Singular Value Decomposition and Entropy Dimension of Fractals

TL;DR

This paper investigates the use of SVD entropy as a measure of fractal complexity, demonstrating its invariance under fractal iteration and its potential applications in communication and sensing fields.

Contribution

It introduces SVD entropy as a novel, invariant complexity measure for fractals, linking it to the Hausdorff dimension and applicable in various wave-based technologies.

Findings

SVD entropy remains invariant under fractal iteration.

SVD entropy distribution shows step-shaped and cluster patterns.

SVD entropy can characterize fractal wave phenomena.

Abstract

We analyze the singular value decomposition (SVD) and SVD entropy of Cantor fractals produced by the Kronecker product. Our primary results show that SVD entropy is a measure of image ``complexity dimension" that is invariant under the number of Kronecker-product self-iterations (i.e., fractal order). SVD entropy is therefore similar to the fractal Hausdorff complexity dimension but suitable for characterizing fractal wave phenomena. Our field-based normalization (Renyi entropy index = 1) illustrates the uncommon step-shaped and cluster-patterned distributions of the fractal singular values and their SVD entropy. As a modal measure of complexity, SVD entropy has uses for a variety of wireless communication, free-space optical, and remote sensing applications.