Beyond holography: the entropic quantum gravity foundations of image processing

TL;DR

Here it is demonstrated that the famous Perona-Malik algorithm for image processing is the gradient flow that maximizes the GfE action in its simple warm-up scenario and provides the geometrical and information theory foundations for the Perona-Malik algorithm.

Abstract

Recently, thanks to the development of artificial intelligence (AI) there is increasing scientific attention in establishing the connections between theoretical physics and AI. Traditionally, these connections have been focusing mostly on the relation between string theory and image processing and involve important theoretical paradigms such as holography. Recently G. Bianconi has formulated the Gravity from Entropy (GfE) approach to quantum gravity in which gravity is derived from the geometric quantum relative entropy (GQRE) between two metrics associated with the Lorentzian spacetime. Here it is demonstrated that the famous Perona-Malik algorithm for image processing is the gradient flow that maximizes the GfE action in its simple warm-up scenario. Specifically, this algorithm is the outcome of the maximization of the GfE action calculated between two Euclidean metrics: the one of the support of the image and the one induced by the image. As the Perona-Malik algorithm is known to preserve sharp contours, this implies that the GfE action, does not in general lead to uniform images upon iteration of the gradient flow dynamics as it would be intuitively expected from entropic actions maximising classical entropies. Rather, the outcome of the maximization of the GfE action is compatible with the preservation of complex structures. These results provide the geometrical and information theory foundations for the Perona-Malik algorithm and might contribute to establish deeper connections between GfE, machine learning and brain research.

Figures (1)

• Figure 1: Schematic representation of the metrics associated to a black-and-white image and the GQRE Lagrangian of the GfE action which provides the foundations of the Perona-Malik algorithm. The flat $2D$ Euclidean manifold $\Omega$ (in green) that offers the support of the image, has points ${\bf r}$ of coordinates ${\bf r}=(x_1,x_2)$ and an Euclidean metric $g_{\mu\nu}=\eta_{\mu\nu}$. The infinitesimal distance $ds$ between points in $\Omega$ is defined by this metric and obeys $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$. The image is associated to the surface $\mathcal{K}$ (in orange) embedded in a flat 3D Euclidean metric,thus every point of $\mathcal{K}$ has coordinates $({\bf r}, \phi({\bf r}))$. The infinitesimal distance $d\hat{s}$ between two points in this surface obeys $d\hat{s}^2=G_{\mu\nu}dx^{\mu}dx^{\nu}$ where $G_{\mu\nu}$, given by Eq.(\ref{['induced']}), is the symmetric rank $2$ tensor that defines the metric induced by the surface $\mathcal{K}$ on the $2D$ flat support of the image $\Omega$. The GfE action, associated to the Lagrangian $\mathcal{L}$, is given by the GQRE between the metrics ${\bf G}$ and $g$. In this work we show that the Perona-Malik algorithm is the gradient flow that maximizes the GfE action. Therefore the Perona-Malik algorithm emerges from the principle of maximization of the GfE action calculated between the metric induced by the image ${\bf G}_{\mu\nu}$ and the metric $g_{\mu\nu}$ of the 2D image support.