Fractal Cross Product: Theory, Differentiable Implementation and Application to Medical Image Analysis

Authors: Noaman Khan, Nihad Hadj Sahraoui, Samir Brahim Belhaouari

arXiv: 2610.03755 [cs.CV]

Submitted: 26 September 2026

Subject: Computer Science > Computer Vision and Pattern Recognition (cs.CV)

Abstract

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. In this work, 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. The formulation recovers the classical generalized cross product exactly at D = k and retains orthogonality, alternation, rotation equivariance, and D-homogeneity, though it 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 (AUROC) 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.

Key Contributions

Theoretical extension of the cross product: The FCP generalizes the classical Euclidean cross product to non-integer homogeneity degrees via a nonlinear radial deformation, while preserving orthogonality, alternation, rotation equivariance, and homogeneity.
Dimension-agnostic scalar construction: The scalar form applies in any ambient dimension m ≥ k, with the oriented vector form requiring codimension one.
Self-similarity law: For exact self-similar frame systems, the construction obeys a scale-balance law anchored to the similarity dimension.
Differentiable image response: A dimensionless, differentiable formulation is derived that enables gradient-based learning and geometric descriptor extraction.
Empirical validation: Numerical experiments recover finite-frame identities to machine precision (Binary64), and raster experiments recover the Sierpiński-triangle fractal dimension of 1.5849625 across three resolutions.
Medical imaging benchmarks: FCP-centered fusion improves AUROC and balanced accuracy on retinal fundus and bone-fracture datasets, with particularly notable gains on the hospital-separated partition of the Retinal Image Database for Optic Nerve Evaluation.

Experimental Results

|---|---|---|---|

| Retinal Image Database for Optic Nerve Evaluation (random partition) | AUROC | 0.7264 | 0.8135 |

| Retinal Image Database for Optic Nerve Evaluation (hospital-separated partition) | AUROC | 0.5843 | 0.6765 |

| FracAtlas | AUROC | 0.7403 | 0.7475 |

| FracAtlas | Balanced accuracy | 0.6186 | 0.6721 |

| FracAtlas | Harmonic mean of precision and sensitivity | 0.3669 | 0.4457 |

All comparisons use a five-seed protocol.

Interpretation and Limitations

The reported gains support the utility of the complete fusion framework combining FCP-based descriptors with complementary features and a fusion head. However, the experiments do not isolate the effect of the FCP component alone; the observed improvements should be attributed to the full pipeline rather than to the FCP construction in isolation. Future work should include controlled ablations to disentangle the contributions of the FCP, auxiliary descriptors, and the fusion head.

Citation

@article{khan2026fractal,
  title={Fractal Cross Product: Theory, Differentiable Implementation and Application to Medical Image Analysis},
  author={Khan, Noaman and Hadj Sahraoui, Nihad and Belhaouari, Samir Brahim},
  journal={arXiv preprint arXiv:2610.03755},
  year={2026}
}

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