulted in better registration quality (Chen et al., 2022).
Beyond CNNs and transformers, (Meng et al., 2024) developed a pure-MLP method, achieving top results on some datasets.

While CNN-based medical image registration has advanced significantly, key limitations persist. First, even when trained on large datasets, these image-to-image models remain vulnerable to domain shifts (Jena et al., 2024) - a problem often addressed through even larger datasets (Tian et al., 2024). Second, while segmentation labels enhance performance in brain MRI (Jena et al., 2024), these methods underperform on unlabeled data (e.g., DIR-Lab lung CT (Castillo et al., 2009)), trailing classical (Vishnevskiy et al., 2016) and INR-based (Harten et al., 2023) approaches. Finally, the CNNs and ViT by design typically cannot infer on several times higher or lower resolution wihout retraining, a limitation partially resolved by resolution-robust neural operators (Drozdov & Sorokin, 2024) at the cost of slightly reduced accuracy.

INRs for Image Registration. In INR-based registration, the deformation field $\Phi$ is modeled as a continuous neural function over the spatial domain $\Omega$, fundamentally differing from discrete image-to-image approaches. While INRs require optimizing a new network for each image pair, making it more computationally demanding than a single forward pass through a pre-trained CNN or transformer, it offers significant advantages. First, by eliminating the need for pretraining on large dataset, the INR-based methods require significantly less computational resources. Second, pair-specific registration algorithms also inherently avoid domain shift issues. Finally, the continuous nature of INRs makes them resolution-independent, allowing seamless application to data at arbitrary resolutions without architectural modifications.

INR-based medical image registration was pioneered by (Wolterink et al., 2022) using SIRENs for lung CT, achieving performance competitive with both classical and learning-based methods. Subsequent work extended this to brain MRI with alternative activations and cycle consistency (Byra et al., 2023), and later to robust bidirectional frameworks with uncertainty quantification (Harten et al., 2023). A fundamentally different (but still instance-specific) approach reformulated registration via neural ODEs in NODEO (Li et al., 2021). Most recently, (Sideri-Lampretsa et al., 2024) hybridized B-spline FFDs with INRs.

As demonstrated in our experiments below, current methods struggle to achieve an optimal balance between essential DIR requirements, including registration accuracy and deformation smoothness, as well as INR-specific constraints arising from the need of per-case network training, such as computational efficiency, memory demands, and seed-dependent stability.

KANs. Modern deep learning relies heavily on MLPs, known for their universal approximation capability (Cybenko, 1989; Hornik et al., 1989).
The Kolmogorov-Arnold approximation theorem for multivariate continuous functions (Kolmogorov, 1956; Arnold, 1957) offers an alternative: any continuous $f:[0,1]^{n}\to\mathbb{R}$ can be expressed as

$$ $f(x_{1},x_{2},\ldots,x_{n})=\sum_{q=1}^{2n+1}\Phi_{q}\left(\sum_{p=1}^{n}\phi_{q,p}(x_{p})\right),$ (2) $$

where $\phi_{q,p}:[0,1]\to\mathbb{R}$ and $\Phi_{q}:\mathbb{R}\to\mathbb{R}$ are continuous univariate functions.

The authors of (Liu et al., 2024c) introduced shallow and deep Kolmogorov-Arnold Networks (KANs) based on ([2](https://arxiv.org/html/2509.22874v1#S2.E2)), where each $\phi_{q,p}$ is implemented as spline-based univariate functions. While theoretically universal, shallow KANs with limited basis functions face convergence issues. Deep KAN architectures with multiple layers address this issue.

Recent work has explored KANs as INRs across domains: (Mehrabian et al., 2024) first adapted KAN layers for standard INR tasks (image fitting, occupancy volumes), while (Li et al., 2025) developed Fourier-KAN for neural sound fields. Several studies (Liu et al., 2024b; Faroughi & Mostajeran, 2025) have demonstrated that KANs outperform MLPs in accurately approximating functions of a physical or mathematical nature, like special functions or solutions to partial differential equations. Image registration can be formulated as finding the solution to an Euler-Lagrange system of a specific target functional (Drozdov & Sorokin, 2024), providing strong justification for using KANs in this context.

## 3 Methodology

### 3.1 Image Registration with INRs

DIR problem can be considered as finding the deformation field $\Phi(x)=x+U(x)$ that minimizes the following objective:

$$ $\hat{\Phi}=\operatorname*{arg\,min}_{\Phi}\mathcal{L}_{\text{data}}(M\circ\Phi,F)+\mathcal{L}_{\text{reg}}(\Phi),$ (3) $$

where $\mathcal{L}_{\text{data}}$ is a similarity measure between the fixed image $F$ and the warped moving image $M\circ\Phi$, and $\mathcal{L}_{\text{reg}}$ is a regularization term imposing constraints on the deformation field $\Phi$. We assume that the fixed and moving images are defined on the same domain, i.e.,
$M,F:\Omega\subset[-1,1]^{n}\to\mathbb{R}$.

### 3.2 INR Architecture

We adopt the Chebyshev KAN (Sidharth et al., 2024) as architecture of INR. This network utilizes Chebyshev polynomials with a fixed grid as basis functions, in contrast of B-splines with an adaptive grid used in the original KAN (Liu et al., 2024c). Chebyshev polynomial of the $n^{\text{th}}$ order is defined analytically as

$$ $T_{n}(x)=\cos(n\cdot\arccos(x)),\>x\in[-1,1].$ (4) $$

Using this definition, we can perform efficient vectorized computations of $T_{n}$ values, thus increasing the network’s speed in comparison to the original B-spline representation.
Forward pass through the Chebyshev KAN’s layer can be mathematically defined as follows:

$$ $[\mathbf{y}(\mathbf{x})]_{b,o}=\sum_{i=1}^{N_{in}}\sum_{d\in\mathcal{D}}T_{d}(\mathbf{x}_{b,i})\cdot\mathbf{C}_{i,o,d},$ (5) $$

where $b$ indexes the samples in the data batch, $i$ indexes the input dimension, $o$ indexes the output dimension, and $d$ indexes the polynomial degree. Here, $\mathbf{x}$ represents the input tensor, $\mathbf{y}$ the output tensor, $\mathbf{C}$ denotes the tensor of learnable coefficients, and $\mathcal{D}\subset\mathbb{N}_{0}$ is a finite set of all basis polynomial degrees, which may be shared across layers or vary between them.

We use normalization with the hyperbolic tangent function (tanh) to map input tensor values in $[-1,1]$ interval before passing into Chebyshev polynomials. To stabilize the training process, we follow the approach of the original KAN model and incorporate a learnable skip-connection in each layer. More precisely, the forward pass through $m^{\text{th}}$ layer of our network is determined by the following expression:

$$ $L_{m}(\mathbf{x})=W_{m}b(\mathbf{x})+\mathbf{y}_{m}(\mathbf{\widetilde{x}}),$ (6) $$

where $\mathbf{x}$ is the input tensor, $\mathbf{\widetilde{x}}$ denotes the inputs after tanh normalization, $\mathbf{y}_{m}(\cdot)$ represents the nonlinear part as defined in Eq. [5](https://arxiv.org/html/2509.22874v1#S3.E5) and $W_{m}$ is the learnable matrix of linear transformation. We have chosen $b(\mathbf{x})=\text{silu}(\mathbf{x})=x/(1+e^{-x})$ as an activation function for linear skip-connection.

Using the definitions above, the architecture of our model can be written as the composition of $N$ layers from Eq. [6](https://arxiv.org/html/2509.22874v1#S3.E6):

$$ $\Phi(\mathbf{x})=(L_{N-1}\>\circ\>L_{N-2}\>\circ\>\dots\>\>L_{0})(\mathbf{x}),\>\mathbf{x}\in\Omega\>.$ (7) $$

Figure: Figure 2: The overall pipeline of the proposed method for deformable image registration. Voxels are sampled from the input volume to serve as input to the KAN-IDIR or RandKAN-IDIR models. The network outputs a deformation field, which is used to warp the moving image, enabling calculation of the data similarity loss. For regularization, the gradient of the deformation field is computed using automatic differentiation.
Refer to caption: img/pipeline_v3.1.png

### 3.3 KANs with Sparse Set of Basis Functions

Sparse approximations were among the most prominent concepts in signal processing and pattern recognition during the pre-deep learning era (Zhang et al., 2015). The core idea of these algorithms is that a target signal can be approximately represented by a sparse linear combination of prototype signals from an overcomplete dictionary (Mallat & Zhang, 1993; Aharon et al., 2006). Inspired by the successful application of sparse representations in the past, we explored whether adopting a similar approach for KANs could enhance the quality and efficiency when applying to the DIR task. In the case of the KAN model, the set of basis functions serves as an overcomplete dictionary, and our goal is to utilize only a few of them during each forward pass of data through the network. Thus, two strategies can be considered: either selecting a fixed sparse set of basis functions for each layer before training or learning the optimal selection dynamically during network optimization. Based on these two strategies together with classical sequential selection of basis functions, we evaluate three distinct models in our subsequent work:

KAN-IDIR: This model employs the classical basis selection scheme with a fixed maximum degree $D$. For each layer, the basis functions are polynomials with degrees from 0 to $D$, i.e., $\mathcal{D}=\{0,1,\dots,D\}$. The set $\mathcal{D}$ of polynomial degree indices is shared across all layers.

RandKAN-IDIR: This model implements basis selection with randomized sampling. Let $\mathcal{D}_{m}$ denote the set of degrees of the basis polynomials for the $m^{\text{th}}$ layer. It is constructed by including the constant term (degree 0) and combining it with a random selection of higher degrees: $\mathcal{D}_{m}=\{0\}\cup\mathcal{S}_{m}$, where $\mathcal{S}_{m}$ is a simple random sample without replacement of size $k$ from the set $\{1,2,\dots,K\}$, independently for each layer $m$. Despite the reduced basis set, experiments (Sec. [4](https://arxiv.org/html/2509.22874v1#S4)) show that the RandKAN-IDIR model maintains competitive registration accuracy while improving efficiency, given an appropriate choice of $k$ and $K$.

A-KAN-IDIR: Adaptively learns basis indices during training (see Section [A.1](https://arxiv.org/html/2509.22874v1#A1.SS1) in Appendix for details). Experiments show no quality improvement over RandKAN-IDIR’s random selection, while being computationally slower, so we excluded this model from evaluations.

Figure: Figure 3: Landmark alignment in DIR-Lab Dataset (Case 7) for fixed, moving, and moved images. Displacements are shown as: blue (method result, moved-to-fixed), green (ground truth, moving-to-fixed), and red (error, moved-to-fixed). Landmarks with errors exceeding 3 mm are highlighted in bold, while others are displayed as transparent. Results reflect each method’s optimal seed (minimum mean TRE across 10 runs). Proposed KAN-IDIR and RandKAN-IDIR achieve the fewest outliers. The best seed-dependent variability of the proposed methods is demonstrated in Figure [5](https://arxiv.org/html/2509.22874v1#A1.F5) in Appendix using worst-case (maximum mean TRE) results.
Refer to caption: img/kp_overlay_collage_case_7_10s_min.png

### 3.4 Training Objective

Our loss function combines similarity and regularization terms (Eq. [3](https://arxiv.org/html/2509.22874v1#S3.E3)). For the similarity term, we use negative normalized cross-correlation (NCC):

$$ $\mathcal{L}_{\text{data}}(I,J)=-NCC(I,J)=-\frac{\sum_{\mathbf{p}\in\Omega_{p}}(I(\mathbf{p})-\bar{I})(J(\mathbf{p})-\bar{J})}{\sqrt{\sum_{\mathbf{p}\in\Omega_{p}}(I(\mathbf{p})-\bar{I})^{2}\sum_{\mathbf{p}\in\Omega_{p}}(J(\mathbf{p})-\bar{J})^{2}}},$ $$

where $\Omega_{p}\subset\Omega$ is the current batch of voxel coordinates, and $\bar{I},\bar{J}\in\mathbb{R}$ are the mean intensities of corresponding images over $\Omega_{p}$.

The regularization term $\mathcal{L}_{\text{reg}}$ combines two components: a smoothness term $\mathcal{L}_{\text{smooth}}$ and a folding prevention term $\mathcal{L}_{\text{Jdet}}$, weighted by parameters $\lambda$ and $\gamma$ respectively. To enforce physically plausible deformations, we employ total variation (TV) regularization (Rudin et al., 1992), which penalizes the L1 norm of the displacement field gradient:

$$ $\mathcal{L}_{\text{smooth}}=\frac{1}{|\Omega_{p}|}\sum_{\mathbf{p}\in\Omega_{p}}||\nabla U(\mathbf{p})||_{1}.$ (8) $$

While total variation (TV) regularization effectively promotes smooth deformation fields, it may still permit folded voxels, i.e., voxels with locally non-diffeomorphic transformations characterized by negative Jacobian determinants ($|J_{\Phi}|<0$). To address this, we incorporate a Jacobian determinant regularization term (Mok & Chung, 2020a) that actively penalizes folding regions:

$$ $\mathcal{L}_{\text{Jdet}}=\frac{1}{|\Omega_{p}|}\sum_{\mathbf{p}\in\Omega_{p}}\sigma(-|J_{\Phi}(\mathbf{p})|+\epsilon),$ (9) $$

where $\sigma(x)=\max(0,x)$ is the ReLU function and $\epsilon>0$ is a small constant added for overcorrection (Li et al., 2021). The complete training objective combines these regularization terms with the data similarity term:

$$ $\mathcal{L}=\mathcal{L}_{data}+\lambda\mathcal{L}_{\text{smooth}}+\gamma\mathcal{L}_{\text{Jdet}}.$ (10) $$

The overall pipeline of the proposed method for deformable image registration is shown in Figure [2](https://arxiv.org/html/2509.22874v1#S3.F2).

## 4 Experiments

### 4.1 Datasets and evaluation metrics

We evaluated our approach using three established DIR datasets spanning three organs and two imaging modalities.

DIR-Lab (Castillo et al., 2009): Ten lung 4D CT scans (inspiration/expiration phases) with 300 human-identified landmarks per pair corresponding to stable anatomical features, capturing complex cardiopulmonary motion posed by the interplay of cardiac and respiratory movements, which significantly exceed the scale of small lung structures. Image resolutions: $256^{2}$-$512^{2}$ in-plane, 94-136 Z-axis slices.

OASIS-1 (Marcus et al., 2007): 414 affinely-aligned brain MRIs with segmentations of 35 major anatomical regions, preprocessed to $160{\times}192{\times}224$ size (Hoopes et al., 2022). We followed Learn2Reg (Hering et al., 2022) evaluation protocols, using 49 test pairs from last 50 subjects for evaluating the compared approaches.

ACDC (Bernard et al., 2018): 150 cardiac MRI pairs (ED/ES phases) with ventricular segmentations. We evaluated on 50 pairs registered bidirectionally (100 pairs total), preprocessed to $128{\times}128{\times}32$ size in accordance with CorrMLP (Meng et al., 2024).

**Table 1: Evaluation of baselines and proposed methods (in bold) on the DIR-Lab dataset. The best result is in bold, and the second-best is in italics. Our models achieve near state-of-the-art performance in lung CT registration with lowest execution times.**
| Methods | pTV | NODEO | SINR | IDIR | ccIDIR | KAN-IDIR (28) | RandKAN-IDIR |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Average TRE, mm $\downarrow$ | 0.95 (1.15) | 3.93 (6.07) | 4.63 (5.26) | 1.07 (1.10) | 1.04 (1.12) | 0.98 (1.10) | 0.99 (1.10) |
| NJD, % $\downarrow$ | 0.6 (–) | 0.0003 (0.0003) | 0.002 (0.002) | 0.002 (0.015) | 0.002 (0.008) | 0.0006 (0.001) | 0.006 (0.016) |
| Runtime, sec $\downarrow$ | 442 | 583.5 | 91.6 | 260.8 | 92.6 | 63.3 | 43.1 |
| VRAM, Gb $\downarrow$ | – | 18.7 | 21.7 | 4.1 | 1.0 | 2.2 | 1.4 |

**Table 2: Cumulative distribution of TRE for DIR-Lab data shown as percentages within thresholds ($\leq 1$-$3$mm) and for outliers ($>3$mm) across instance-specific DL methods. Proposed method names and best values in bold.**
| Threshold | NODEO | SINR | IDIR | ccIDIR | KAN-IDIR (28) | RandKAN-IDIR |
| --- | --- | --- | --- | --- | --- | --- |
| $\leq$1mm $\uparrow$ | 39.05% | 18.14% | 56.51% | 57.07% | 60.13% | 60.28% |
| $\leq$2mm $\uparrow$ | 57.23% | 34.56% | 79.78% | 79.97% | 81.69% | 81.43% |
| $\leq$3mm $\uparrow$ | 73.09% | 55.59% | 97.52% | 97.95% | 98.14% | 98.09% |
| $>$3mm $\downarrow$ | 26.91% | 44.41% | 2.48% | 2.05% | 1.86% | 1.91% |

For all datasets, we evaluated four quantitative metrics: registration accuracy, measured by Target Registration Error (TRE) for DIR-Lab and by Dice score (DSC) and 95^th percentile Hausdorff distance (HD95) for OASIS-1 and ACDC; deformation regularity, quantified as the percentage of folded voxels (NJD); GPU memory consumption; and runtime. Experiments ran on an Intel Xeon Gold 6226R CPU / NVIDIA A6000 server under exclusive load to ensure runtime precision measurement. For INR-based methods, metrics were averaged over ten random seeds (mean$\pm$std), following (Harten et al., 2023; Wolterink et al., 2022).

### 4.2 Implementation Details

We implemented our method using the PyTorch library (Paszke et al., 2019). All KAN-based models consist of two hidden layers, each with 70 neurons. For the parameters $\lambda$, $\gamma$ and $\epsilon$ in Eq. [9](https://arxiv.org/html/2509.22874v1#S3.E9)-[10](https://arxiv.org/html/2509.22874v1#S3.E10), we adopt the values $\lambda=0.4$, $\gamma=15$ and $\epsilon=0.1$ unless otherwise specified. The Adam optimizer (Kingma & Ba, 2014) was employed for model training with an initial learning rate of $10^{-4}$. The learning rate remained constant for the first 50% of the iterations, after which it decreased following a cosine annealing schedule. Each model was trained for a total of 1500 iterations, with 10,000 points randomly sampled from the domain of interest on each iteration. The domain of interest is defined by a segmentation mask that separates the organ from the background. For the DIR-Lab dataset, we followed the approach from (Wolterink et al., 2022) and used the pretrained segmentation model to obtain the lung mask (Hofmanninger et al., 2020). The OASIS-1 dataset provides reference masks for anatomical segments, enabling us to use the union of all non-background segmentations as the resulting mask. For the ACDC dataset, we used all image voxels for training. For comprehensive baseline implementation details we refer readers to the Appendix (Section [A.3](https://arxiv.org/html/2509.22874v1#A1.SS3)).

**Table 3: Results of experiments on the OASIS-1 dataset. For learning-based methods, runtime and memory consumption are reported for both inference and training in the format inference/training. Our KAN-IDIR and RandKAN-IDIR hit the highest registration quality among INR-based methods, with nearly the fastest execution times and low VRAM consumption. Proposed method names and best values in bold, second-best in italics.**
| Methods | VM | TM | NODEO | SINR | IDIR | ccIDIR | KAN-IDIR (28) | RandKAN-IDIR |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| DSC $\uparrow$ | 0.795 (0.02) | 0.811 (0.02) | 0.790 (0.17) | 0.729 (0.04) | 0.762 (0.18) | 0.779 (0.16) | 0.793 (0.16) | 0.792 (0.17) |
| HD95 $\downarrow$ | 1.94 (0.29) | 1.81 (0.30) | 1.98 (0.38) | 2.65 (0.47) | 2.22 (0.45) | 2.19 (0.44) | 1.96 (0.32) | 1.97 (0.32) |
| NJD, % $\downarrow$ | 0.564 (0.11) | 0.526 (0.14) | 0.010 (0.01) | 0.020 (0.01) | 0.042 (0.03) | 0.315 (0.12) | 0.048 (0.03) | 0.049 (0.04) |
| Runtime $\downarrow$ | 0.1 s / 2.1 days | 0.2 s / 3.7 days | 68.1 s | 20.6 s | 260.8 s | 92.6 s | 63.3 s | 43.1 s |
| VRAM, Gb $\downarrow$ | 8.7 / 14.1 | 19.8 / 22.0 | 2.7 | 17.5 | 4.4 | 1.0 | 2.2 | 1.4 |

**Table 4: Results of experiments on the ACDC dataset. Runtime and memory consumption are omitted due to space constraints. KAN-based models achieve competitive performance with superior deformation regularity. Proposed method names and best values in bold, second-best in italics.**
| Methods | CorrMLP | NODEO | SINR | IDIR | ccIDIR | KAN-IDIR (8) | KAN-IDIR (28) | RandKAN-IDIR |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| DSC $\uparrow$ | 0.810 | 0.795 (0.095) | 0.811 (0.07) | 0.814 (0.09) | 0.792 (0.093) | 0.814 (0.09) | 0.809 (0.09) | 0.810 (0.09) |
| HD95 $\downarrow$ | – | 7.87 (3.17) | 6.75 (3.11) | 7.38 (3.38) | 8.23 (3.15) | 7.39 (3.17) | 7.87 (3.25) | 7.88 (3.28) |
| NJD, % $\downarrow$ | 0.389 | 0.011 (0.045) | 0.002 (0.002) | 0.111 (0.166) | 0.051 (0.066) | 2e-5 (2e-4) | 5e-5 (4e-4) | 4e-5 (2e-4) |

### 4.3 Results

Our evaluation compares against: instance-specific NODEO (Li et al., 2021); INR-based methods (IDIR (Wolterink et al., 2022), ccIDIR (Harten et al., 2023), SINR (Sideri-Lampretsa et al., 2024)); dataset-specific SOTA (pTV (Vishnevskiy et al., 2016) for DIR-Lab, CorrMLP (Meng et al., 2024) for ACDC); and established DIR baselines (VoxelMorph (Balakrishnan et al., 2019a), TransMorph (Chen et al., 2022) for OASIS-1).

We denote KAN-IDIR ($D$) as the model configuration with a maximum degree of $D$. For the RandKAN-IDIR model, we use a configuration with $k=12$ and $K=84$ in all experiments unless otherwise specified.

#### Results on DIR-Lab Dataset.

Table [1](https://arxiv.org/html/2509.22874v1#S4.T1) presents the experimental results obtained on the DIR-Lab dataset. KAN-IDIR and RandKAN-IDIR achieve the second and third best TRE, respectively, outperformed only by the classical pTV method, which remains SOTA for this dataset against all classical and DL-based approaches. However, our methods are significantly faster: RandKAN-IDIR registers pairs in under one minute versus pTV’s several minutes. Additionally, the performance difference between RandKAN-IDIR and KAN-IDIR is minimal, with only 0.01 mm average TRE gap, while RandKAN-IDIR achieves approximately 30% faster execution (43.1 s versus 63.3 s).
Notably, while NODEO and SINR perform competitively on other modalities, they show poor lung CT performance with TRE $\approx 4$ times higher than our methods.

Figure [1](https://arxiv.org/html/2509.22874v1#S1.F1) depicts the distribution of registration errors across all landmark pairs in the DIR-Lab dataset, while Table [2](https://arxiv.org/html/2509.22874v1#S4.T2) presents the percentage of errors within specified thresholds. It demonstrates that our methods not only achieve superior average registration quality but also exhibit improved quality in terms of error distribution and seed-dependent variability. Figure [3](https://arxiv.org/html/2509.22874v1#S3.F3) visualizes the overlay of the human-defined landmarks used to calculate TRE showing the qualitative behavior of the evaluated INR-based methods.

#### Results on OASIS-1 Dataset.

For the OASIS-1 dataset, we also include the learning-based methods such as VoxelMorph and TransMorph in the comparison. Table [3](https://arxiv.org/html/2509.22874v1#S4.T3) shows the results of experiments on the OASIS-1 dataset. While learning-based methods outperform case-specific approaches in registration quality, they demand substantial computational resources for training ($10^{6}\times$ greater training time and $10\times$ higher VRAM consumption on both training and inference). Compared to other INR-based methods, KAN-IDIR and RandKAN-IDIR achieve the best and second-best results in terms of DSC and HD95, while maintaining a moderate level of folded voxels in the deformation field. Furthermore, among INR-based methods, our models achieve nearly the fastest runtimes and memory consumption of no more than 2.2 GB, enabling their execution on consumer-grade GPUs.

#### Results on ACDC Dataset.

Table [4](https://arxiv.org/html/2509.22874v1#S4.T4) presents the experimental results for the ACDC dataset. Most methods exhibit similar performance levels, which we attribute to the relatively small number of voxels available for optimization, promoting a more stable training process.
The proposed KAN-IDIR model with $D=8$ shares the highest DSC value with IDIR and achieves the absolute best NJD result – only 0.00002% of foldings. Our DSC value overcomes the current state-of-the-art DSC reported in (Meng et al., 2024). Similarly, RandKAN-IDIR provides competitive registration quality while exhibiting excellent deformation field regularity.

### 4.4 Ablation Studies and Parameter Tuning

#### Loss Function Weights.

In this study, we emphasize the importance of Jacobian determinant regularization in our method. Figure [4](https://arxiv.org/html/2509.22874v1#S4.F4) (left) illustrates the results for the RandKAN-IDIR model on the OASIS-1 dataset across 26 different loss function weight configurations. Combinations with $\gamma=0$ (smallest circles) exhibit poorer deformation regularity compared to others, even when increasing the $\lambda$ weight of the TV regularizer. Our selected configuration, with $\lambda=0.4$ and $\gamma=15$, lies on the Pareto frontier, achieving a balance between Dice score and the amount of foldings.

#### Choice of k k and K K parameters.

Figure [4](https://arxiv.org/html/2509.22874v1#S4.F4) (right) shows the results for the RandKAN-IDIR model on the OASIS-1 dataset with different values of $k$ and $K$. Overall, performance improves monotonically with increasing $k$ up to 60, beyond which the model encounters convergence issues. For a fixed $k$, models with randomized sampling (results for $K>k$) and sufficiently large value of $K$ achieve better registration quality than models with sequential basis selection (results on the main diagonal with $K=k$). The improvement is especially notable for smaller values of $k$. For example, the model with $k=12$ and $K=84$ outperforms the model with $k=K=12$ (equivalent to KAN-IDIR with $D=12$) by 1.5% DSC and matches the performance of the model with $k=K=24$. Moreover, our models can be scaled to reach slightly higher performance than reported in Table [3](https://arxiv.org/html/2509.22874v1#S4.T3) – up to 0.797 DSC – at the cost of slower runtime and increased VRAM consumption.

Figure: Figure 4: Left: Ablation study of loss weights for RandKAN-IDIR model on the OASIS-1 dataset. Configurations on the Pareto frontier are highlighted in blue. Right: Ablation study of $k$ and $K$ parameters for RandKAN-IDIR model on the OASIS-1 dataset.
Refer to caption: img/abl.png

## 5 Conclusion

This paper presents KAN-IDIR and RandKAN-IDIR, the first models to incorporate KANs into INR-based deformable image registration.
By integrating randomized basis sampling into KAN-IDIR, RandKAN-IDIR reduces computational complexity while preserving near-equivalent registration quality. Both models deliver best or best-in-class performance across three medical imaging datasets of different organs and modalities, achieving superior deformation regularity and minimal computational overhead compared to other INR-based methods. Additionally, KAN-IDIR and RandKAN-IDIR demonstrate improved seed-dependent stability compared to their counterparts, ensuring consistent performance across diverse scenarios.

Looking ahead, our framework enables new research directions in KAN architecture optimization for broader medical image analysis tasks. The method’s computational efficiency, seed-dependent stability, and instance-specific adaptability make it particularly suitable for clinical adoption, especially when working with limited datasets.

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## Appendix A Appendix

**Table 5: Comparison of A-KAN-IDIR and RandKAN-IDIR on DIR-Lab, OASIS-1, and ACDC datasets. The best result per metric is highlighted in bold.**
| Method | DIR-Lab | OASIS-1 | ACDC |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Avg. TRE $\downarrow$ | NJD $\downarrow$ (%) | DSC $\uparrow$ | HD95 $\downarrow$ | NJD $\downarrow$ (%) | DSC $\uparrow$ | HD95 $\downarrow$ | NJD $\downarrow$ (%) |  |
| A-KAN-IDIR | 1.02 | 0.006 | 0.786 | 2.01 | 0.135 | 0.811 | 7.76 | 8e-5 |
| RandKAN-IDIR | 0.99 | 0.006 | 0.792 | 1.97 | 0.049 | 0.810 | 7.88 | 4e-5 |

### A.1 Description of A-KAN-IDIR vs RandKAN-IDIR

A-KAN-IDIR architecture.
Here we describe our A-KAN-IDIR model with adaptive learning of basis function indices.

We assign a learnable weight to each function in the basis function set, with each weight reflecting the importance of its corresponding basis function. The forward pass through the A-KAN-IDIR layer is formulated as follows:

$$ $[\mathbf{y}(\mathbf{x})]_{b,o}=\sum_{i=1}^{N_{in}}\sum_{d\in\mathcal{D}_{w}}\text{sigmoid}(w_{d})\cdot T_{d}(\mathbf{x}_{b,i})\cdot\mathbf{C}_{i,o,d},$ (11) $$

where $w_{d}$ represents the learnable weight, constrained to $(0,1)$ with the sigmoid function, and $\mathcal{D}_{w}$ denotes the set of indices for basis functions selected at each iteration. To construct $\mathcal{D}_{w}$, we employ the Noisy Top-$k$ algorithm, which selects the indices corresponding to the top-$k$ values of the noisy weight vector:

$$ $\mathcal{D}_{w}=\text{TopkIndices}(\{w_{i}+n_{i}\mid w_{i}\in\mathbf{w}\}),$ (12) $$

where $\mathbf{w}$ denotes the vector of all weights, and $n_{i}$ represents Gaussian noise added to each weight $w_{i}$ to prevent convergence to local minima early in the training process. We model $n_{i}$ as Gaussian noise with a standard deviation that decreases from 0.3 at the start of training to zero by 75% of the iterations. During the final 25% of iterations, we freeze the learnable weights and train only the Chebyshev polynomial coefficients.

To achieve a richer parameterization of learnable basis function weights, we adopt the Deep Image Prior (Ulyanov et al., 2018) approach, where the target image is learned from a fixed noise vector. We learn the weight vector $\mathbf{w}$ for each layer from a unique fixed vector $\mathbf{z}$, sampled from a standard Gaussian distribution. The mapping from $\mathbf{z}$ to $\mathbf{w}$ is learned using a two-layer MLP. The values of $\mathbf{z}$, $\mathbf{w}$, and the MLP weights are layer-specific and not shared across layers.

The overall forward pass pipeline for A-KAN-IDIR layer is as follows:

- 1.
Pass $\mathbf{z}$ to MLP and obtain the $\mathbf{w}$ values: $\mathbf{w}=\text{MLP}(\mathbf{z})$;
- 2.
Sample noise vector $\mathbf{n}$;
- 3.
Select set of indices $\mathcal{D}_{w}$ according to Eq [12](https://arxiv.org/html/2509.22874v1#A1.E12);
- 4.
Constrain values of $\mathbf{w}$ with indices from $\mathcal{D}_{w}$ with sigmoid function;
- 5.
Perform forward pass according to Eq. [11](https://arxiv.org/html/2509.22874v1#A1.E11).

Figure: Figure 5: Landmark alignment in DIR-Lab Dataset (Case 7) for fixed, moving, and moved images. Displacements are shown as: blue (method result, moved-to-fixed), green (ground truth, moving-to-fixed), and red (error, moved-to-fixed). Landmarks with errors exceeding 3 mm are highlighted in bold, while others are displayed as transparent. Results reflect each method’s suboptimal seed (maximum mean TRE across 10 runs). Proposed KAN-IDIR and RandKAN-IDIR still achieve the fewest outliers, while the other methods demonstrate significant increase.
Refer to caption: img/kp_overlay_collage_case_7_10s_max.png

**Table 6: Evaluation of TRE ($\downarrow$) in mm for baselines and proposed methods on DIR-Lab dataset. Best result is highlighted in bold, second-best – in italic.**
| Case | pTV | NODEO | SINR | IDIR | ccIDIR | KAN-IDIR<br>(D=28, Ours) | RandKAN-IDIR<br>(k=12, K=84, Ours) |
| --- | --- | --- | --- | --- | --- | --- | --- |
| 4DCT 01 | 0.76 (0.90) | 0.88 (0.94) | 1.59 (1.33) | 0.76 (0.94) | 0.83 (0.94) | 0.77 (0.91) | 0.78 (0.92) |
| 4DCT 02 | 0.77 (0.89) | 1.08 (1.62) | 1.81 (1.90) | 0.76 (0.94) | 0.78 (0.93) | 0.77 (0.93) | 0.76 (0.92) |
| 4DCT 03 | 0.90 (1.05) | 1.38 (2.13) | 3.36 (3.10) | 0.94 (1.02) | 1.02 (1.10) | 0.89 (1.02) | 0.91 (1.03) |
| 4DCT 04 | 1.24 (1.29) | 2.93 (3.64) | 3.82 (3.55) | 1.32 (1.27) | 1.37 (1.36) | 1.30 (1.28) | 1.29 (1.26) |
| 4DCT 05 | 1.12 (1.44) | 3.30 (4.53) | 4.76 (4.12) | 1.23 (1.47) | 1.25 (1.51) | 1.13 (1.41) | 1.13 (1.41) |
| ‘<br>4DCT 06 | 0.85 (0.89) | 4.52 (5.55) | 4.58 (3.30) | 1.09 (1.03) | 1.06 (1.09) | 0.94 (1.01) | 0.94 (1.01) |
| 4DCT 07 | 0.80 (1.28) | 7.00 (7.93) | 6.74 (6.41) | 1.12 (1.00) | 0.97 (0.98) | 0.93 (0.96) | 0.94 (0.97) |
| 4DCT 08 | 1.34 (1.93) | 10.47 (10.34) | 12.10 (8.71) | 1.21 (1.29) | 1.13 (1.40) | 1.11 (1.26) | 1.13 (1.28) |
| 4DCT 09 | 0.92 (0.94) | 4.06 (4.42) | 3.44 (2.51) | 1.22 (0.95) | 1.02 (0.93) | 1.02 (0.96) | 1.03 (0.96) |
| 4DCT 10 | 0.82 (0.89) | 3.65 (5.31) | 4.10 (4.12) | 1.01 (1.05) | 0.96 (0.98) | 0.97 (0.99) | 0.99 (0.98) |
| Average | 0.95 (1.15) | 3.93 (6.07) | 4.63 (5.26) | 1.07 (1.10) | 1.04 (1.12) | 0.98 (1.10) | 0.99 (1.10) |

**Table 7: Ablation study of basis functios number D in KAN-IDIR model on the DIR-Lab dataset. KAN-IDIR model with $D=28$ delivers best registration quality. Best in bold.**
| D | 8 | 12 | 16 | 20 | 24 | 28 | 36 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| Avg. TRE, mm $\downarrow$ | 1.11 | 1.04 | 1.01 | 0.99 | 0.99 | 0.98 | 0.98 |

Comparison of A-KAN-IDIR vs RandKAN-IDIR.
Table [5](https://arxiv.org/html/2509.22874v1#A1.T5) presents a comparative evaluation of A-KAN-IDIR and RandKAN-IDIR across the DIR-Lab, OASIS-1, and ACDC datasets, with metrics including Average TRE, DSC, HD95, and NJD. RandKAN-IDIR outperforms A-KAN-IDIR in most metrics across the datasets, achieving better registration quality and deformation regularity. However, on the ACDC dataset, A-KAN-IDIR achieves a slightly higher Dice score and better HD95. Additionally, A-KAN-IDIR exhibits approximately 20% higher runtime (e.g., 51.2 s vs. 43.1 s on OASIS-1) and VRAM consumption (e.g., 2.64 Gb vs. 2.2 Gb on OASIS-1) compared to RandKAN-IDIR. These results suggest that A-KAN-IDIR’s adaptive basis selection mechanism introduces higher computational overhead compared to RandKAN-IDIR’s simpler randomized basis sampling, which provides superior registration quality and efficiency for deformable image registration in our experimental setup.

### A.2 Additional Experimental Results

Visual Appearance of Registration Results for the Suboptimal Seed.
The visual appearance of the registration results that confirms the best performance of the proposed models in terms of seed-dependent stability is demonstrated in Figure [5](https://arxiv.org/html/2509.22874v1#A1.F5). The proposed RandKAN-IDIR and KAN-IDIR maintain consistent outlier counts across seeds, with results similar to the optimal seed in Figure [3](https://arxiv.org/html/2509.22874v1#S3.F3).

DIR-Lab Per-Case Results.
In Table [6](https://arxiv.org/html/2509.22874v1#A1.T6) we show the full per-case results for DIR-Lab dataset, following the practice for this dataset in Wolterink et al. (2022); Harten et al. (2023).

#### Ablation for maximum polynomial degree.

In Table [7](https://arxiv.org/html/2509.22874v1#A1.T7), we evaluate the performance of the KAN-IDIR ($D$) model on the DIR-Lab dataset across various values of $D$. The results indicate that a value of $D=16$ is sufficient for KAN-IDIR to achieve state-of-the-art performance among INR-based methods on the DIR-Lab dataset. For $D=28$, the performance reaches a plateau, with minimal further improvements observed for higher degrees.

### A.3 Baselines Implementation Details and Parameters Selection

For IDIR (Wolterink et al., 2022), ccIDIR (Harten et al., 2023), NODEO (Li et al., 2021), and SINR (Sideri-Lampretsa et al., 2024) we extended the original implementations to all three datasets: OASIS-1, and ACDC for IDIR and ccIDIR; DIR-Lab, OASIS-1, and ACDC for SINR; DIR-Lab and ACDC for NODEO). We ensured that our modifications preserved performance and adhered strictly to the original implementations.

For SINR and NODEO, we identified and fixed several substantial bugs in both codebases that prevented execution on datasets beyond those in the original papers. Pending paper acceptance, we will submit these fixes as pull requests to the respective repositories.

IDIR, ccIDIR, and NODEO were executed using their default parameter configurations without modification.

For SINR Sideri-Lampretsa et al. (2024), we revised the original stopping criterion (NJD $>$ 0.9%) due to observed cases where NJD values initialized high and decreased during optimization. After testing iterations at 300, 500, and 1500, we standardized to 500 iterations as no significant improvements were observed at higher counts. Notably, SINR required additional cropping to landmark min-max coordinates on DIR-Lab to accommodate 48GB GPU memory constraints—the only method necessitating such adaptation.

VoxelMorph (Balakrishnan et al., 2019b) and TransMorph (Chen et al., 2022) were trained without segmentation-based weak supervision. We adjusted their train-test splits to use the last 49 pairs (versus the original 19) to enhance statistical robustness and maintain consistency with other experiments.

For pTV (Vishnevskiy et al., 2016) and CorrMLP (Meng et al., 2024), we directly reproduced the authors’ reported values as these represent current state-of-the-art implementations for their respective datasets.

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 6
presentation: 3
questions: - The authors should compare also compare their network with other architectures like WIRE, FINER,... How do they compare to their chosen approach? Justify your choice!
soundness: 3
strengths: The paper is well written in all sections and their architecture is clearly explained The authors report a performance improvement with a lower resource consumption. They shared their implementation and used publicly available datasets which is a plus for the reproducibility, and they promise to make their code publicly available. In the analysis the authors provide multiple ablations across various settings and parameter ranges and also show some qualitative examples of their method compared to others.
summary: The authors consider the task of deformable image registration using the well known method INRs to model the deformation/displacement field.

Contrary to previous work they use an other network architecture for the INR, in this case a Chebyshev KAN. They explored the use of fixed, randomized and adaptive selection of polynomial basis functions for their network. They evaluate their task on three datasets, including 4D lung CTs, brain MRIs and cardiac MRIs. They show that for the selected datasets their method slightly outperforms previous work with other network architectures, while consuming a little bit less of computational resources.
weaknesses: The authors only investigate one type of networks architectures (the Chebyshev KANs) and compare them against other existing types from other publications. Since this was the only focus of this paper, it would have been nice to see some broader comparison across other network architectures that have been proposed for INRs. One of the previous work they base their work on (Wolternik, 2022) used SIREN, but since, there have many other architectures have been published (WIRE, FINER etc) that easily could have been explored in the same work, so the motivation for only focusing this particular architecture remains unclear.

---

confidence: 5
contribution: 2
initial_rating: 4
presentation: 3
questions: How robust is the method to multimodal intensity differences (e.g., CT→MR registration)? Have you tried cross-modality experiments?
soundness: 3
strengths: Comprehensive evaluation: Evaluated on three modalities/organ types with multiple metrics (TRE/DSC/HD95/NJD), per-case tables, seed variability, and ablations on basis size and loss weights.

Seed-stability analysis: The paper explicitly studies seed-dependent variability and demonstrates improved robustness, an important but often neglected INR property.
summary: This paper proposes KAN-IDIR and RandKAN-IDIR, the first integration of Kolmogorov–Arnold Networks (KANs) into implicit neural representation (INR) based deformable image registration.
weaknesses: 1. In figure 1, the visual differences between the first two panels and the last two panels in Figure 1 are not explained; it is difficult to understand what each subfigure is intended to illustrate.
2. The contribution reads largely as “apply Chebyshev KAN architecture to INR for registration” with no substantial methodological modifications targeted specifically to registration (beyond RandKAN sampling). This reduces perceived novelty.
3. In figure 3, the regularization loss should be imposed to the learned displacement, not on the addition of the displacement and the input coordinate.
4. In Table 2, the results of pTV method are not reported.  
5. The claim of “best or best-in-class across all datasets” is overstated: classical pTV remains best on DIR-Lab TRE in Table 1, and dataset-trained methods outperform on some metrics (OASIS).
6. It is unclear whether reported runtimes are end-to-end (training per instance) or inference only, and how they scale with image resolution.
7. The paper adopts Chebyshev-based KANs but provides limited ablation comparing different KAN variants or other INR bases (e.g., SIREN, Fourier features, B-spline KAN). It is therefore unclear whether Chebyshev KAN is the best or merely sufficient.

---

confidence: 4
contribution: 2
initial_rating: 2
presentation: 1
questions: 1.  How is sparse approximation related to pruning literature, both within and outside the scope of KANs and neural networks?
2.  How do spline-based KANs compare to the Chebyshev KANs in terms of performance (excluding speed)?
soundness: 2
strengths: - Using KANs over traditional, MLP-based architectures for INRs is a refreshing and novel perspective for deformable image registration.
- Based on the authors' presentation, their proposed method outperforms all baseline models for all datasets.
summary: This paper introduces a KAN-based Implicit Neural Representation (INR) framework for deformable image registration and compares its performance to SIREN-based INRs. Moreover, the authors ablate architectural variants of the KAN network with respect to its (learnable) basis function.
weaknesses: ### Presentation of the method within the context of other works (Introduction/Related Work):

The structure of the introduction and related work does not effectively motivate the usage of KANs in the context of deformable image registration, and it lacks the placement of the proposed framework within this domain.

To be precise:
* The authors provide a lengthy introduction to state-of-the-art cohort-based registration frameworks, such as VoxelMorph, yet they focus on iterative settings. They should instead focus on placing their proposed method within this context.
* The motivation for choosing KAN over an MLP in the context of deformable registration is not clearly stated—what specific features do KANs offer that MLPs do not?
* While seminal works like [1] have employed SIREN, a plethora of other INR architectures have been proposed since—which may be equally relevant (to KANs) for this task—such as those utilizing Gaussian activation functions, FINER, BACON, MFN, etc. [2-6]. Since the primary contribution of this paper is to use KANs instead of MLP-based INRs, the authors should discuss (and later ablate) these architectures in this context, and not focus only on SIREN. For instance, [2] remains of particular relevance since they group ReLU activations to be a spline, which may be adjacent to this work, even though the authors do not use spline-based KANs.

Moreover, the KAN literature has also grown significantly over the past year. Since one of the authors' main contributions is the introduction of their RandKAN model, it would be beneficial to explain the elements of different KAN variants with more rigor and also contrast their performance in experiments. For instance, why did the authors choose the Chebyshev KAN architecture over the traditional KAN (Liu et al.) using B-splines, given that methods like FFD [7] typically use B-Splines? Is there any intuition for this? While the authors state that this increases the speed, did they ablate this decision?

### Experimental Setup:

* The authors use entirely different regularization than methods like IDIR [1]. The motivation for using other regularization (e.g., no bending energy) remains unjustified. A fair experimental setup would integrate all architectures into the same framework and evaluate all models for the same model footprint on the problem, requiring hyperparameter tuning for all architectures within the same framework. For instance, the KAN models are much smaller, which may cause them to overfit much less to noise and thus exhibit much less folding. How does folding behave once the model size is increased?
* Missing classical, iterative registration pipelines for DIRLAB experiments.
* The authors report that default parameters were used for, e.g., IDIR for all datasets, resulting in suboptimal performance since IDIR's hyperparameters were optimized for lung registration, and not for brain (OASIS) or cardiac datasets.
* No qualitative results are provided for the OASIS and ACDC datasets.

Given these substantial weaknesses, I recommend a thorough revision of the Introduction/Related Work and the experimental setup for this manuscript.

[1] Wolterink JM, Zwienenberg JC, Brune C. Implicit neural representations for deformable image registration. MIDL'22

[2] Liu Z, Zhu H, Zhang Q, Fu J, Deng W, Ma Z, Guo Y, Cao X. Finer: Flexible spectral-bias tuning in implicit neural representation by variable-periodic activation functions. CVPR'24

[3] Fathony R, Sahu AK, Willmott D, Kolter JZ. Multiplicative filter networks. ICLR2020

[4] Lindell DB, Van Veen D, Park JJ, Wetzstein G. Bacon: Band-limited coordinate networks for multiscale scene representation. CVPR22

[5] Ramasinghe S, Lucey S. Beyond periodicity: Towards a unifying framework for activations in coordinate-mlps. ECCV'22

[6] Shenouda J, Zhou Y, Nowak RD. ReLUs are sufficient for learning implicit neural representations. ICML'24

[7] Rueckert D, Sonoda LI, Hayes C, Hill DL, Leach MO, Hawkes DJ. Nonrigid registration using free-form deformations: application to breast MR images. TMI'02

---

confidence: 4
contribution: 2
initial_rating: 2
presentation: 2
questions: Please see the weaknesses.

**Preliminary Justification:**

The paper presents a conceptually interesting approach to a new way of representing the transformation of registration based on KANs.

However, I believe that the paper quality is insufficient to be accepted in ICLR this year. I believe the authors have to put substantial effort into enhancing the motivation, the literature review, the storyline, experiments, and discussion, as well as to include more baselines (classical methods) and qualitative results. 

Given these shortcomings, I recommend rejection in its current form. Nonetheless, the underlying idea is promising, and I encourage the authors to strengthen the paper and resubmit it to a future venue. Finally, given the structure of the proposed work, I believe that this would be a better fit in a computer vision conference.
soundness: 2
strengths: 1. The paper utilizes a new architecture for pairwised INR-based registration that was not used before.
2. It proposed a novel way of learning KAN activations that might be useful out of the context of deformable registration.
3. The performance is evaluated against 3 mono-modal diverse datasets measuring the accuracy, regularity and speed.
4. The language of the paper is clear.
summary: The paper introduces a novel way of representing the transformation of deformable image registration in the context on INRs architectures, using Kolmogorov-Arnold-Networks (KANs). It proposes a new strategy of learning KANs that reduces the the number of learned basis function lowering the overall cost while maintaining competitive performance. The performance is evaluated on a 3 mono-modal datasets and compared against several INR-based methods and 2 cohort-based methods.
weaknesses: 1. The introduction and related work are rather weak and unfocused failing to motivate the topic effectively. First of all, there is not a clear motivation on what makes KANs a better candidate as a network architecture compared to MLPs in the context of registration. The literature review both in the case of Registration and INRs is not thorough making it difficult to understand what is state-of-the-art in the field.
2. The discussion about iterative optimization methods (classical methods) for registration is somehow missing from the discussion. In my opinion INRs are a sub-category of iterative optimization methods. As a result, I believe they should be mentioned in the introduction and also compared against during the evaluation.
3. There are several inaccuracies in the text, namely:
    1. the paper refers to INR methods as instance-specific optimization methods. However, since registration is happening in pairs I believe that the right term is pairwise optimization methods.
    2. Line 130: *“Second, while segmentation labels enhance performance in brain MRI (Jena et al., 2024), these methods underperform on unlabeled data (e.g., DIR-Lab lung CT (Castillo et al., 2009)), trailing classical (Vishnevskiy et al., 2016) and INR-based (Harten et al., 2023) approaches.”*
        
        I believe that this is intuitive and that doesn’t diminish their utility. Moreover, I am not sure this is a fair claim since in the case of brains the deep learning methods are trained on segmentation losses and evaluated using dice score (optimization for the metric) while in the case of DIRLab even if they are trained using segmentation labels they are evaluated using TRE on landmarks (the network was not optimized for this metric).
        
    3. Line 127 - 129: “First, even when trained on large datasets, these image-to-image models remain vulnerable to domain shifts (Jena et al., 2024) - a problem often addressed through even larger datasets (Tian et al., 2024).”
    
    I am not sure why this is a limitation. If larger datasets are available they can be used to alleviate the domain shift issue mentioned by the authors.
    4. Line 149: *“A fundamentally different (but still instance-specific) approach reformulated registration via neural ODEs in NODEO (Li et al., 2021).”*
    
    Why do the authors believe that this is a fundamentally different approach than the INRs?
4. I appreciate that the authors experimented with different seeds, but I believe that any learning based method is suffering from the performance discrepancy when choosing different seeds. Is there any reason why this has not been applied to Vmorph and Tmorph?
5. I am not sure I understand the motivation behind the analysis in figure 1 (e.g. What does 54 vs 64 for INR refer to?) This figure and its caption lack clarity.
6. Although there are figures for the lung registration and the landmark errors there are no visualizations of the deformation fields and for the other modalities. Maybe the authors should consider adding them at least in the appendix since visual inspection is a very important way of evaluating whether the final result is accurate.
7. The paper is using a combination of 2 regularization terms:
    1. Where the other methods trained on the same objective? If this is not the case then the evaluation is not very fair and it makes sense why the proposed method is delivering lower folding compared to its counterparts.
8. In the evaluation section I believe that the dice score that is reported is the mean over all structures. It would be usedul to see the mean and std and only demonstrate the seed stability as an ablation over only one application.
9. For completeness I believe that VRAM and speed should also be included in the table for the cardiac data.
10. There is not publicly available repository.

**Minor points**

1. Is there any drawback due to the choice of Chebychev polynomials instead of the original splines?
2. Is there any intuition why the different KAN strategies behave differently?
3. It is not clear whether the INR methods are fitted till convergence or over fixed steps and whether the learning rate and omega of siren was tuned to the application. Finally, the choice of the batch size affects both the memory requirements, the convergence and the performance. It would be nice to know how all these hyperparameters were chosen to understand whether the evaluation is fair.


Decision: Reject