## 1 Introduction
Based on the hypothesis of Tobler's First Law of Geography [33], the pixels around the mixed pixel are regarded as having the same inner parts as the mixed pixel, the same inner part spectrum, and a similar spectral mixing mode. These pixels in adjacent land surface regions contribute to the solving of inner part values. However, when near things are less related to each other on the land surface, the inner parts are usually different around the adjacent pixel regions. This phenomenon often leads to linear systems having no unique solution, called an \"underdetermined problem\" [34]. The downscaling approach is incapable in this situation because the pixel value cannot be decomposed into an inner part value [22]. Therefore, understanding the relationship between the mixed pixel and its region on the land surface and handling these cases may be the key to avoiding the underdetermined problem of linear spectral unmixing. In general, the mixed pixel and its adjacent land surface region are considered homogeneous, as [PERSON]'s First Law of Geography describes [33]. It implies that the inner parts of mixed pixels are included in the pixels of adjacent regions to a certain extent, and the adjacent region affords enough pixels to the linear system of unmixing pixels in inner part value solving. It can be deduced that the acquisition of pixels from a certain-sized adjacent region is an orthodox way to accomplish the unmixing process. On this account, the image scene with a certain extent size was adopted to access the pixels around the adjacent region. For instance, [PERSON] et al. (1995) introduced two sets of 3-by-3-pixel windows to extract the land cover given via TM classification and the AVHRR pixel response [29]. [PERSON] et al. (1997) adopted a sub-scene with 3-by-3 pixels in the AVHRR image to construct the system of equations for each pixel [31]. [PERSON] et al. (1999) performed the unmixing of coarse pixels in a moving window mode with a selected size, in which the contextual information of surrounding coarse pixels was used [35]. [PERSON] et al. (2008), [PERSON] et al. (2013), and [PERSON] et al. (2013) applied a sliding window to the classified image to select the coarse neighborhood pixels [32; 36; 37]. These approaches provide the neighborhood pixels for the unmixing process directly and effectively and have a wide range of applications. With the development of research, the underdetermined problem caused by the unbalance of inner part numbers and neighborhood pixel numbers in the unmixing process draws attention, and two major means are taken to address this problem. On one hand, it is achieved by reducing the number of inner parts in the window. [PERSON] et al. (2015) considered the relation between the numbers of the inner part k and the sliding window size n [22]. The inner part with a low proportion in the window is discarded when k is larger than n\({}^{2}\) to allow a more accurate decomposition. On the other hand, it is achieved by raising the number of neighborhood pixels by enlarging the window size. [PERSON] et al. (2015 and 2016) deemed the window size for each land cover class to be different and developed a correlation-coefficient-based method to obtain the best window size [38; 39]. These manners are important for the unmixing process and show great improvement. However, they are not the fundamental solutions. During the complex distribution of objects on the land surface, various land cover types contribute to each coarse pixel, and the relationship between the inner parts of the pixel and adjacent region pixels exceeds the qualitative definition of homogeneity or heterogeneity. The unmixing process is based on the pixel unit, aiming to solve the inner part values within the local land region. Therefore, setting the optimal window size for each land cover type is still not enough for the whole image pixels to unmix, which limits the performance of recent research facing the underdetermined problem [40].