*Article* **Joint Local Abundance Sparse Unmixing for Hyperspectral Images**

**Mia Rizkinia 1,2,†,\* and Masahiro Okuda 1,†**


Received: 20 October 2017; Accepted: 22 November 2017; Published: 27 November 2017

**Abstract:** Sparse unmixing is widely used for hyperspectral imagery to estimate the optimal fraction (abundance) of materials contained in mixed pixels (endmembers) of a hyperspectral scene, by considering the abundance sparsity. This abundance has a unique property, i.e., high spatial correlation in local regions. This is due to the fact that the endmembers existing in the region are highly correlated. This implies the low-rankness of the abundance in terms of the endmember. From this prior knowledge, it is expected that considering the low-rank local abundance to the sparse unmixing problem improves estimation performance. In this study, we propose an algorithm that exploits the low-rank local abundance by applying the nuclear norm to the abundance matrix for local regions of spatial and abundance domains. In our optimization problem, the local abundance regularizer is collaborated with the *<sup>L</sup>*2,1 norm and the total variation for sparsity and spatial information, respectively. We conducted experiments for real and simulated hyperspectral data sets assuming with and without the presence of pure pixels. The experiments showed that our algorithm yields competitive results and performs better than the conventional algorithms.

**Keywords:** sparse unmixing; hyperspectral; local abundance; nuclear norm
