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

Noaman Khan, Nihad Hadj Sahraoui, Samir Brahim Belhaouari

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…