**5. Discussion**

## *5.1. Sensitivity Test*

We evaluated the performance of the proposed J-LASU algorithm when *λ*, *γ*, and *ρ* were not set to the optimal values. In the experiment, when a parameter was adjusted from 0 to 10, the other parameters were set to their optimal values. When the parameter increases from 0 to the optimal value, the RMSEs decrease and the SREs increase gradually. When it reaches a higher value, the results worsen. Hence, we can conclude that each parameter influences the performance of J-LASU.

To clearly evaluate the contribution of the LA regularizer, we conducted an experiment of our optimization problem with *ρ* = 0, which means no contribution of the LA regularizer. Figure 11 represent the RMSE of this condition at the three levels of SNR compared to those of J-LASU, where *ρ* > 0. For each simulated data set, other parameters were set to the optimum values. For all data, it was observed that when *ρ* = 0, the RMSE was higher than the condition when the LA regularizer was used. In other words, adding our LA regularizer with an optimal regularization parameter will contribute improvement in RMSE.

We found that improvement in visual quality corresponds to the additional low-rank regularization. Figure 12 shows visual improvement due to the abundance regularizer. The abundance maps in the figure belong to endmember 5 of the *DS* data set and endmember 7 of the *FR 2* data set. For the *FR* abundance maps, one can see that after applying our LA regularizer with an optimal *ρ*, the active abundances have higher intensities. The active abundances in the left-edge of the map and around the speckles clearly appear, although in lower intensities than in the true abundance map. For the *DS* data set, when *ρ* is set to the optimal value, the small squares are preserved better than when *ρ* = 0.

**Figure 11.** Effect of the LA regularizer represented by improvement in RMSE when *ρ* > 0.

**Figure 12.** Effect of LA regularizer represented by improvement in RMSE when *ρ* > 0 for (**a**) *DS* data set and (**b**) *FR 2* data set. (**a1**) and (**b1**) Before, (**a2**) and (**b2**) after, (**a3**) and (**b3**) true abundance.

#### *5.2. Effect of Block Size*

The coverage of the local region affects the optimization results. In this region, the highly correlated abundance of the endmembers is taken into account by the local abundance nuclear norm. We conducted experiments to find the optimum size of the sliding block. We also observed the effect of the block size. Figure 13 shows the RMSE and SRE when the block size was adjusted in the *DS* data set. From the curves, we could determine that the radius of spatial similarity in the abundance map affects the optimum size of the sliding block. The distribution of spatial similarity in the *DS* data set, as shown in Figure 8, has a distinct pattern in which every 5 × 5 pixel has the same abundance value, giving the optimum block size in turn. However, the correlation does not hold for the data in which the spatial similarity is not represented in a square region, e.g., the *FR* and Cuprite data sets. In this circumstance, some trials were conducted prior to the experiment. After the trials, we found that the optimum size is 5 pixels. Hence, we selected [5 5 5] as the optimum block size for all data.

**Figure 13.** RMSE and SRE in relation to block size.
