**2. Methods**

#### *2.1. Nonlinear Band Dimensionality Expansion*

An early attempt to expand the original set of band images is to utilize nonlinear functions, for example, auto-correlation and cross-correlation, an idea derived from [16,20]. This type of NBE process is referred to as correlation band expansion process (CBEP). Combining these new CBEP-generated band images with the original set of band images produces a hyperspectral image with sufficient band images.

The CBEP presented in this section is an NBE process using correlation functions to generate new band images from the original set of multispectral images. Its original idea was developed in [16,20].

#### *Correlation Band Expansion Process* (CBEP)

Step 1. First-order band image: {**<sup>B</sup>***l*}*<sup>L</sup> l*=1 = set of original band images Step 2. Second-order correlated band images:


Step 3. Third order correlated band images

(i) . **B**<sup>3</sup> *l*/*L l*=1= set of auto-correlated band images


Step 4. Other nonlinear correlated band images


It should be noted that, according to the nonlinear functions described in Steps (1)–(4), the band images generated by CBEP contain only nonlinear spectral information but not spatial information. In what follows, we develop an iterative CEM (ICEM) to address this issue where spatial information can be captured by using a Gaussian filter and feed it back to expand images currently being processed to create a new set of image data cubes.

## *2.2. Iterative CEM*

ICEM, presented in this section, is implemented in conjunction with CBEP in an iterative manner. More specifically, it utilizes CBEP to create new band images via an NBE process. Once CBEP process is completed, a new set of image data cubes is generated for CEM to perform subpixel target detection. To obtain class spatial information, a Gaussian filter is introduced in the CEM-detected maps so that spatial contextual information of data sample vectors can be captured by a Gaussian filter. The resulting Gaussian-filtered CEM-detection abundance fractional map is fed back to create a new band incorporated into NBE to form a new hyperspectral cube which will be further used for re-processing CEM again. The same process is repeated over and over again until a stopping rule is satisfied. This repeated implementation of CEM via feedback loops in an iterative fashion is ICEM.

Specifically, at each iteration, say *k*th iteration, a Gaussian filter is used to blur -- -**B**| (*k*) CEM which is the absolute value of CEM-detection abundance fractional map, **B**(*k*) CEM. This Gaussian-filtered band image, - - -**B**| (*k*) GF(CEM) provides spatial classification information as similar filters used in [23] and will be further fed back to **Ω**(*k*) NBE to create a new set of hyperspectral images, **Ω**(*k*+<sup>1</sup>) NBE = **Ω**(*k*) NBE ∪ , - - -**B**| (*k*) GF(CEM) - to be used by CEM again for next iteration. The same procedure is continued. To terminate the process, an automatic stopping rule is designed. It applies Otsu's method [21] to - - -**B**| (*k*) GF(CEM) to produce a binary classification map, **B**(*k*) binary that will be used to calculate DSI [22]. If two consecutive DSI values are within an error threshold, ICEM will be terminated and - - -**B**| (*k*) CEM and **B**(*k*) binary will be the desired final real-valued WMH lesion detection map for visual inspection and binary value detection maps of WMH lesions for quantitative analysis.

In the following, we describe detailed step-by-step implementation of ICEM in grea<sup>t</sup> detail.
