Maksim Alexandrovich Penkin

Medical image segmentation requires spatial transformations that remain effective across heterogeneous image statistics and boundary conditions. We introduce FunKAN, a KAN-inspired operator that acts directly on spatial feature maps: analytical Hermite basis maps are evaluated on an input-conditioned deformed grid, scored against learned channel-wise templates, mixed by a softmax over modes, and aggregated across channels. A Hilbert-space functional approximation result motivates the function-space viewpoint, while the exact finite FunKAN operator is treated separately and is not claimed to inherit a universal-approximation theorem. We embed FunKAN in a U-shaped architecture (U-FunKAN) and evaluate it under the validation-only selection protocol on BUSI, GlaS, and CVC-ClinicDB. The frozen model is tested once per seed after three-seed training. U-FunKAN improves mean test IoU over a same-protocol U-Net by 6.35 percentage points on BUSI and 1.85 points on CVC-ClinicDB, while GlaS performance is comparable, with U-FunKAN 0.50 points lower. U-FunKAN uses fewer parameters than the retrained U-Net (15.95 M vs. 31.03 M) but is substantially slower in measured GPU latency, so we do not claim universal computational efficiency. The results support dataset-dependent gains from functional spatial parameterization rather than uniform dominance.

Medical image enhancement and segmentation are critical yet challenging tasks in modern clinical practice, constrained by artifacts and complex anatomical variations. Traditional deep learning approaches often rely on complex architectures with limited interpretability. While Kolmogorov-Arnold networks offer interpretable solutions, their reliance on flattened feature representations fundamentally disrupts the intrinsic spatial structure of imaging data. To address this issue we propose a Functional Kolmogorov-Arnold Network (FunKAN) -- a novel interpretable neural framework, designed specifically for image processing, that formally generalizes the Kolmogorov-Arnold representation theorem onto functional spaces and learns inner functions using Fourier decomposition over the basis Hermite functions. We explore FunKAN on several medical image processing tasks, including Gibbs ringing suppression in magnetic resonance images, benchmarking on IXI dataset. We also propose U-FunKAN as state-of-the-art binary medical segmentation model with benchmarks on three medical datasets: BUSI (ultrasound images), GlaS (histological structures) and CVC-ClinicDB (colonoscopy videos), detecting breast cancer, glands and polyps, respectively. Experiments on those diverse datasets demonstrate that our approach outperforms other KAN-based backbones in both medical image enhancement (PSNR, TV) and segmentation (IoU, F1). Our work bridges the gap between theoretical function approximation and medical image analysis, offering a robust, interpretable solution for clinical applications.

A way to improve the quality of magnetic resonance image processing using the Kolmogorov–Arnold networks for deep feature filtering in the convolutional neural network is studied. Recently proposed Kolmogorov–Arnold networks are inspired by the representation theorem of the same name from real analysis and approximation theory. It states that every multivariate continuous function on a compact set can be represented as a superposition of continuous single-variable functions. However, the need to use gradient descent while training the network imposes restrictions on the parameterization of such single-variable functions—they must be at least differentiable; for this reason, in practice, they are searched in a linear span of B‑Splines or some other differentiable basis functions. In this study, we propose an adaptive method for the selection of basis functions by the model itself during training thus mitigating the rule of thumb choice of the basis functions. The method is based on the self-attention mechanism successfully used in state-of-the-art transformer networks. The proposed approach is tested on magnetic resonance image enhancement for images taken from IXI dataset; and it demonstrates better average PSNR and TV values over the synthetic testing dataset. Without loss of generality, the system of basis functions included B-splines, Chebyshev polynomials and Hermite functions.

Gibbs-ringing artifact is a common artifact in medical image processing. It is usually caused by image enhancement procedures, like image deblurring, or physically caused in magnetic resonance imaging (MRI), where the k-space is a finite sampling of underlying signals. Under-sampling in Fourier space is often employed to reduce imaging time for real-time applications. In addition, ringing effect is regularly accompanied by noise boosting due to a low-grade sensor, resulting in lowering image quality to a large extent. In this work we propose hybrid approach for suppressing the stated artifacts, which utilizes recently proposed deep learning Fourier Neural Operator over the classical solvers: Perona-Malik diffusion and Kellner algorithm.

Preparation of input data for an artificial neural network is a key step to achieve a high accuracy of its predictions. It is well known that convolutional neural models have low invariance to changes in the scale of input data. For instance, processing multiscale whole-slide histological images by convolutional neural networks naturally poses a problem of choosing an optimal processing scale. In this paper, this problem is solved by iterative analysis of distances to a separating hyperplane that are generated by a convolutional classifier at different input scales. The proposed method is tested on the DenseNet121 deep architecture pre-trained on PATH-DT-MSU data, which implements patch classification of whole-slide histological images.

Preparation of input data for an artificial neural network is a key step to achieve a high accuracy of its predictions. It is well known that convolutional neural models have low invariance to changes in the scale of input data. For instance, processing multiscale whole-slide histological images by convolutional neural networks naturally poses a problem of choosing an optimal processing scale. In this paper, this problem is solved by iterative analysis of distances to a separating hyperplane that are generated by a convolutional classifier at different input scales. The proposed method is tested on the DenseNet121 deep architecture pretrained on PATH-DT-MSU data, which implements patch classification of whole-slide histological images.

We present a novel method of blind image deblurring, capable of handling spatially variant blur caused by various reasons such as camera shake and object motions. Due to the spatially variant nature of the complex blur, our deblurring method includes both spatially variant and spatially invariant components. Namely, we utilize hybrid neural network consisting of a convolutional neural network and a spatially variant recurrent neural network trained in end-to-end manner. Recurrent neural network is used as a deconvolution operator performed on feature maps extracted from the input image by the convolutional neural network, while per-pixel weights of the recurrent neural network are also produced by the same convolutional neural network, making it solve both tasks at the same time. Such hybrid and multi-tasking design allows us to achieve both low computational complexity and high deblurring quality. Quantitative and qualitative evaluations on the widely used GoPro dataset demonstrate that the proposed method provides a good trade-off in terms of complexity and quality against state-of-the-art algorithms.

The paper deals with problem of optimal input scale search for deep classification neural networks. It is shown that state-of-the-art deep neural networks are not stable to input image scale, leading to quality degradation. The paper demonstrates relevance of the topic on classical image classification DL-pipeline. Unlike previous researchers, who aim to build entire complex invariant neural nets, we claim that computing optimal input transformations (e.g. scale) is a more perspective way for successful neural networks real-life applications. Thus, a new scale search algorithm for DL image classification is proposed in the paper, based on empirical hierarchical analysis of activation values.

Suppression of ringing artifacts in images is a well-known image restoration problem. Gibbs-ringing artifacts occur when, in the process of magnetic resonance imaging, the source data from the frequency domain are mapped onto the spatial domain by using the discrete Fourier transform. The artifacts are caused by the incompleteness of these data, which, in turn, is due to cutting off the high frequencies of the Fourier spectrum. In this paper, we propose a hybrid method for Gibbs-ringing artifact suppression in magnetic resonance images that combines a deep learning model and a classical non-machine-learning algorithm for Gibbs-ringing artifact suppression based on optimal subvoxel shifts.