Singular Value Decomposition and Entropy Dimension of Fractals

Xiaojing Weng, Altai Perry, Michael Maroun, Luat T. Vuong

20222022 International Conference on Image Processing, Computer Vision and Machine Learning (ICICML)15 citationsDOI

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.

Topics & Concepts

Fractal dimensionMathematicsDimension (graph theory)Singular value decompositionEntropy (arrow of time)FractalComputer scienceStatistical physicsPure mathematicsAlgorithmMathematical analysisPhysicsThermodynamicsMathematical Dynamics and FractalsChaos-based Image/Signal EncryptionChaos control and synchronization