Endmember Extraction on the Grassmannian

TL;DR

This paper introduces a novel algorithm for extracting endmembers represented as subspaces on the Grassmannian, improving analysis of high-dimensional data like hyperspectral images by identifying boundary subspaces.

Contribution

The paper presents a new method for endmember extraction on the Grassmannian, addressing noise and boundary detection in high-dimensional subspace data.

Findings

Successfully applied to synthetic data demonstrating boundary detection.

Effective on AVIRIS Indian Pines hyperspectral image.

Algorithm outperforms traditional point-based methods.

Abstract

Endmember extraction plays a prominent role in a variety of data analysis problems as endmembers often correspond to data representing the purest or best representative of some feature. Identifying endmembers then can be useful for further identification and classification tasks. In settings with high-dimensional data, such as hyperspectral imagery, it can be useful to consider endmembers that are subspaces as they are capable of capturing a wider range of variations of a signature. The endmember extraction problem in this setting thus translates to finding the vertices of the convex hull of a set of points on a Grassmannian. In the presence of noise, it can be less clear whether a point should be considered a vertex. In this paper, we propose an algorithm to extract endmembers on a Grassmannian, identify subspaces of interest that lie near the boundary of a convex hull, and demonstrate the use of the algorithm on a synthetic example and on the 220 spectral band AVIRIS Indian Pines hyperspectral image.