Riemannian information gradient methods for the parameter estimation of ECD: Some applications in image processing

TL;DR

This paper introduces a Riemannian optimization method called the information stochastic gradient (ISG) for efficient parameter estimation of elliptically-contoured distributions (ECD) in large-scale image processing tasks, outperforming traditional fixed-point methods.

Contribution

The paper develops the ISG method based on the Riemannian information gradient, enabling faster and more memory-efficient ECD parameter estimation for large datasets, with applications in image re-colorization and texture classification.

Findings

ISG achieves comparable accuracy to MLE with less computational time.

The offline IDG method is effective for moderate-sized datasets.

Numerical simulations demonstrate the methods' superior performance in practical applications.

Abstract

Elliptically-contoured distributions (ECD) play a significant role, in computer vision, image processing, radar, and biomedical signal processing. Maximum likelihood. estimation (MLE) of ECD leads to a system of non-linear equations, most-often addressed using fixed-point (FP) methods. Unfortunately, the computation time required for these methods is unacceptably long, for large-scale or high-dimensional datasets. To overcome this difficulty, the present work introduces a Riemannian optimisation method, the information stochastic gradient (ISG). The ISG is an online (recursive) method, which achieves the same performance as MLE, for large-scale datasets, while requiring modest memory and time resources. To develop the ISG method, the Riemannian information gradient is derived taking into account the product manifold associated to the underlying parameter space of the ECD. From this information gradient definition, we define also, the information deterministic gradient (IDG), an offline (batch) method, which is an alternative, for moderate-sized datasets. The present work formulates these two methods, and demonstrates their performance through numerical simulations. Two applications, to image re-colorization, and to texture classification, are also worked out.