Residuum-Condition Diagram and Reduction of Over-Complete Endmember-Sets

TL;DR

This paper introduces condition-residuum-diagrams to analyze and reduce over-complete endmember sets in hyperspectral unmixing, improving stability while maintaining low reconstruction error, and aiding in determining the optimal number of endmembers.

Contribution

The paper presents a novel visualization tool and a greedy reduction algorithm for over-complete endmember sets, enhancing stability without sacrificing unmixing accuracy.

Findings

Condition-residuum-diagrams provide insights into spectral unmixing behavior.

The reduction scheme improves set stability while preserving low reconstruction error.

Demonstrated benefits on multiple hyperspectral datasets.

Abstract

Extracting reference spectra, or endmembers (EMs) from a given multi- or hyperspectral image, as well as estimating the size of the EM set, plays an important role in multispectral image processing. In this paper, we present condition-residuum-diagrams. By plotting the residuum resulting from the unmixing and reconstruction and the condition number of various EM sets, the resulting diagram provides insight into the behavior of the spectral unmixing under a varying amount of endmembers (EMs). Furthermore, we utilize condition-residuum-diagrams to realize an EM reduction algorithm that starts with an initially extracted, over-complete EM set. An over-complete EM set commonly exhibits a good unmixing result, i.e. a lower reconstruction residuum, but due to its partial redundancy, the unmixing gets numerically unstable, i.e. the unmixed abundances values are less reliable. Our greedy reduction scheme improves the EM set by reducing the condition number, i.e. enhancing the set's stability, while keeping the reconstruction error as low as possible. The resulting set sequence gives hint to the optimal EM set and its size. We demonstrate the benefit of our condition-residuum-diagram and reduction scheme on well-studied datasets with known reference EM set sizes for several well-known EE algorithms.