2.2.2. *L*2-NMF

Due to the needs of application, the *L*2 constraint can be adopted to generate smooth results other than sparse results. For areas in hyperspectral images that are evenly mixed with signatures, we also need the *L*2 regularizer to promote evenness in the abundances of pixels in these areas. In [30], Pauca et al. explore the use of *L*2 regularizer in NMF algorithm. The cost function with *L*2 regularization term is expressed as:

$$f(W, H) = \frac{1}{2} \left\| X - WH \right\|\_F^2 + \mu \left\| H \right\|\_2 \tag{9}$$

where *H*2 = *P*,*N* ∑ *p*,*n*=1 *<sup>H</sup>*2*pn*.

> The objective in (9) is nonincreasing under the multiplicative update rules:

$$\mathcal{W} = \mathcal{W}.\*(XH^T)./\mathcal{W}HH^T\tag{10}$$

$$H = H.\*(\mathcal{W}^T X)./(\mathcal{W}^T \mathcal{W} H + \mathcal{Z}\mu H) \tag{11}$$

#### **3. Proposed NMF with Data-Guided Constraints for Hyperspectral Unmixing**

## *3.1. Sparsity Analysis*

The phenomena of sparsity in abundances commonly exists in hyperspectral images [31]. Sparsity is an inherent property which refers to a representative occasion where mixed pixels could be represented by a few endmember signatures. Accordingly, sparseness constraints such as the *L*1/2 regularizer help to obtain unique solutions and lead to better answers in scenes with obvious sparsity. However, in hyperspectral images there exist pixels located in transition regions which are evenly mixed and own low sparsity levels. Imposing a sparseness constraint over the entire image may not contribute to the unmixing accuracy of those evenly mixed pixels. Therefore, we also adopt an *L*2 regularizer to promote the evenness of pixels' abundance vectors, achieving an effect on abundances opposite to that of the *L*1/2 regularizer. Through imposing the *L*2 regularizer on a pixel's abundance vector, extreme abundance values are reduced and the sparseness level of abundances tends to be lower. In our method, each pixel enjoys a individual constraint related to its own sparsity level of abundance. Figure 1 represents the well known Cuprite dataset collected by an airborne visible/infrared imaging spectrometer (AVIRIS) sensor over the Cuprite mining site and the corresponding sparseness map of this scene. To evaluate the sparsity levels of pixels, the sparsity level of the *n*th pixel's abundances is defined as [37]

$$supersness(H\_n) = \frac{\sqrt{P} - \left(\sum\_{1}^{P} \left| H\_{pn} \right|\right) / \sqrt{\sum\_{1}^{P} H\_{pn}^{2}}}{\sqrt{P} - 1} \tag{12}$$

where *Hn* denotes the abundance vector of *n*th pixel, *P* denotes the number of endmembers, and *Hpn* denotes the (*p*, *n*)th element of *H*. As shown in Figure 1b, some regions mainly composed by one or a few materials possess high sparsity levels, while some other regions show low sparsity levels where minerals are evenly mixed there. The estimated sparsity levels of pixels range from 0.14 to 1. For hyperspectral data consisting of regions with various sparsity levels, using a simple kind of constraint on the whole image does not meet the practical situation and may not lead to a well-defined result.

**Figure 1.** (**a**) Airborne visible/infrared imaging spectrometer (AVIRIS) hyperspectral data of the Cuprite mining district in Nevada, USA; (**b**) Estimated sparseness map from the obtained abundance.

To solve this problem, we propose the DGC-NMF algorithm which is designed to impose constraints precisely according to the data's sparsity levels in different regions. However, the sparsity levels of abundances are previously unknown since the ground truth of abundance is not available. In the proposed DGC-NMF algorithm, we firstly carry out an unmixing process based on NMF with no constraint to derive the sparseness map of data. No sparseness constraint is employed in this unmixing process to avoid the distinctive sparsity information of a pixel's abundances being interfered with by a sparseness constraint without verification. This method of estimating sparseness maps may be a biased way. However, it is still a good choice for estimating sparsity levels since the ground truth of real hyperspectral data is not available and a small sparseness error is tolerable in our proposed DGC-NMF algorithm. To demonstrate the accuracy of the sparseness estimation, experiments are conducted to make comparison between estimated and real sparsity levels of data. Figure 2 shows that the estimated sparsity levels fit the real sparsity levels well under various sparsity levels. The estimated sparseness values could correctly reflect the general trend of the real sparsity levels. Figure 3 presents the real and the estimated sparseness map of synthetic data. The estimated map coincides with the real map well. For regions possessing high or low sparsity levels, the estimate sparseness map also shows high or low values. The estimated sparseness values can represent the real sparsity levels of pixels well. We also conduct experiments in Section 4.1 to compare DGC-NMF with the real sparseness map and DGC-NMF with the estimated sparseness map. The results also validate that it is practical to estimate sparsity levels via the unmixing result of NMF algorithm.

**Figure 2.** The real average sparsity level and the estimated average sparsity level.

**Figure 3.** (**a**) The real sparseness map of synthetic data; (**b**) The estimated sparseness map of synthetic data.
