发表机构
College of Science and Engineering, Hamad Bin Khalifa University（哈马德·本·哈利法大学科学与工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出广义分形叉积（FCP），一种具有非整数次数的非线性径向变形，保留经典叉积性质，并在医学图像融合中显著提升ROC-AUC等指标，验证其有效性。
AI 中文摘要
广义欧几里得叉积的幅值是一个Gram体积，其常见缩放下的次数由张成框架的整数维数决定。我们将广义分形叉积（FCP）定义为具有指定正次数$D$（可能为非整数）的非线性径向变形。标量构造适用于任意环境维度$m\geq k$，而其规范定向的向量形式要求余维数为1。当$D=k$时，它精确地恢复经典广义叉积，并保留正交性、交错性、旋转等变性和$D$-齐次性，但通常不是多重线性的。对于精确自相似的框架系统，该构造还满足相似维数下的尺度平衡定律。我们推导出一个可微的、无量纲的图像响应，以及通过跨斑块宽度回归原始角Gram响应获得的非圆形经验累积指数。Binary64计算将有限框架恒等式恢复至舍入误差，而光栅实验在三个分辨率下恢复了谢尔宾斯基三角形值1.5849625。在五次种子的医学图像比较中，以FCP为中心的融合将随机和医院分离的视神经评估视网膜图像数据库分区上的平均受试者工作特征曲线下面积分别从0.7264提高到0.8135，从0.5843提高到0.6765，在FracAtlas上从0.7403提高到0.7475。在FracAtlas上，平衡准确率从0.6186提高到0.6721，精确率和灵敏度的调和平均数从0.3669提高到0.4457。这些结果支持完整融合框架的有效性，但并未将FCP的效果与其互补描述符和融合头的影响分离开来。
英文摘要
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 $m\geq k$, 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.
