M-Net: Integrating Spectral Features and Physical Field Operators into Deep Learning for Medical Image Segmentation
Authors: Jing Zhu, Ye Wang, Fumin Wang
Organizations: School of Physics, Xi’an Jiaotong University, Xi’an 710049, China · University of Sydney, Sydney, NSW 2006, Australia
Abstract
Purpose: Deep learning-based medical image segmentation has achieved remarkable success, yet purely data-driven approaches often fail to exploit the rich mathematical structure inherent in medical images. We investigate whether explicit mathematical inductive biases, specifically matrix spectral analysis and vector calculus operators, can enhance segmentation beyond data-driven learning alone. Methods: We propose M-Net (Math-Augmented Network), which integrates three complementary mathematical priors into U-Net: (1) continuous spectral features derived from the condition number of centered local pixel matrices, providing a differentiable measure of texture ill-conditioning; (2) physical field operators (divergence and a discrete curl-like boundary irregularity operator) computed from image gradient fields, capturing focal intensity extrema and edge non-smoothness; and (3) a Math-Attention Gate (MAG) that adaptively fuses mathematical features with CNN-extracted deep features at skip connections. Results: Experiments on three benchmarks (LiTS, KiTS, and BraTS) show that M-Net achieves Dice scores of 78.42%, 76.15%, and 83.67%, outperforming baseline U-Net by 12.37%, 3.52%, and 5.55% on liver, kidney, and brain tumor segmentation, respectively. Ablations reveal that the condition-number feature contributes a 2.14% gain over binary invertibility features, while MAG adds 1.45% over simple concatenation. Conclusion: M-Net establishes that mathematical inductive biases provide effective complementary information for medical image segmentation. The continuous condition-number feature offers superior gradient information over discrete alternatives, and MAG preserves these priors throughout the network. This work opens avenues for integrating linear algebra and vector calculus into deep architectures for medical imaging.
This work demonstrates a full reproduction and extension of MNet, a hybrid 2D/3D convolutional network designed for anisotropic medical image segmentation. The original architecture was re-implemented within the nnU-Net framework to verify its reported performance and robustness to variable voxel spacing, known as anisotropy. Experiments were conducted on PROMISE prostate MRI and a controlled subset of LiTS liver CT under matched preprocessing and compute constraints. The reproduced MNet achieved a Dice similarity coefficient (DSC) of 89.0 +/- 0.9% on PROMISE, within 0.8% of the published result, and 94.3 +/- 1.9% / 54.6 +/- 3.1% for liver and tumor segmentation on LiTS, respectively. Two lightweight extensions were further introduced: (1) a learned Fusion Gating mechanism enabling adaptive 2D-3D feature blending, and (2) a VMamba state-space module for efficient long-range depth modelling. The Spatial Gating variant improved DSC by +0.8% with less than 3% inference overhead, while VMamba improved performance consistency, reducing PROMISE Dice variation to +/- 0.7% and achieving the strongest LiTS liver performance at 95.8% Dice. Both extensions preserved MNet robustness to anisotropy, with delta Dice = 1.5% across 1-4 mm voxel spacing. Overall, the study confirms MNet reproducibility and demonstrates that adaptive fusion and state-space modelling have the potential to further strengthen segmentation reliability under anisotropic conditions. However, further tests are required to provide definitive conclusions.
Deep learning has achieved remarkable performance in medical image segmentation, yet it suffers from critical limitations: mathematical intractability, substantial parameter requirements, and lack of clinical interpretability. We propose RadiomicNet, a novel two-stream hybrid architecture that enhances standard deep learning by integrating handcrafted radiomics features directly into the segmentation learning process. The key contribution is the Radiomics Attention Gate (RAG), which leverages Gray-Level Co-occurrence Matrix (GLCM) and Local Binary Pattern (LBP) features to modulate skip-connection attention in a lightweight MobileNetV2-based encoder-decoder, providing ante-hoc interpretability without post-hoc approximations. A novel Radiomics Consistency Loss further enforces alignment between texture complexity and prediction uncertainty, reducing Expected Calibration Error (ECE) from 0.142 to 0.118. RadiomicNet achieves a Dice Similarity Coefficient (DSC) of 0.763 +/- 0.231 on the Breast Ultrasound Images (BUSI) dataset and 0.854 +/- 0.112 on Kvasir-SEG, outperforming U-KAN by 1.2% and 1.8%, respectively (p < 0.05, Wilcoxon signed-rank test), with only 3.27M parameters, 9.5x fewer than standard U-Net and 4.3x fewer than U-KAN. Gradient-based feature importance analysis reveals that GLCM dissimilarity (15.24%), GLCM energy (14.56%), and LBP entropy (11.49%) are the dominant radiomics cues, providing clinically meaningful explanations for segmentation decisions. The proposed approach demonstrates that compact, interpretable models grounded in domain knowledge can deliver state-of-the-art segmentation performance with substantially reduced computational overhead.
Many dense prediction networks rely on additive feature transformations and model higher-order feature interactions only implicitly. Product units provide an explicit mechanism for multiplicative feature modeling, but their logarithmic--exponential formulation can cause numerical instability, which has limited their use in deep dense prediction networks. In this work, we propose Product-Unit U-Net (PU-UNet), a residual U-Net that integrates stable product-unit residual blocks into rich low-resolution stages for medical image segmentation. The proposed formulation combines smooth positivity mapping with log-domain clipping, enabling stable multiplicative feature learning with negligible computational overhead. On ISIC 2018, Kvasir-SEG, and BUSI, PU-UNet achieves Dice scores of 0.942, 0.959, and up to 0.925, respectively. Compared with a matched Residual U-Net baseline, PU-UNet consistently improves Dice and IoU while keeping parameters, FLOPs, and inference latency nearly unchanged, and reduces the image-level false-positive rate on normal BUSI cases from 0.077 to zero. Ablation studies suggest that the gains are associated with product-unit interactions, are strongest under low-resolution placement, and benefit from the proposed stabilization design. These results suggest that stable product-unit residual learning can be an effective way to enhance U-Net-style segmentation networks with explicit multiplicative interactions.