Adaptive Method for Selecting Basis Functions in Kolmogorov–Arnold Networks for Magnetic Resonance Image Enhancement

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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, further gradient descent application imposes restrictions on the inner Kolmogorov functions to be at least differentiable, that’s why, 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 of basis functions selection by the model itself during training, mitigating the rule of thumb choice of that basis functions. The method is based on the attention mechanism, successfully used in state-of-the-art transformers. The proposed approach is tested on magnetic resonance images enhancement on IXI dataset and demonstrates the best average PSNR and TV over the synthetic testing dataset. Without loss of generality, the system of basis functions included: B-splines, Chebyshev polynomials and Hermite functions.

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作者简介

M. Penkin

Faculty of Computational Mathematics and Cybernetics, Moscow State University

编辑信件的主要联系方式.
Email: penkin97@gmail.com
俄罗斯联邦, Moscow, 119991

A. Krylov

Faculty of Computational Mathematics and Cybernetics, Moscow State University

Email: kryl@cs.msu.ru

Laboratory of Mathematical Methods of Image Processing

俄罗斯联邦, Moscow, 119991

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2. Fig. 1. (a) – T1-weighted MRI image of the brain in low resolution (64 × 64), (b) – the result of the MRI image quality improvement algorithm (suppression of Gibbs oscillations, blur and noise), (c) – Gibbs oscillations on a step function simulating a sharp contour in the image.

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3. Fig. 2. Diagram of the Kolmogorov–Arnold network.

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4. Fig. 3. Scheme of the proposed architecture with adaptive selection of basis functions of Kolmogorov–Arnold networks (MBA–KAN).

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5. Fig. 4. Scatterplot of mean PSNR and total variation values for 2617 test images depending on the chosen architecture of the deep feature description filter of the convolutional neural network.

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