Polyakov Action on (ρ,G)-Equivariant Functions Application to Color Image Regularization

Abstract

We propose a new mathematical model for color images taking into account that color pixels change under transformation of the light source. For this, we deal with (ρ,G)-equivariant functions on principal bundles, where ρ is a representation of a Lie group G on the color space RGB. We present an application to image regularization, by minimization of the Polyakov action associated to the graph of such functions. We test the groups ${\rm I\!R}^{+\ast}$, DC(3) of contractions and dilatations of ${\rm I\!R}^3$ and SO(3) with their natural matrix representations, as well as ${\rm I\!R}^{+\ast}$ with its trivial representation. We show that the regularization has denoising properties if the representation is unitary and segmentation properties otherwise.