U-FunKAN: Binary Med Image Segmentation

• U-FunKAN is a binary segmentation model that integrates functional Kolmogorov-Arnold layers with a U-Net–style encoder–decoder to preserve 2D spatial structures.
• The architecture employs Fourier-Hermite parameterization to project functional mappings, yielding interpretable spectral weights and dynamic spatial offsets.
• Empirical results on BUSI, GlaS, and CVC-ClinicDB show that U-FunKAN outperforms conventional models in IoU and F1, while maintaining efficiency for clinical use.

U-FunKAN is a binary medical image segmentation architecture introduced within the broader FunKAN framework in "FunKAN: Functional Kolmogorov-Arnold Network for Medical Image Enhancement and Segmentation" (Penkin et al., 16 Sep 2025). It embeds functional Kolmogorov-Arnold layers into a U-Net–style encoder–decoder and is motivated by a specific limitation of prior KAN-based image models: the reliance on flattened feature representations, which the paper argues fundamentally disrupts the intrinsic spatial structure of imaging data. Within that framework, U-FunKAN is presented as a state-of-the-art binary medical segmentation model and is benchmarked on BUSI, GlaS, and CVC-ClinicDB, targeting breast tumors, glands, and polyps, respectively (Penkin et al., 16 Sep 2025).

1. Position within the FunKAN framework

The FunKAN paper addresses two medical image processing problems: image enhancement and segmentation. FunKAN is the general functional Kolmogorov-Arnold architecture proposed for image processing, while U-FunKAN is the segmentation-specific instantiation obtained by embedding functional layers into a U-Net–style encoder–decoder. The paper therefore treats U-FunKAN not as an independent theoretical construct, but as the architectural realization of the FunKAN formalism for binary medical segmentation (Penkin et al., 16 Sep 2025).

The motivating contrast is explicit. Traditional deep learning approaches are described as often relying on complex architectures with limited interpretability, whereas conventional KAN formulations are described as interpretable but structurally mismatched to imaging data because they flatten spatially organized inputs. U-FunKAN is designed to preserve intrinsic $2$D image structure while retaining the interpretability associated with Kolmogorov-Arnold decompositions. This suggests that its central novelty lies less in replacing encoder–decoder segmentation practice than in redefining the internal function class used at the bottleneck.

The paper situates U-FunKAN alongside both classical U-Net variants and KAN-based backbones. In the reported segmentation experiments, the comparison set includes U-Net, Att-U-Net, U-Net++, U-Mamba, and U-KAN. The stated comparative target is therefore broad: conventional CNN segmentation models, attention-augmented variants, state-space-inspired segmentation, and earlier KAN-based designs.

2. Functional Kolmogorov-Arnold formulation

The theoretical basis begins with the classical Kolmogorov-Arnold representation theorem for a continuous multivariate function $f:[0,1]^n\to\mathbb{R}$, stated as

$$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$

where each inner map $\phi_{ji}:[0,1]\to\mathbb{R}$ and outer map $\zeta_j:\mathbb{R}\to\mathbb{R}$ is continuous. U-FunKAN is built on the paper’s hypothesized formal generalization of this representation to continuous functionals on $H^n$, where $H$ is a separable Hilbert space, such as an $L^2$-space of images (Penkin et al., 16 Sep 2025).

That functional extension is stated informally as

$$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$

where $\varphi_{ji}:H\to\mathbb{R}$ are continuous linear functionals and $f:[0,1]^n\to\mathbb{R}$0 are continuous outer functions. By the Riesz representation theorem, each $f:[0,1]^n\to\mathbb{R}$1 is identified with an inner product:

$f:[0,1]^n\to\mathbb{R}$2

In the deep-network construction, these approximations are layered as

$f:[0,1]^n\to\mathbb{R}$3

with

$f:[0,1]^n\to\mathbb{R}$4

The consequence for segmentation is structural. Instead of treating image features as flattened coordinate vectors, U-FunKAN treats them as elements of a function space and parameterizes cross-channel transformations through continuous linear functionals. A common point of confusion is to assimilate U-FunKAN to a conventional KAN applied to image tensors; the paper’s explicit claim is the opposite, namely that the model is defined by extending Kolmogorov-Arnold representations onto functional spaces rather than by reusing flattened feature representations.

3. Fourier-Hermite parameterization and interpretable inner mappings

To make the functional maps trainable, U-FunKAN projects each $f:[0,1]^n\to\mathbb{R}$5 onto the first $f:[0,1]^n\to\mathbb{R}$6 Hermite basis functions $f:[0,1]^n\to\mathbb{R}$7. The paper gives the approximation

$f:[0,1]^n\to\mathbb{R}$8

which induces

$f:[0,1]^n\to\mathbb{R}$9

The trainable coefficients $$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$0 are stored in an attention matrix $$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$1, described as directly interpretable because row $$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$2 contains the spectral weights used to map channel $$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$3 into the next layer (Penkin et al., 16 Sep 2025).

This spectral construction is a defining property of U-FunKAN. The inner mapping is expressed through a small number of Hermite modes, and in the segmentation instantiation the bottleneck uses $$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$4 Hermite bases. The paper describes this as spectral interpretability: every FunKAN block’s inner mapping is expressed as a small number of modes, enabling post-hoc analysis of which spatial frequencies each channel uses. A plausible implication is that interpretability is attached not merely to attention coefficients in an abstract sense, but specifically to a basis expansion over a known functional basis.

The architecture also includes spatial adaptivity through learned offset fields $$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$5. These offsets deform the sampling grid from $$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$6 to $$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$7, and the paper states that the resulting fields can be visualized to show where the network “looks” hardest, for example along edges. This combines spectral filtering with a deformation mechanism rather than treating them as separate design choices.

4. Encoder-decoder realization

U-FunKAN embeds the functional layers into a standard U-Net–style encoder–decoder. The input $$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$8 has shape $$f(x_1,\dots,x_n)=\sum_{j=1}^{2n+1}\zeta_j\left(\sum_{i=1}^n \phi_{ji}(x_i)\right),$$9 or $\phi_{ji}:[0,1]\to\mathbb{R}$0, followed by an embedding convolution $\phi_{ji}:[0,1]\to\mathbb{R}$1 channels. The encoder then comprises four lifting stages with channel sizes $\phi_{ji}:[0,1]\to\mathbb{R}$2, $\phi_{ji}:[0,1]\to\mathbb{R}$3, $\phi_{ji}:[0,1]\to\mathbb{R}$4, and $\phi_{ji}:[0,1]\to\mathbb{R}$5. At each stage, the paper specifies a strided $\phi_{ji}:[0,1]\to\mathbb{R}$6 convolution with ReLU pre-activation for downsampling to $\phi_{ji}:[0,1]\to\mathbb{R}$7, together with a residual connection in the corresponding channel space (Penkin et al., 16 Sep 2025).

The bottleneck is the architectural locus of the FunKAN machinery. It consists of a sequence of three FunKAN blocks, each with $\phi_{ji}:[0,1]\to\mathbb{R}$8 channels and $\phi_{ji}:[0,1]\to\mathbb{R}$9 Hermite bases. For each block, a small residual network computes dynamic offsets $\zeta_j:\mathbb{R}\to\mathbb{R}$0, the sampling grid is deformed to $\zeta_j:\mathbb{R}\to\mathbb{R}$1, and each output channel $\zeta_j:\mathbb{R}\to\mathbb{R}$2 is computed by the functional spectral expansion across the $\zeta_j:\mathbb{R}\to\mathbb{R}$3 inner channels. A skip connection from the input of the FunKAN block is added in pre-activation form.

The decoder mirrors the encoder. For stages $\zeta_j:\mathbb{R}\to\mathbb{R}$4, it applies nearest-neighbour $\zeta_j:\mathbb{R}\to\mathbb{R}$5 upsampling, a $\zeta_j:\mathbb{R}\to\mathbb{R}$6 convolution with ReLU pre-activation to recover $\zeta_j:\mathbb{R}\to\mathbb{R}$7 channels, and a skip connection from the corresponding encoder stage. Restoration is performed by a $\zeta_j:\mathbb{R}\to\mathbb{R}$8 convolution producing $\zeta_j:\mathbb{R}\to\mathbb{R}$9 or $H^n$0 output logits. Each FunKAN block preserves spatial resolution $H^n$1 across its $H^n$2 channels, replacing the usual convolution-plus-nonlinearity pattern with the functional spectral expansion.

Architecturally, this means that U-FunKAN retains a familiar multi-scale segmentation scaffold while relocating most of the theoretical innovation to the bottleneck. This suggests a deliberate division of labor: conventional hierarchical feature aggregation in the encoder–decoder and functional approximation in the latent stage.

5. Optimization protocol and evaluation datasets

For segmentation, the paper defines the loss as

$H^n$3

where $H^n$4 is binary cross-entropy and $H^n$5 is the standard soft-Dice loss. Optimization uses Adam with $H^n$6, $H^n$7, and $H^n$8. The learning-rate schedule is manual: epochs $H^n$9 at $H$0, reduction to $H$1 on plateau, then $H$2 until convergence. The batch size is $H$3. Data augmentation on segmentation tasks consists of random horizontal and vertical flips, $H$4 rotations, and transpositions, each with $H$5, while BatchNorm in residual and offset blocks is used to ensure stable gradients (Penkin et al., 16 Sep 2025).

The segmentation benchmarks are three binary medical datasets.

Dataset	Samples	Target
BUSI	780 images total; 647 with annotated benign/malignant masks $H$6	Tumor region
GlaS	165 H&E stained RGB images $H$7	Glandular structures
CVC-ClinicDB	612 frames $H$8	Pixel-wise polyp annotation

For GlaS, the split is $H$9 train/test with seed $L^2$0. For CVC-ClinicDB, the split is $L^2$1 randomized train/test. In all cases, images are resized to uniform resolution, normalized to $L^2$2, and augmented as specified above. The reported metrics are Intersection-over-Union and F1 score.

The dataset composition matters because it spans ultrasound, histology, and endoscopy. The paper explicitly frames this as cross-modality evaluation, and a plausible implication is that the architecture is intended to test whether functional spectral mappings remain stable across markedly different image statistics and anatomical textures.

6. Empirical performance, interpretability, and clinical relevance

The quantitative comparison reports the following U-FunKAN scores: on BUSI, $L^2$3 IoU and $L^2$4 F1; on GlaS, $L^2$5 IoU and $L^2$6 F1; on CVC-ClinicDB, $L^2$7 IoU and $L^2$8 F1 (Penkin et al., 16 Sep 2025). The paper states that U-FunKAN achieves the highest IoU on all three datasets, with particularly strong gains on BUSI $L^2$9, and that in GlaS it also attains the best F1. The comparative table shows that the CVC F1 is not the highest reported F1 across all methods, but the IoU remains the best in that benchmark.

The baseline values contextualize those improvements. On BUSI, U-KAN reports $$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$0 IoU and $$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$1 F1, while U-Mamba reports $$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$2 IoU and $$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$3 F1. On GlaS, U-KAN reports $$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$4 IoU and $$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$5 F1. On CVC-ClinicDB, U-KAN reports $$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$6 IoU and $$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$7 F1. The paper further states that, compared to prior KAN-based backbones, specifically U-KAN and UKAGNet, U-FunKAN is both more accurate and more efficient, with $$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$8 Gflops and $$f(\chi_1,\dots,\chi_n),\ \chi_i\in H \approx \sum_{j=1}^m \zeta_j\left(\sum_{i=1}^n \varphi_{ji}(\chi_i)\right),$$9 M parameters.

The qualitative findings are described in anatomical terms. On BUSI, irregular tumor margins are captured without the spill-over common in plain U-Nets. In GlaS, gland lumina and epithelial borders are sharply segmented even in low-contrast regions. For CVC-ClinicDB, polyps under specular highlights and bleeding artifacts remain correctly outlined. These observations are consistent with the paper’s emphasis on boundary delineation and robustness to artifacts rather than merely aggregate overlap scores.

Interpretability is presented along two axes. First, the attention matrices $\varphi_{ji}:H\to\mathbb{R}$0 expose spectral weights over Hermite modes, which the paper describes as revealing which spatial frequencies each channel uses. Second, the offset fields $\varphi_{ji}:H\to\mathbb{R}$1 provide a visualizable account of spatial adaptivity. Clinical relevance is then argued from architecture and efficiency together: by preserving intrinsic $\varphi_{ji}:H\to\mathbb{R}$2D structure, combining multi-scale skip connections with spectral filtering, and operating with approximately $\varphi_{ji}:H\to\mathbb{R}$3 M parameters and $\varphi_{ji}:H\to\mathbb{R}$4 Gflops per $\varphi_{ji}:H\to\mathbb{R}$5 image, the model is described as suitable for real-time execution on modern GPUs and even on certain CPU-accelerators. The paper further states that U-FunKAN handles variable anatomy, image artifacts such as noise and low contrast, and modality shifts across ultrasound, histology, and endoscopy without retraining. This suggests that the authors regard the method not only as an interpretable segmentation backbone, but also as a deployment-oriented model whose functional formulation is intended to remain stable across heterogeneous clinical image domains.