The magnitude of the generalized Euclidean cross product is a Gram volume whose degree under common scaling is fixed by the integer dimension of the spanning frame. We formulate a generalized Fractal Cross Product (FCP) as a nonlinear radial deformation with a prescribed positive degree D, which may be non-integer. The scalar construction applies in any ambient dimension m≥k, while its canonically oriented vector form requires codimension one. It recovers the classical generalized cross product exactly at D=k and retains orthogonality, alternation, rotation equivariance, and D-homogeneity, but is generally not multilinear. For exact self-similar frame systems, the construction also obeys a scale-balance law at the similarity dimension. We derive a differentiable, dimensionless image response and a non-circular empirical accumulation exponent obtained by regressing raw angular Gram responses across patch widths. Binary64 calculations recover the finite-frame identities to roundoff, while raster experiments recover the Sierpiński-triangle value 1.5849625 at three resolutions. In five-seed medical-imaging comparisons, FCP-centered fusion increased mean area under the receiver operating characteristic curve from 0.7264 to 0.8135 and from 0.5843 to 0.6765 on the random and hospital-separated Retinal Image Database for Optic Nerve Evaluation partitions, respectively, and from 0.7403 to 0.7475 on FracAtlas. On FracAtlas, balanced accuracy increased from 0.6186 to 0.6721 and the harmonic mean of precision and sensitivity from 0.3669 to 0.4457. These results support the utility of the complete fusion framework, but do not isolate the effect of FCP from that of its complementary descriptors and fusion head.
Deep learning (DL) methods are currently being explored to restore images from sparse-view-, limited-data-, and undersampled-based acquisitions in medical applications. Although outputs from DL may appear visually appealing based on likability/subjective criteria (such as less noise, smooth features), they may also suffer from hallucinations. This issue is further exacerbated by a lack of easy-to-use techniques and robust metrics for the identification of hallucinations in DL outputs. In this work, we propose performing Fourier Ring Correlation (FRC) analysis over small patches and concomitantly (s)canning across DL outputs and their reference counterparts to detect hallucinations (termed as sFRC). We describe the rationale behind sFRC and provide its mathematical formulation. The parameters essential to sFRC may be set using predefined hallucinated features annotated by subject matter experts or using imaging theory-based hallucination maps. We use sFRC to detect hallucinations for three undersampled medical imaging problems: CT super-resolution, CT sparse view, and MRI subsampled restoration. In the testing phase, we demonstrate sFRC's effectiveness in detecting hallucinated features for the CT problem and sFRC's agreement with imaging theory-based outputs on hallucinated feature maps for the MR problem. Finally, we quantify the hallucination rates of DL methods on in-distribution versus out-of-distribution data and under increasing subsampling rates to characterize the robustness of DL methods. Beyond DL-based methods, sFRC's effectiveness in detecting hallucinations for a conventional regularization-based restoration method and a state-of-the-art unrolled method is also shown.
Prabhat Kc, Rongping Zeng, Nirmal Soni +1
Center for Devices and Radiological Health in FDA, Silver Spring, MD, 20993, USA
Focal cortical dysplasia (FCD) type II is an important structural cause of drug-resistant focal epilepsy, but its small size, heterogeneous appearance, and subtle MRI characteristics make automated segmentation challenging. Conventional multimodal networks commonly concatenate T1-weighted (T1w) and fluid-attenuated inversion recovery (FLAIR) images, requiring subsequent layers to learn useful cross-modal relationships implicitly. We propose CRIL-U-Net, a 3D U-Net incorporating a Compact Ratio-Interaction Learning module that combines local spatial features, voxel-wise cross-modal mixing, and bidirectional ratio-inspired interactions. CRIL-U-Net was compared with a conventional 3D U-Net and an input self-attention U-Net using five-fold cross-validation on 85 FCD subjects and 25 healthy controls. Each architecture was trained independently using Dice-binary cross-entropy (Dice-BCE) and Focal Tversky-Focal (FTF) losses. With FTF, CRIL-U-Net achieved the highest mean Dice score (0.196 +/- 0.262), compared with 0.136 +/- 0.224 for the U-Net and 0.135 +/- 0.214 for the attention comparator. It produced nonzero lesion overlap in 44 of 85 cases, compared with 36 for the U-Net. Under FTF, CRIL-U-Net significantly outperformed both comparison architectures after false-discovery-rate correction. These findings suggest that compact cross-modal representation learning can improve FCD segmentation within a controlled U-Net setting when combined with an imbalance-aware objective, although the remaining zero-overlap rate of 48.2% highlights the need for further validation and methodological development.
Soumen Ghosh, Amit Soni Arya, Tilottama Goswami +3
The University of Queensland Brisbane, Australia · Bennett University Greater Noida, India · University College of Engineering, Osmania University Hyderabad, India +2
Multi-modal medical image segmentation leverages complementary diagnostic information, yet fusion can underperform single-modality baselines when spatially aligned inputs differ in quality. Here, "corruption" primarily denotes resolution-induced degradation rather than misalignment or complete modality absence, while synthetic noise is evaluated only as an auxiliary setting. We identify a critical optimization-inference inconsistency: degraded modalities can receive weak training updates yet substantially affect predictions, indicating active interference with fusion. We attribute this failure to resampling-induced feature corruption and optimization bias, where noisy features propagate through skip connections and encourage unreliable modality selection. We therefore propose CoReFuse-Med, a Corruption-aware Rebalanced Fusion framework that suppresses corruption during feature transmission and rebalances modality contributions during high-level fusion. Experiments on EPVS, BraTS, and WMH, including multiple Z-axis slice-retention ratios and an auxiliary noise test, demonstrate improved accuracy and robustness under modality-quality discrepancies. Our code is available at https://github.com/lrever/CoReFuse.
Yuchen Pei, Xiaoyu Hu, Yixiong Zou +5
Central China Normal University Wuhan, China · Huazhong University of Science and Technology Wuhan, China · Guangzhou University of Chinese Medicine Guangzhou, China +2