---
title: 'RandKAN-IDIR: Efficient INR Registration'
url: https://www.emergentmind.com/topics/randkan-idir
type: topic
---

# RandKAN-IDIR: Efficient INR Registration

RandKAN-IDIR is an instance-specific deformable image registration method that integrates Kolmogorov-Arnold Networks into implicit neural representations for continuous deformation modeling. It was introduced alongside KAN-IDIR as the first integration of KANs into deformable image registration with implicit neural representations, with the specific aim of reducing the computational burden and seed-dependent instability that characterize prior INR-based registration methods. Its defining mechanism is a randomized basis sampling strategy that retains registration quality while reducing the number of active basis functions, thereby lowering runtime and memory costs without introducing an additional basis-index learning stage [2509.22874].

## 1. Conceptual position within deformable image registration

Deformable image registration seeks a smooth deformation field that aligns a moving image to a fixed image. In the formulation used for RandKAN-IDIR, the deformation is represented as a continuous coordinate-to-displacement mapping,
$$
\Phi(\mathbf{x}) = \mathbf{x} + U(\mathbf{x}),
$$
with the displacement field \(U\) parameterized by a KAN-based implicit neural representation. The optimization target is
$$
\hat{\Phi} = \arg\min_{\Phi} \mathcal{L}_{\text{data}}(M \circ \Phi, F) + \mathcal{L}_{\text{reg}}(\Phi),
$$
where the similarity term is negative normalized cross-correlation and the regularization combines Total Variation with a penalized Jacobian determinant to discourage non-diffeomorphic or folded voxels [2509.22874].

The method occupies a specific position between classical iterative registration and dataset-trained deep models. The motivating paper states that learning-based methods such as CNNs and transformers offer fast inference but often require large training datasets and can struggle to match classical iterative approaches on some organs and modalities. INRs are presented as a promising alternative because they parameterize deformations as continuous mappings from coordinates to displacement vectors, but they do so at the cost of instance-specific optimization. RandKAN-IDIR is therefore designed as a more computationally efficient and more stable INR variant rather than as a conventional pre-trained registration network [2509.22874].

## 2. KAN parameterization and the randomized-basis variant

The KAN component is motivated through the Kolmogorov-Arnold theorem, described in the source paper as allowing any multivariate continuous function to be expressed as sums of univariate functions of linear combinations of the inputs. In this setting, KANs use learnable univariate activations, and the implementation discussed for image registration is based on Chebyshev polynomial bases. The layer computation is written as
$$
[\mathbf{y}(\mathbf{x})]_{b,o} = \sum_{i=1}^{N_{in}} \sum_{d \in \mathcal{D}_m} T_d(\mathbf{x}_{b,i}) \cdot \mathbf{C}_{i,o,d},
$$
where \(T_d\) is the Chebyshev polynomial of degree \(d\), \(\mathbf{C}\) denotes learnable coefficients, and \(\mathcal{D}_m\) is the set of basis-function degrees used in layer \(m\) [2509.22874].

The distinction between KAN-IDIR and RandKAN-IDIR is entirely in the construction of \(\mathcal{D}_m\). KAN-IDIR uses a fixed contiguous degree set \(\{0,1,\dots,D\}\). RandKAN-IDIR instead defines
\[
\mathcal{D}_m = \{0\} \cup \mathcal{S}_m,
\]
where \(\mathcal{S}_m\) is a random sample without replacement of \(k\) higher-degree indices drawn from \(\{1,2,\dots,K\}\). The constant basis is always retained; the remaining selected bases are fixed for the remainder of optimization for that image pair. This construction is intended to preserve the expressivity of a sufficiently rich basis family while reducing over-parameterization, evaluation cost, and coefficient storage [2509.22874].

| Aspect | KAN-IDIR | RandKAN-IDIR |
|---|---|---|
| Basis selection | Fixed sequence, degrees \(0:D\) | Random \(k\) higher degrees per layer plus degree 0 |
| Training complexity | Baseline | Lower due to fewer bases |
| Learning stability | High | Highest; least seed-to-seed variability |

## 3. Randomized basis sampling as a sparsification strategy

The randomized basis sampling strategy is introduced as a practical alternative to activating all high-degree bases or learning basis indices adaptively. The rationale given in the source is twofold. First, high-degree KANs with all bases active are computationally expensive and may over-parameterize the network. Second, by analogy with sparse dictionary learning, a signal may be represented efficiently using only a subset of basis atoms. RandKAN-IDIR operationalizes that intuition by fixing a sparse but randomly chosen subset of polynomial degrees in each layer [2509.22874].

The reported effect is reduced computation and improved robustness without a measurable loss of registration quality. The paper states that randomized basis selection reduces runtime and VRAM by approximately \(30\%\)–\(40\%\) with no registration-quality loss, and that the RandKAN-IDIR model with randomized basis sampling slightly outperforms the model with learnable basis function indices while eliminating the latter’s additional training-time complexity. This makes the randomized variant not merely a cost-reduction heuristic but the preferred KAN-based INR configuration in the reported experiments [2509.22874].

A central technical point is that the method remains instance-specific. RandKAN-IDIR does not replace per-case optimization with amortized inference; rather, it reduces the complexity of that optimization. This is significant because the same paper identifies instance-specific optimization and seed-dependent learning stability as the critical bottlenecks of INR-based deformable registration. RandKAN-IDIR is proposed specifically to address those bottlenecks rather than to change the underlying optimization regime [2509.22874].

## 4. Empirical behavior across lung CT, brain MRI, and cardiac MRI

The evaluation reported for RandKAN-IDIR spans three datasets: DIR-Lab for lung CT, OASIS-1 for brain MRI, and ACDC for cardiac MRI. Comparisons include other instance-specific INR methods such as IDIR, ccIDIR, NODEO, and SINR; dataset-trained deep models such as VoxelMorph, TransMorph, and CorrMLP; and the classical method pTV. Across these datasets, KAN-IDIR and RandKAN-IDIR are reported to achieve the highest accuracy among INR-based methods, with RandKAN-IDIR additionally showing superior learning stability across multiple random seeds [2509.22874].

On DIR-Lab, RandKAN-IDIR achieves a target registration error of \(0.99\) mm, compared with \(0.98\) mm for KAN-IDIR, \(1.07\) mm for IDIR, and \(0.95\) mm for pTV. The same experiment reports that RandKAN-IDIR is approximately \(30\%\) faster than KAN-IDIR, with \(43.1\) s versus \(63.3\) s per case, and that it shows the fewest outliers together with the best seed-dependent stability. The result places RandKAN-IDIR very close to the strongest classical baseline while remaining the most stable INR configuration in the study [2509.22874].

On OASIS-1, RandKAN-IDIR is described as matching KAN-IDIR in Dice similarity coefficient and surpassing all other INR methods, while also obtaining the lowest HD95 among INR methods. Its negative Jacobian determinant rate is reported as acceptably low, although classical or large deep learning models have the lowest folding rates at the cost of \(10^6\times\) more training. The same section reports low memory consumption, specifically \(1.4\) GB, and the shortest runtime among INR methods. On ACDC, RandKAN-IDIR is characterized as highly competitive in DSC and HD95 while providing the best deformation regularity [2509.22874].

Taken together, these findings define RandKAN-IDIR less as a universal best-in-class method than as a particularly strong point in the design space of INRs: near-KAN-IDIR accuracy, reduced computational overhead, fewer outliers, and improved seed robustness. The paper’s own comparative framing emphasizes that it slightly outperforms the adaptive-basis A-KAN-IDIR variant on most metrics while avoiding an extra optimization component [2509.22874].

## 5. Relation to IDIR and evidence from mammography benchmarking

RandKAN-IDIR is best understood as a development within the broader IDIR lineage. MGRegBench, a public mammography registration benchmark, evaluates IDIR rather than RandKAN-IDIR, but its results are useful for understanding the failure modes and desiderata that the KAN-based variants target. In that study, IDIR was adapted for 2D mammography data, trained across 5 random seeds to address sensitivity to initialization, and evaluated by selecting the best-performing seed for each image pair because averaging was found sensitive to outlier seeds. The reported training configuration uses batch size \(2{,}500\), \(2{,}500\) epochs, learning rate \(10^{-4}\), ReLU activation, and a \(0.01\) weight on the Jacobian deformation regularization term [2512.17605].

MGRegBench reports that, on its evaluation set, IDIR attains \(2.46\%\) rTRE, MSE \(418.70\), SSIM \(0.811\), MI \(1.260\), CC \(0.908\), NJD \(0\), DSC \(0.950\), VRAM usage \(8.4\) GB, and runtime \(29.96\) s. In the same benchmark, IDIR is slightly worse in rTRE than ANTs/SyN at \(2.10\%\), VoxelMorph and TransMorph at \(2.14\%\), and Affine+MRN at \(2.03\%\), but it uniquely achieves zero foldings. The benchmark summary highlights this as perfect deformation regularity and emphasizes very good correspondence of internal features, while also noting breast boundary misalignment and a substantial computational burden [2512.17605].

This juxtaposition is informative. MGRegBench identifies exactly the issues of seed sensitivity and runtime/memory cost that the RandKAN-IDIR paper makes central, whereas the KAN-based work reports improved learning stability across random seeds and substantial reductions in computational overhead. This suggests that RandKAN-IDIR should be read not as a departure from INR-based registration, but as a response to recurrent INR bottlenecks exposed in other modalities and benchmarks [2509.22874].

## 6. Interpretation, naming, and common sources of confusion

A common source of confusion is the prefix “Rand.” In RandKAN-IDIR, it refers to randomized basis sampling in KAN layers, not to the Rand Index or its modified forms used to compare partitions. The Modified Rand Index and Modified Wallace indices address a different problem entirely: comparison of partitions over non-equal but overlapping sets of units, including newcomers and outgoers, with optional asymmetry for splitting and merging effects [1805.07996]. RandKAN-IDIR is unrelated to that line of work except in name similarity.

Methodologically, RandKAN-IDIR is also distinct from standard dataset-trained registration networks. The paper explicitly compares it with CNN-, transformer-, and MLP-based approaches, but its operational regime is instance-specific optimization. Its reported advantage is therefore not elimination of optimization cost, but a better trade-off between representational capacity, regularity, stability, memory footprint, and runtime within the INR family [2509.22874].

The limitations are correspondingly specific. RandKAN-IDIR remains an optimization-based method and is therefore slower than feed-forward registration networks in absolute inference terms. Its benefits are most clearly expressed against other INR-based approaches: highest accuracy among INR methods across the evaluated anatomies and modalities, superior seed stability, and lower computational overhead. A plausible implication is that its strongest use case is the setting in which one wants the continuity and flexibility of coordinate-based registration, but cannot tolerate the full cost and instability of denser KAN or MLP-based INR parameterizations [2509.22874].

Source: https://www.emergentmind.com/topics/randkan-idir