Fuzzy Geospatial Object-Based Membership Function Downscaling

Abstract

The area-to-point kriging method (ATPK) is an important technology of downscaling without auxiliary information in remote sensing. However, it uses a constant semivariogram to downscale geospatial variables, which ignores the spatial heterogeneity between the geospatial objects. To deal with this kind of heterogeneity, this study proposes a fuzzy geospatial object-based ATPK method, which mainly consists of three steps: the extraction of fuzzy geospatial objects, the estimation of semivariograms for each object, and the downscaling of each object by ATPK with the corresponding semivariogram. Two groups of membership functions acquired from Worldview-2 and Sentinel-2 are used to test the proposed approach. Six classic downscaling algorithms are compared, and the results of two experiments show a better performance than the classical methods.

1. Introduction

Fuzziness is an inherent uncertainty prevalent in the geographic world. Almost all natural geographical objects have a vague boundary, such as meadows, animal habits, wetlands, and city influence zones [1,2,3]. The extensive ambiguity between fuzzy geospatial objects pose many challenges when processing multispectral remote sensing data. Since the introduction of the fuzzy set theory by Zadeh [4], geographers have widely applied it in the study of geographical issues [5,6,7,8]. Fuzzy methods use the concept of membership function (MF) to describe the vagueness of geographic objects. MFs can be generated from remote sensing images by fuzzy clustering or classification methods [7,9,10,11], and the membership values in the MF of each object (such as pixels, image objects, and super-pixels) express the possibility that each object partially belonging to different classes. Specifically, membership values (denoted as ${\mu }_c\left(x\right)$) are assigned to an object $x$ for all considered $c$ classes after fuzzy clustering or classification tasks. A value of ${\mu }_c\left(x\right)=0$ indicates that the object does not completely belong to a class, while a value ${\mu }_c\left(x\right)=1$ indicates that it completely belongs to the class. The closer the value is to 1, the higher membership degree, and vice versa [7].

Medium-resolution remote sensing images, such as Sentinel-2 and Landsat remote sensing images, can be used to support global land services, including vegetation monitoring and water coverage [12,13,14]. Because of their availability and low cost, these data have received more and more attention in remote sensing research and applications. However, due to the limitation of the coarse spatial resolution of these images, many mixed pixels exist, leading to a limited ability to express local detail information, such as small objects. Therefore, medium-resolution images and their products (such as classification results and membership degrees) often need to be downscaled, such as data fusion [15,16,17,18,19,20] and sub-pixel mapping [19,21,22,23,24,25,26,27], to improve the resolution of remote sensing images. Downscaling aims to improve the spatial resolution of the image and reproduce more information that is not available in the original data [28,29,30]. Generally, two classes of downscaling can be distinguished based on predicting a continuous or categorical variable [30]. For downscaling continuous variables, the methods can be divided into downscaling without and with fine-scale auxiliary information [28,31,32,33]. The downscaling method involved in this study is a downscaling method without fine-scale auxiliary information.

The downscaling method without auxiliary information mainly contains two forms of resampling and spatial statistical analysis [28]. The univariate downscaling technique considers only a single spatial variance function and uses interpolation to achieve the downscaling of geospatial variables. However, interpolation methods, such as Bilinear interpolation [34], inverse distance-weighted methods [35], and function-based interpolation [36], do not consider the shape of the data scale and the spatial correlation information. When downscaling continuous variables, geostatistical methods consider the spatial correlation between geological data. However, simple kriging and ordinary kriging do not solve the problem of modifiable regional units when downscaling [37].

To address this challenge, Kyriakidis [35] developed the area-to-point kriging method (ATPK). The principle of ATPK is that the unknown point value is a linearly weighted sum of the data on the area where it is located and the surrounding area. ATPK is distinguished from conventional centroid-based kriging, which ignores spatial support. ATPK explicitly considers the size of the support and predicts variables from regional supports to points by semivariogram deconvolution, with parameterized random function models and kriging [36]. Additionally, ATPK can precisely honor the observed areal data and provide predictions on finer support (point or area) than those of the original data, ensuring the coherence of prediction [36]. For example, the sum of downscaled predictions within any area is equal to the original aggregated count. Therefore, ATPK was originally proposed for downscaling data with irregular geographic units, where the size and shape supported by observations vary by geographic location, such as population and disease data across counties. After elaborating on the technique in the general case of census data, Kyriakidis and Yoo [37] further developed the technique for direct application to remote sensing imagery and demonstrated that the method reproduces point histograms, point semivariograms, and coarse resolution data. Although the basic model is easily applicable to remote sensing data, in the initial stage of the development of the ATPK method, the method was mostly used for downscaling data such as regional population [37] and soil types [36]. Therefore, the number of applications of ATPK in remote sensing was quite limited.

Based on the characteristics of the ATPK method, many researchers have extended and developed the ATPK method in recent years and proposed many methods for downscaling remote sensing data. Liu et al. [38] proposed an area-to-point regression kriging (ATPRK) method to decompose population density from irregular census units into land use areas within them, where remote sensing images (i.e., IKONOS images) were used to obtain auxiliary information. ATPRK shows its potential to classify irregular geographic unit data. Combining the advantages of regression kriging and ATPK, Wang et al. [16] proposed an ATPRK method that can be used for remote sensing image downscaling and proved that the method has perfect coherence properties and can produce high-quality sharpened images. Zhang et al. [20] combined the fuzzy c-means segmentation algorithm with the ATRPK algorithm to propose an image pan-sharpening method for the OATPRK that can always achieve the sharpening of multi-spectral images with the highest precision.

Geostatistics uses variance functions as a tool to describe spatial variability, and accurate predictions can only be made using variance functions that truly reflect the structure of spatial variability [39]. Although the ATPK method has its unique advantages in remote sensing image downscaling, it is known that the ATPK method requires an accurate semivariogram to support it [40]. Nevertheless, the existing research on spatial downscaling based on the ATPK method has focused on using the overall semivariogram of spatial variables to estimate the spatial variation and relevant information of geographical objects [16,20,31]. However, this approach may not be effective as the global image is not stationary, which can limit the performance of extraction algorithms. Some researchers have used geostatistical methods for downscaling studies in the form of moving windows to consider the variation across different spatial geographic objects, but this method requires setting the size of the window to be sufficient to estimate the variation, and cannot consider individual geographic objects independently [15]. Universal kriging is one of the spatial interpolators from geostatistics, a tool that deals with spatial autocorrelation using a probabilistic approach of regionalized variables [39,41]. As we know, in the real world, remote sensing images cover different spatial patterns of geographic objects or different terrain type areas, and the spatial boundary changes in these geographic objects are different, so the variogram for each feature object is also different [42,43,44,45,46]. However, the existing ATPK method and its optimization method do not fully consider this spatial heterogeneity feature, i.e., they do not consider the unique shape, size, and spatial correlation of objects.

Therefore, in response to the above problems, this study takes into account the heterogeneity of fuzzy geospatial objects and proposes a fuzzy geospatial object-based ATPK membership function downscaling. A fuzzy geospatial object is a geospatial object with fuzzy boundaries, distinguishing it from a typical fuzzy image object. This unique characteristic has been identified in previous research [42,43,47]. In the study, first of all, the MFs with the coarse spatial resolution are produced by the fuzzy random forest method using medium spatial remote sensing images, i.e., Sentinel-2. Secondly, the extraction of fuzzy geospatial objects is obtained combining slope analysis and connected component labelling algorithm. Thirdly, two different semivariogram calculation methods are proposed for the different fuzzy geospatial objects. Then, the ATPK method is used to downscale each fuzzy geospatial object based on its semivariogram. Finally, the downscaling results are compared and analyzed based on other downscaling methods with this method.

Compared to classical ATPKs, the method proposed in this study takes into account the heterogeneity of different fuzzy geospatial objects and enables ATPK downscaling for semivariograms of different fuzzy geospatial objects. More importantly, the MFs used in this study can identify the boundaries of fuzzy geospatial objects and extract them more easily. The contributions of this study are as follows:

• Obtaining a low-error MF image for each land use type based on the fuzzy random forest method.
• Using the threshold segmentation method and gradient analysis method to study fuzzy objects and topology (internal, boundary, and external); a connected component labelling algorithm is used to label each fuzzy geospatial object.
• Studying different methods of calculating semivariograms for geographic objects at different scales: (1) based on fuzzy geospatial object boundaries and (2) based on small-scale fuzzy geospatial objects.
• Developing an ATPK downscaling method for fuzzy spatial variables with composite multiple semivariograms.

The organization of this paper is as follows. Section 2 introduces the principles of the new downscaling approach in detail, which includes three parts: the extraction of fuzzy geospatial objects, the calculation of semivariogram for each geographic object, and ATPK downscaling methods. Section 3 presents the study area, study data, and experiment set. Section 4 presents the discussion. The conclusion is summarized in Section 5.

2. Methodology

Four main steps were adopted for the fuzzy geospatial object level ATPK, as shown in Figure 1. Firstly, the MFs of medium-resolution remote sensing imagery were generated with a coarse spatial resolution using the fuzzy random forest method. Secondly, fuzzy geospatial objects were extracted by combining slope analysis and a connected component labelling algorithm. Thirdly, different semivariogram calculation methods were used for each fuzzy geospatial object with different scale sizes. Lastly, each MF image was downscaled using the ATPK method according to the semivariogram of each fuzzy geospatial object.

2.1. Generating Membership Function by the Fuzzy Random Forest

It has been proved that random forest class probabilities are consistent estimates of the true probabilities [48]. According to Marcio et al. [49], land use and land cover area can be estimated from class membership probability for the random forest classification. Therefore, in this study, this conclusion was used to generate a membership map of each land use type for downscaling. The method mainly consists of two steps, mainly random forest classification algorithm and class proportion estimates.

In the fuzzy random forest algorithm, the land cover type is first predicted using the random forest classification, and then class proportions in the prediction dataset are estimated by Equation (1) counting the number of pixels in each classification and dividing by the number of pixels in the prediction dataset.

$$\widehat{{PC}_C}=\frac{{\sum }_N{I}_i(class=c)}{N}$$

(1)

where $N$ is the number of pixels in the population, $c$ is the class index, and $I_i\left(class=c\right)$ is the number of classes $c$ for pixel $i$. Equation (2) averages the class membership probabilities estimates of each class over all pixels.

$$\widehat{{PP}_C}=\frac{{\sum }_N{P}_i(class=c)}{N}$$

(2)

where $P_i(class=c)$ is the class membership probability of class $c$ for pixel $i$.

Through the above steps, the membership function of each land cover can be obtained.

2.2. Fuzzy Geospatial Object Extraction

In the extraction of fuzzy geospatial objects, threshold segmentation and slope analysis were used to study the fuzzy objects and their topological components (internal, external, and boundary); later, a connected component labeling algorithm was used to label each fuzzy geospatial object.

2.2.1. Slope Analysis

Slope analysis determines the steepness at each cell of the raster surface, with the smaller slope values indicating a flatter terrain, and vice versa. Slope analysis is most commonly used in elevation dataset processing. In this paper, slope analysis was used to identify geographic boundaries in membership maps. The slope calculation formula is as follows:

$$\theta =\arctan {(({\frac{dz}{dx})}^2+({\frac{dz}{dy})}^2)}^{\frac{1}{2}}\times \frac{180}{\pi }$$

(3)

where $\theta$ is the slope (°), between 0° and 90°; $\frac{dz}{dx}$ is the gradient change rate of the membership value in the $x$ direction; and $\frac{dz}{dy}$ represents the gradient change rate of the membership value in the $y$ direction [50]. $\frac{180}{\pi }$ is the conversion factor for the radian rotation angle.

As far as the characteristics of the membership map are concerned, the membership value of each type of membership map is 1 within the general geographic object, and the membership value at the boundary gradually transitions from 1 to 0. Therefore, within the geographic object, there is no membership gradient change, and the slope value is 0, while in the edge area of the geographic object, there is a gradient change in which the membership value gradually transitions from 1 to 0, and the slope value is greater than 0. The values of the center cell and its eight neighbors determine the horizontal and vertical gradients. It is thus possible to extract edge regions of geographic objects, as shown in Figure 2.

For general fuzzy geospatial objects, gradient variation occurs only at the edge region. Therefore, slope analysis can be used to identify the edge regions of fuzzy geospatial objects. The edge regions can be extracted by combining the connected component-labeling algorithm and the slope analysis method. Since there is no gradient change within the general fuzzy geospatial objects, it may affect the changing trend of the semivariogram. Therefore, only the boundary regions with gradient changes are extracted for the subsequent semivariogram calculation. On the other hand, for small-scale fuzzy geospatial objects, there is a gradient variation in MF values due to their small number of pixels and the fact that a pixel may represent more than one type of feature. Hence, slope analysis can extract the entire fuzzy geospatial object regions.

2.2.2. Connected Component Labeling

Since the range of MF value is from 0 to 1, the greater the MF value, the greater the possibility that the object belongs to this class, and vice versa. Since very small MF values of less than 0.05 are generated when generating MF images, the thresholds [0.05, 0.95] were used in this study for each type of geographical object MF for normalization. The pixel whose MF value is less than 0.05 was assigned a MF value of 0, and the pixel whose MF value is greater than 0.95 was assigned a MF value of 1. The purpose is to reduce the interference of very small MF value pixels among fuzzy geospatial objects so that the interconnected regions form one independent fuzzy geospatial object, which can be labeled by the connected component-labeling algorithm.

The connected component labeling (CCL) technique is used in image processing to identify pixels in an image [51,52] and mark the object pixels in the binary image. It enables each individual connected domain to form a marked block, and relevant image feature information, such as the area, bounding box, and centroid of these regions, can be obtained. Once the connected components in an image have been labeled, they can be further analyzed individually. In this study, the MF was first binarized, and then the 8-connectivity algorithm was used to obtain the connectivity region of each fuzzy geospatial object.

2.3. Determing Semivariograms of Fuzzy Geospatial Objects

The semivariogram is an important tool for analyzing the spatial heterogeneity of regionalized variables. Previous studies have shown that the variogram provides a concise description of the scale and pattern of spatial variability, which is essential for geostatistics [53]. In this paper, the semivariogram was used to consider the spatial heterogeneity of different geographical objects.

Unlike previous studies that calculated only the global semivariogram of images, this study calculated the semivariogram for each extracted fuzzy geospatial object. To consider the spatial characteristics of fuzzy geospatial objects at different scales, two methods were used to construct semivariograms: (1) the semivariogram construction method based on the boundary of geographic objects and (2) the semivariogram construction method for small-scale geographic objects.

2.3.1. The Semivariogram Construction Method Based on the Boundary of Fuzzy Geospatial Objects

For larger-scale fuzzy geospatial objects, there is no gradient change in the value of the MF within the geospatial object. As a result, the changing trend of the semivariogram may be affected. To address this issue, only the boundary parts with gradient changes were considered in the calculation of the semivariogram of larger-scale fuzzy geospatial objects.

To calculate the semivariogram of larger-scale fuzzy geospatial objects, we first used the slope analysis algorithm to identify the edge regions of each fuzzy geospatial object where the gradient of fuzzy MF changes. We then used the CCL to obtain a label for each individual fuzzy geospatial object. Finally, we combined these two results to obtain the MF value of each fuzzy geospatial object’s edge region, which was used to calculate its semivariogram.

Suppose $l$ denotes the $lth$ fuzzy geospatial object in the coarse resolution MF $(l=\mathrm{1,2},3,...,L)$. The semivariogram of the geographic object of the MF can be defined as half of the variance of the value difference between the value $Z\left(\mathbf{x}\right)$ of the pixel $\mathbf{x}$ in the fuzzy geospatial object at point $\mathbf{x}$ and $\mathbf{x}+h$, where $h$ represents the lag distance between pixels. For remote sensing images, the separation distance is extracted at regular intervals, as shown in Figure 3. In this study, a lag distance $h\in \left[\mathrm{1,20}\right]$ was used when calculating the semivariogram for the boundary of each fuzzy geospatial object. The semivariogram $\gamma \left(h\right)$ for each coarse resolution fuzzy geospatial object was defined as the variance of the difference in values between all pairs of membership pixels at distance $h$:

$${\mathsf{\gamma }}_{Cl}\left(h\right)=\frac{1}{2}{E[\left(Z_{Cl}\left(\mathbf{x}\right)-Z_{Cl}\left(\mathbf{x}+h\right)\right)}^2]$$

(4)

where $C$ represents the coarse resolution membership map and $l$ represents the $lth$ fuzzy geospatial object in a certain membership function.

For each fuzzy geospatial object, there are $m\left(h\right)$ pairs of pixel membership values in a fuzzy geospatial object, separated by the same pixel interval $h$. The semivariogram estimator of each geographic object at lag $h$ is given by

$${\widehat{\gamma }}_{Cl}\left(h\right)=\frac{1}{2m\left(h\right)}{\sum _{i=1}^{m\left(h\right)}[Z_{Cl}\left({\mathbf{x}}_i\right)-Z_{Cl}({\mathbf{x}}_i+h)]}^2$$

(5)

This algorithm was used to calculate the semivariogram of each extracted fuzzy geospatial object, which was then downscaled using ATPK method. The quantity $\widehat{\gamma }$ is an estimate of the semivariance $\gamma$ [54], which is a useful measure of the variance between spatially separated pixels [55]. The larger the $\widehat{\gamma }$, the less similar the pixels are.

2.3.2. The Semivariogram Construction Method for Small-Scale Fuzzy Geographical Spatial Objects

Due to the small number of pixels contained in the small-scale fuzzy geospatial objects, the semivariogram has a large uncertainty. Therefore, we studied the method of constructing the semivariogram for small-scale fuzzy geospatial objects. For small-scale fuzzy geospatial objects, there is a gradient variation in MF values due to their small number of pixels and the fact that a pixel may represent more than one type of feature. In the slope analysis of coarse resolution MF, small-scale fuzzy geospatial objects do not have regions where the gradient changes to zero, and these regions are regarded as small-scale geographic objects. The semivariogram for small-scale fuzzy geospatial objects were calculated as in Equations (4) and (5).

Since the number of pixels in small-scale geographic objects is small, we set $h\in \left[\mathrm{1,10}\right]$ when calculating their semivariogram.

2.4. Geospatial Object-Based Membership Function ATPK Downscaling

In this paper, we downscaled the membership maps to the target fine spatial resolution using ATPK.

ATPK refers to prediction on a support that is smaller than that of the original data [30]. It differs from traditional centroid-based kriging, which ignores spatial support and always treats it as equivalent to observational support. ATPK specifies the supported size, spatial correlation, and PSF of the sensor [56]. ATPK explicitly considers the size of the support and predicts variables from regional supports to points by semivariogram deconvolution, with parameterized random function models and kriging.

The principle of ATPK is as follows:

Suppose $Z_{Cl}\left({\mathbf{x}}_i\right)$ is the membership value of the coarse resolution pixel $C$ centered at ${\mathbf{x}}_i$($i=1,\dots ,M$, where $M$ is the number of pixels) for the $lth$ geographic object in membership map Z. Based on ATPK, the membership value $Z_{Fl}\left(\mathbf{x}\right)$ of all fine pixel F of the $lth$ fine resolution geographic object is predicted by the downscaling process. For a fine pixel centered at ${\mathbf{x}}_0$, the pixel value predicted by ATPK is a linear combination of the measurements of neighboring coarse pixels.

$${\mathrm{Z}}_{Fl}\left({\mathbf{x}}_0\right)=\sum _{i=1}^N{\lambda }_i{Z}_{Cl}\left({\mathbf{x}}_i\right),\mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}:\sum _{i=1}^N{\lambda }_i=1$$

(6)

In Equation (6), $N$ is the number of coarse pixels in the neighboring system, such as the $N=5\times 5$ windows of pixels. ${\lambda }_i$ is the weight for the $ith$ neighbor centered at ${\mathbf{x}}_i$. As $Z_{Cl}\left({\mathbf{x}}_i\right)$ is already known, the estimation of $Z_{Fl}\left(\mathbf{x}\right)$ requires the estimation of ${\lambda }_i$, and this can be solved by the following kriging matrix. Thus, the spatial correlation between coarse pixels is accounted for in ATPK.

The task becomes the estimation of weights $\{{\lambda }_1,...,{\lambda }_N\}$ in Equation (7). They are calculated by minimizing the prediction error variance, and the corresponding ordinary kriging system is [57,58]:

$$\left[\begin{array}{cccc}{\gamma }_{CCl}({\mathbf{x}}_1,{\mathbf{x}}_1)& \dots & {\gamma }_{CCl}({\mathbf{x}}_1,{\mathbf{x}}_N)& 1\\ ⋮& \ddots & ⋮& ⋮\\ \begin{array}{c}{\gamma }_{CCl}({\mathbf{x}}_N,{\mathbf{x}}_1)\end{array}& \dots & \begin{array}{c}\begin{array}{c}{\gamma }_{CCl}({\mathbf{x}}_N,{\mathbf{x}}_N)\end{array}\end{array}& 1\\ 1& \dots & 1& 0\end{array}\right]\left[\begin{array}{c}{\lambda }_1\\ \begin{array}{c}⋮\\ \begin{array}{c}{\lambda }_N\\ \mu \end{array}\end{array}\end{array}\right]=\left[\begin{array}{c}{\gamma }_{FCl}({\mathbf{x}}_0,{\mathbf{x}}_1)\\ \begin{array}{c}⋮\\ \begin{array}{c}{\gamma }_{FCl}({\mathbf{x}}_0,{\mathbf{x}}_N)\\ 1\end{array}\end{array}\end{array}\right]$$

(7)

where ${\gamma }_{CCl}\left({\mathbf{x}}_i,{\mathbf{x}}_j\right)$ is the coarse-to-coarse semivariogram calculated by the point support covariance between coarse pixels located at $x_i$ and $x_j$ for coarse resolution geographic object $l$, ${\gamma }_{FCl}\left({\mathbf{x}}_i,{\mathbf{x}}_j\right)$ is the fine-to-coarse semivariogram between fine and coarse pixels centered at $x_0$ and $x_j$, and $\mu$ is the Lagrange multiplier [59].

The PSF is used to transform fine spatial resolution data into coarse spatial resolution. Let $s$ denote the Euclidean distance between the centroids of any two pixels, and ${\gamma }_{FFl}\left(s\right)$ represents the fine-to-fine semivariogram between two fine pixels. Suppose $h_c\left(s\right)$ is the point spread function (PSF) of the satellite sensor. The fine-to-coarse semivariogram ${\gamma }_{FCl}\left(s\right)$ and the coarse-to-coarse semivariogram ${\gamma }_{CCl}\left(s\right)$ in Equation (7) are calculated by convolving ${\gamma }_{FFl}\left(s\right)$ with the PFS, i.e.,

$${\gamma }_{FCl}\left(\mathbf{s}\right)={\gamma }_{FFl}\left(\mathbf{s}\right)\ast h_{\mathrm{c}}\left(\mathbf{s}\right)$$

(8)

$${\gamma }_{CCl}\left(\mathbf{s}\right)={\gamma }_{FFl}\left(\mathbf{s}\right)\ast h_{\mathrm{c}}\left(\mathbf{s}\right)\ast h_{\mathrm{c}}\left(-\mathbf{s}\right)$$

(9)

where $\ast$ is the convolution operator and $\mathbf{s}$ represents the distance between point A within a pixel and point B within another pixel, which is denoted as $-\mathbf{s}$ to indicate that it is the opposite direction of the distance from point B to point A (which is denoted as s).

Assuming that the pixel value is the average of the fine pixel values within it, the PSF is defined as follows:

$${\mathrm{h}}_{\mathrm{c}}\left(\mathbf{x}\right)=\left\{\begin{array}{c}\frac{1}{S_c}, ifx\in C\left(x\right)\hfill \\ 0, \mathrm{otherwise}\hfill \end{array}\right.$$

(10)

where $C\left(\mathbf{x}\right)$ is the spatial support of pixel C centered at $x$ and $S_C=\left|C\right(\mathbf{x}\left)\right|$ is the surface/size of pixel $C$. Using the PSF defined in Equation (10), the calculation in Equations (8) and (9) can be further simplified:

$${\gamma }_{FCl}\left({\mathbf{x}}_{0,}{\mathbf{x}}_{\mathrm{j}}\right)=\frac{1}{\sigma }\sum _{m=1}^{\sigma }{\gamma }_{FFl}\left({\mathbf{s}}_m\right)$$

(11)

$${\gamma }_{CCl}\left({\mathbf{x}}_{i,}{\mathbf{x}}_j\right)=\frac{1}{{\sigma }^2}\sum _{m=1}^{\sigma }\sum _{m\prime =1}^{\sigma }{\gamma }_{FF}\left({\mathbf{s}}_{mm\prime }\right)$$

(12)

In Equations (11) and (12), $\sigma =S_C/S_F$ is the pixel size ratio (zoom factor) between the coarse and fine pixels and $s_m$ is the distance between the centroid ${\mathbf{x}}_0$ of fine pixel $F$ and the centroid of any fine pixel within the coarse pixel $C$ centered at ${\mathbf{x}}_{\mathrm{j}}$.

The fine-to-fine semivariogram ${\gamma }_{FFl}\left(\mathbf{s}\right)$ in Equations (11) and (12) is derived by the deconvolution (also called deregularization in geostatistics) of each coarse resolution geospatial object semivariogram, denoted by ${\widehat{\gamma }}_{Cl}\left(h\right)$ in Equation (5). In semivariogram modeling, the fitting function is typically characterized by three parameters: nugget, sill, and range. However, the computational cost increases linearly with the number of nugget candidates. To ease the computational burden, we adopted the assumption of Atkinson et al. [60] and Pardo-Igúzquiza et al. [15,61] in this paper: there is zero nugget effect (absence of local noise) in the fine-to-fine semivariogram, which refers to the spatial autocorrelation between observations that are very close together. To select the punctual sill and range intervals, we set the interval for punctual sill selection between 1 and 3 times that of the areal sill, and the interval for punctual range selection was set between 0.5 and 2.5 times that of the areal range. The step was set to 0.1.

The above methodology differs from the previous downscaling method that calculates a semivariogram for the entire remote sensing image. In this study, the membership map was used to extract each geospatial object unit and downscale each unit using its unique semivariogram. Compared to traditional interpolation methods, geostatistical methods can take into account the shape and intrinsic spatial correlation of data. Additionally, ATPK explicitly considers the size of the support and predicts variables from regional supports to points by semivariogram deconvolution, with parameterized random function models and kriging. Therefore, when downscaling using the ATPK method, the semivariogram of each geographic object can be used to obtain more accurate downscaling results. By using the ATPK method to downscale remote sensing images and combining it with the semivariogram of each fuzzy geospatial object, it is possible to obtain more accurate downscaling results.

2.5. Proof of Methodological Validity

The semivariogram is an important tool for analyzing the spatial heterogeneity of regionalized variables. In this paper, we used standardized image with 512 × 512 pixels to verify the effect of the semivariogram on the ATPK downscaling method and the effectiveness of the proposed method. The original 512 × 512 resolution image was used as the reference image for accuracy verification, and it was upscaled by a zoom factor of S = 2 to produce a coarse image with 256 × 256 resolution. Unlike geospatial object extraction, for grayscale images, we segmented the Lena image into 19 objects based on grayscale values, as shown in Figure 4b.

To obtain the downscaling result, the semivariogram of each image object was computed using the method presented in this paper. Then, the semivariogram of each object was calculated using the small-scale object method. Finally, the ATPK method was used to downscale each object on its semivariogram, resulting in the downscaling of each image object. The effectiveness of the proposed method was verified using six classical downscaling methods, namely Bilinear, Cubic, ICBI (Iteractive Curvature Based Interpolation) [62], INEDI (Improved New Edge-Directed Interpolation) [63], RBF (Radial Basis Function Interpolation) [23], and the classical ATPK downscaling methods based on global semivariogram. Four evaluation indicators were used, namely root-mean-square error (RMSE), correlation coefficient (CC), peak signal-to-noise ratio (PSNR), and universal image quality index (UIQI) [64]. In addition to these evaluation indices, the proposed method was also evaluated using the reduction in remaining error (RRE) index, which measures the improvement over other methods. The RRE is calculated as follows:

$$RRE=\left|\frac{{RE}_i-{RE}_0}{{RE}_i}\right|\times 100\%$$

(13)

where ${RE}_i$ and ${RE}_0$ are the mean squared error (or remaining errors) of the compared downscaling method and the method proposed, respectively. The RRE index indicates the percentage of reduction in error achieved by the proposed method compared to other methods.

Figure 4 presents the outcomes of the different downscaling algorithms that were employed to process the standardized image. During the experiments, we partitioned the Lena image into 19 objects by considering their gray values, in order to draw a comparison with the fuzzy geospatial objects. For each segmented object, a semivariogram was computed and utilized in the ATPK downscaling approach.

From an overall perspective, the method proposed in this paper can better downscale coarse resolution images.

Table 1. Quantitative assessment of the downscaling results for standardized images.

Quantitative Assessment		Bilinear	Cubic	ICBI	INEDI	RBF	ATPK	The Proposed Method
RMSE		0.0276	0.0276	0.0358	0.0352	0.0243	0.0224	0.0214
	RRE	22.46%	22.46%	40.22%	39.20%	11.93%	4.46%
CC		0.9926	0.9946	0.9872	0.9877	0.9944	0.9954	0.9958
	RRE	0.32%	0.12%	0.87%	0.82%	0.14%	0.04%
PSNR		79.32	80.62	77.04	77.18	80.44	81.14	81.52
	RRE	2.77%	1.11%	5.82%	5.63%	1.35%	0.47%
UIQI		0.9541	0.9646	0.9374	0.9398	0.966	0.9690	0.9716
	RRE	1.83%	0.73%	3.65%	3.38%	0.58%	0.27%

The bold values indicate the most accurate result in each case.

shows the accuracy evaluation results of various downscaling methods. Compared to Bilinear, Cubic, ICBI, INEDI, RBF, and the classical ATPK, the RMSE value increased by 22.46%, 22.46%, 40.22%, 39.20%, 11.93%, and 4.46%, respectively. It is clear that the method proposed in this paper is also effective for downscaling ordinary grayscale images.

3. Experiments

To verify the validity of this study, a total of two groups of experiments were set up. For the simulation experiment, the Worldview-2 MF images were used, while Sentinel-2 membership images were used for the real data experiment. In both groups of experiments, the results of downscaling the semivariogram for the entire region were compared and analyzed with the results of downscaling the semivariogram for geographic objects. The accuracy of both sets of experiments was evaluated using Worldview-2 2 m resolution membership images.

3.1. Study Area

This research was performed in the Beidagang Wetland core area, located in southern Tianjin city, China, where the basic elements are mainly vegetation and water bodies, as shown in Figure 5. Due to sea level regression and historical river impacts, the soil salinity is relatively high, the predominant plant species are Phragmites australis, Typha angustifolia, and Suaeda salsa. The water bodies include tidal flats, swamps, and rivers, and Suaeda salsa is classified separately as it appears as a red hue in the RGB composite color images, unlike other green vegetation. The land cover classification is divided into five types: water body, bare land, vegetation, Suaeda salsa, and withered vegetation. Wetlands have intertwined and fuzzy features with no clear boundary between vegetation and water bodies. Therefore, accurate boundary distinction of wetland land cover types is a meaningful method to study.

3.2. Study Data

3.2.1. Worldview-2 Dataset

The Worldview-2 image used in this study contains eight multispectral bands (coastal blue, blue, green, yellow, red, red-edge, near-infrared 1 (NIR1), and near-infrared 2 (NIR2)) with a resolution of 1.8 m. It was resampled to 2 m resolution for analysis. The Worldview-2 dataset covers the Beidagang Wetland core area of Tianjin province, China, and was acquired on 16 September 2020. Radiometric and atmospheric corrections were performed using the ENVI 5.3 software.

3.2.2. Sentinel-2 Dataset

Sentinel-2 image comprises 13 spectral channels in the visible, near-infrared, and short-wave infrared parts of the spectrum. Among these 13 bands, the spatial resolution of the visible bands 2–4 and NIR band 8 (spectral region ranging from 0.705 to 0.885 µm) is 10 m. For this study, we used the 10 m resolution bands. The high-resolution multispectral image from the Sentinel-2 satellite was acquired on 16 September 2020 and was free of clouds.

3.3. Experiment Setup

In the simulation experiments, the Worldview-2 dataset was upscaled by a zoom factor of S = 2 to produce a coarse image with 4 m resolution. The original 2 m MFs were used as the reference image. These MFs also served as a reference image for the Sentinel-2 image downscaled to 2 m. Four indices were used to quantitatively evaluate the performance of the downscaling methods in both the simulation and real experiments. These indices included RMSE, CC, PSNR, and UIQI.

Six classical downscaling methods were used for comparison against the method proposed in this study in both the simulation and real experiments. These methods included Bilinear, Cubic, ICBI, INEDI, RBF, and the classical ATPK downscaling methods based on global semivariogram.

3.4. Simulation Experiment

The study area is covered mainly by water bodies, bare land, vegetation, Suaeda salsa, and withered vegetation. Therefore, in the experiment, the membership maps of these five types of ground cover (i.e., water bodies, bare land, vegetation, Suaeda salsa, and withered vegetation) were downscaled. Figure 6 shows the MFs for the five land cover types, which were zoomed to a resolution of 4 m by the Worldview-2 2 m resolution MFs. As can be seen from the membership maps, the boundaries between the different land cover classes are not well-defined and are therefore difficult to distinguish, which necessitates the use of downscaling techniques. For the downscaling of the five coarse images in this experiment, the zoom factor was set to S = 2 to restore the fine spatial resolution images. The scaling factor S = 2 was chosen so that the original 2 m resolution membership maps could be used for accuracy verification after downscaling. The six downscaling methods (i.e., Bilinear, Cubic, ICBI, INEDI, RBF, and the classical ATPK) were used to evaluate the performance of the method proposed in this study.

To better illustrate the effectiveness of the method proposed in this study, we selected five sub-areas within the study area to display the membership images of different types of ground objects, as shown in Figure 7. Figure 8 displays the results of the various downscaling algorithms applied to the Worldview-2 dataset. The visual comparison of the seven downscaling results in each membership map in Figure 8 indicates that the proposed method is suitable for effectively downscaling coarse images.

Moreover, the proposed method provides the most satisfactory downscaling results among the seven methods. Specifically, in the case of the water body membership image shown in Figure 8, the proposed method and the downscaling method based on semivariogram computation for different fuzzy geospatial objects presented in this paper resulted in a smaller MF value for the water body boundary, indicating that the gradient value between the fuzzy geospatial objects is larger. This result demonstrates that the proposed method can identify the boundary area of ground objects better. Figure 8 shows that, when compared to other methods such as Bilinear, Cubic, ICBI, INDEI, and RBF, the ATPK downscaling method based on the semivariogram of the entire image yields a smaller boundary membership value. However, the boundary membership value is still slightly larger than the downscaling method proposed in this study, which is based on the semivariogram of fuzzy geospatial objects. Furthermore, the downscaling result based on the semivariogram of the whole image may produce a MF value of less than 1 in the inner edge area of the water body, which is not observed in the downscaling result based on the semivariogram of the fuzzy geospatial object. Additionally, the other methods cause the fuzzy geographical object boundary to have a small change gradient after downscaling and its boundary expands outward to varying degrees. In contrast, the method proposed in this study, accurately extracts the boundary range of small water bodies by compressing their internal range and preventing outward expansion of their boundaries.

Table 2. Quantitative assessment of downscaling results for the Worldview-2 dataset.

Quantitative Assessment	Land Cover	Bilinear	Cubic	ICBI	INEDI	RBF	ATPK	The Proposed Method
RMSE	water body	0.1204	0.1128	0.1307	0.1293	0.1039	0.0914	0.0876
	bare land	0.2150	0.2296	0.2244	0.2225	0.1891	0.1635	0.1586
	vegetation	0.1852	0.1776	0.1948	0.1931	0.1636	0.1433	0.1405
	Suaeda salsa	0.1701	0.1540	0.1767	0.1757	0.1561	0.1415	0.1364
	withered vegetation	0.1675	0.1788	0.1739	0.1732	0.1399	0.1386	0.1325
	RRE	23.66%	22.30%	27.36%	26.82%	12.76%	3.43%
CC	water body	0.9873	0.9897	0.9851	0.9848	0.9901	0.9926	0.9925
	bare land	0.9408	0.9503	0.9351	0.9362	0.9544	0.9663	0.9665
	vegetation	0.9586	0.9657	0.9552	0.9560	0.9686	0.9769	0.9764
	Suaeda salsa	0.9224	0.9312	0.9163	0.9173	0.9351	0.9472	0.9495
	withered vegetation	0.9102	0.9204	0.9030	0.9040	0.9251	0.9402	0.9444
	RRE	2.36%	1.54%	2.91%	2.83%	1.19%	0.13%
PSNR	water body	71.4179	72.2634	70.7026	70.7991	72.6974	73.8118	74.1788
	bare land	69.6563	70.3965	69.2836	69.3555	70.7715	72.0309	72.2966
	vegetation	67.7296	68.4427	67.2887	67.3638	68.8061	69.9567	70.1270
	Suaeda salsa	70.1872	70.6849	69.8570	69.9061	70.9320	71.7848	72.1068
	withered vegetation	71.1344	71.6384	70.8075	70.8445	71.8869	72.7814	73.1718
	RRE	3.36%	2.39%	4.01%	3.91%	1.91%	0.42%
UIQI	water body	0.7416	0.7533	0.7386	0.7398	0.7584	0.7737	0.9544
	bare land	0.7408	0.7587	0.7394	0.7411	0.7656	0.7964	0.9385
	vegetation	0.8215	0.8381	0.8173	0.8196	0.8459	0.8660	0.9520
	Suaeda salsa	0.7036	0.7269	0.7053	0.7059	0.7365	0.7876	0.8873
	withered vegetation	0.6683	0.6924	0.6731	0.6724	0.7026	0.7596	0.8728
	RRE	25.59%	22.42%	25.61%	25.45%	21.13%	15.73%

The bold values indicate the most accurate result in each case.

presents the quantitative assessment of seven downscaling methods based on various evaluation indices, including RMSE, CC, PSNR, and UIQI indices. The optimal results for each land type are shown in bold. In addition to these evaluation indices, the proposed method was also evaluated using the reduction in remaining error (RRE) index, which measures the improvement over other methods. The RRE is calculated as follows:

$$RRE=\left|\frac{\sum _{i=1}^5\frac{{RE}_i-{RE}_0}{{RE}_i}}{5}\right|\times 100\%$$

(14)

where ${RE}_i$ and ${RE}_0$ are the mean squared error (or remaining errors) of the compared downscaling method and the method proposed, respectively. The RRE index indicates the percentage of reduction in error achieved by the proposed method compared to other methods.

According to the visual comparison, both geostatistical approaches, the ATPK downscaling method based on global semivariogram and ATPK downscaling method based on fuzzy geospatial object semivariogram, outperform the other five methods in terms of RMSE, CC, PSNR, and UIQI for all MF images. Only the CC index of vegetation in the ATPK downscaling method is slightly better than the method proposed in this paper, while all other accuracy evaluation indexes are the most effective in the method proposed in this paper. Comparing the method proposed in this study to Bilinear, Cubic, ICBI, INEDI, RBF, and ATPK, the RMSE value increased by 23.66%, 22.30%, 27.36%, 26.82%, 12.76%, and 3.43%, respectively. Additionally, the UIQI value increased by 25.59%, 22.42%, 25.61%, 25.45%, 21.13%, and 15.73%, respectively, demonstrating the superiority of the method proposed in this study for downscaling the Worldview-2 dataset.

3.5. Real Experiment

To verify the effectiveness of the method described in this paper on real data, an experiment was conducted to downscale the 10 m membership images of Sentinel-2 to 2 m in the same experimental area. The accuracy of the downscaled images was then verified using Worldview-2 membership images.

Figure 9 shows a 10 m resolution image of Sentinel-2 without downscaling. Five sub-areas were selected from the study area to display MF images of different types of ground objects, as shown in Figure 10. For this experiment, the Sentinel-2 MF images were downscaled using a zoom factor of S = 5 to reduce the fine spatial resolution images. By setting the scaling factor S = 5, the Worldview-2 2 m resolution membership maps could be used to verify accuracy after downscaling. Figure 11 shows the downscaling results using Bilinear, Cubic, ICBI, INEDI, RBF, the classical ATPK, and the proposed method.

Table 3. Quantitative assessment of the downscaling results for the Sentinel-2 datasets.

Quantitative Assessment	Land Cover	Bilinear	Cubic	ICBI	INEDI	RBF	ATPK	The Proposed Method
RMSE	water body	0.3208	0.3562	0.3164	0.3328	0.3559	0.3399	0.2594
	bare land	0.5065	0.5372	0.5003	0.5251	0.5368	0.5320	0.3775
	vegetation	0.4147	0.4404	0.4092	0.4283	0.4403	0.4389	0.3256
	Suaeda salsa	0.3466	0.3583	0.3390	0.3588	0.3580	0.3802	0.3256
	Withered vegetation	0.4147	0.4265	0.4017	0.3388	0.4243	0.4603	0.3646
	RRE	16.84%	21.31%	15.23%	15.15%	21.19%	22.73%
CC	water body	0.9132	0.8928	0.9151	0.9073	0.8929	0.9033	0.9453
	bare land	0.7489	0.7160	0.7532	0.7348	0.7163	0.7278	0.8434
	vegetation	0.8136	0.7893	0.8176	0.8048	0.7893	0.7940	0.8510
	Suaeda salsa	0.6945	0.6708	0.7027	0.6819	0.6712	0.6576	0.7446
	withered vegetation	0.5978	0.5711	0.6113	0.5734	0.5732	0.5488	0.6965
	RRE	8.89%	12.89%	7.86%	11.07%	12.78%	13.57%
PSNR	water body	63.1836	62.2741	63.3016	62.8632	62.2813	62.6803	65.2063
	bare land	63.5236	63.0138	63.6307	63.2105	63.0190	63.0977	66.0132
	vegetation	61.0748	60.5537	61.1925	60.7954	60.5557	60.5822	62.1168
	Suaeda salsa	63.6399	63.3537	63.8337	63.3395	63.3602	62.8372	64.1825
	withered vegetation	64.3320	64.0890	64.6077	63.8388	64.1321	63.4262	65.4493
	RRE	2.28%	3.10%	2.02%	2.84%	3.08%	3.30%
UIQI	water body	0.5481	0.5206	0.5486	0.5454	0.5201	0.5344	0.8601
	bare land	0.5142	0.4978	0.5136	0.5108	0.4974	0.5086	0.8160
	vegetation	0.5717	0.5392	0.5713	0.5682	0.5388	0.5559	0.7181
	Suaeda salsa	0.4778	0.4583	0.4784	0.4757	0.4582	0.4644	0.6366
	withered vegetation	0.4478	0.4344	0.4483	0.4461	0.4340	0.4382	0.6925
	RRE	45.82%	52.12%	45.78%	46.58%	52.24%	49.14%

The bold values indicate the most accurate result in each case.

reports the accuracy assessment in terms of RMSE, CC, PSNR, and UIQI indices for the seven downscaling methods of coarse membership images for Sentinel-2.

It is important to note that, before downscaling, the pixels of the MF images with a of 10 m resolution could not accurately fit the boundary of the fuzzy geospatial object, resulting in a rough outline. However, after downscaling, the edge of the membership images with 2 m resolution more closely fits the fuzzy geospatial object boundary, resulting in a smoother outline of the object.

Although the Bilinear, Cubic, INEDI, and ICBI methods can also achieve better downscaling visual effects compared to the proposed methods, the boundary range of the ground object will spread outward by these methods, making it difficult to obtain accurate fuzzy geospatial object boundaries. The RBF downscaling method cannot produce a smoother boundary when downscaling the MF images with a large zoom factor (i.e., S = 5), resulting in a poor visual effect. The classic ATPK downscaling method can lead to the phenomenon that the MF value of the internal pixel whose MF value should be 1 of the fuzzy geospatial object is less than 1 in the downscaling result. Moreover, for small-scale fuzzy geospatial objects, this method caused the coverage area of fuzzy geospatial objects to shrink. Compared with other methods, the method proposed in this paper can more accurately downscale the coarse-resolution membership images and obtain a downscaling result that is closer to the boundary of the actual fuzzy geospatial object.

Table 3 presents a quantitative assessment of the seven downscaling methods in terms of RMSE, CC, PSNR, and UIQI indices for the Sentinel-2 membership images downscaled with a zoom factor of S = 5 in comparison with the Worldview-2 experiment. The accuracy results of each downscaling method exhibit varying degrees of decline. However, the proposed method for downscaling in this paper achieved the best results in all accuracy evaluation indicators, especially UIQI. Compared to the UIQI value of the proposed method, the UIQI values of the Bilinear, Cubic, ICBI, INEDI, RBF, and the classic ATPK methods increased by 45.82%, 52.12%, 45.78%, 46.58%, 52.24%, and 49.14%, respectively. This demonstrates the superiority of the proposed method for downscaling larger-scale resolution images.

4. Discussions

4.1. Characteristics and Advantages of the Methods

The experimental results shown in Section 3.4 and Section 3.5 indicate that the ATPK method presented in this paper has potential for downscaling, particularly for small-scale fuzzy geospatial objects. In simulation experiments, involving the downscaling of five coarse spatial resolution membership images with different land cover patterns, the method presented in this paper provided better fitting to the boundary of fuzzy geospatial objects and produces smoother results than the Bilinear, Cubic, ICBI, INEDI, RBF, and the classic ATPK methods. Moreover, the evaluation of the downscaling results for accuracy was also the best for the method proposed in this paper. In the real experiment using Sentinel-2 10 m coarse spatial resolution, where the boundary of fuzzy geospatial objects was rough, the new method accurately reconstructed small-scale fuzzy geospatial objects.

The difference between classic ATPK and the proposed method is that the former is a global interpolation, while the latter considers the semivariogram of each fuzzy geospatial object. The classic ATPK method is implemented based on a geostatistical theoretical framework that considers the spatial configuration of the entire study area and estimates the weight of each observed data point based on the global spatial covariance. In this study, however, instead of estimating the weight of each observation data point based on the global spatial covariance of the previous classic ATPK method, the weight is estimated for the different spatial covariance of each fuzzy geospatial object. Satellite sensor images cover areas with different spatial variation models due to different spatial distributions of fuzzy geospatial objects and terrain types. Therefore, it is a logical step to try to incorporate these changes in spatial variability into the image downscaling process.

The paper proposed a method that consisted of three main stages: fuzzy geospatial object identification, calculating the semivariogram for each fuzzy geospatial object, and ATPK downscaling. In the fuzzy geospatial object identification stage, different fuzzy geospatial object extraction methods were chosen for different scales of fuzzy geospatial objects. For large-scale fuzzy geospatial objects, MF images only have MF value changes at the edges of fuzzy geospatial objects. Therefore, only the edge regions of fuzzy geospatial objects were extracted for identifying the fuzzy geospatial object and calculating their semivariogram. By applying slope analysis to the MF image, the edge regions of the fuzzy geospatial objects with MF value changes can be precisely identified. Combined with each fuzzy geospatial object identified by the CCL algorithm, the boundary region of each fuzzy geospatial object can be extracted. For small-scale fuzzy geospatial objects, the number of pixels contained has a greater impact on the trend of semivariogram changes. Therefore, in the ATPK downscaling of small-scale fuzzy geospatial objects, the semivariogram was calculated using the entire fuzzy geospatial object, while the downscaling of large-scale fuzzy geospatial objects was performed with the extracted edges of the fuzzy geospatial object.

In this paper, Wang et al.’s deconvolution method [14] was used to estimate the point-scale semivariogram. This method utilizes statistical information in the coarse image without any a priori information. The focus of the ATPK method is on accurately inferring the point-scale variogram. Compared with the traditional surface-scale downscaling method, ATPK takes into account the shape and size of spatial data scales, exploits the spatial interrelationships of spatial data, and interpolates downscaled results with qualitative preservation.

4.2. Influence of Semivariograms

To verify the influence of the semivariogram on the downscaling results, 15 water objects of various shapes and sizes were selected within the study area, and the corresponding semivariogram of each object was calculated. Additionally, the global semivariogram within the selected area was compared. Figure 12a shows the distribution of selected water bodies in the study area, and Figure 12b shows the semivariogram of each water body. Figure 12d shows the semivariogram of the global membership map.

Different methods were used to calculate the semivariograms for fuzzy geospatial objects at different scales. Large-scale fuzzy geospatial objects only calculate their edge regions, while small-scale fuzzy geospatial objects calculate their full domains. Figure 12a shows that water bodies of different sizes and shapes have completely different semivariograms. The global semivariogram represents the average variability across the region. The comparison of the global semivariogram with the semivariogram of a specific fuzzy geospatial object shows how the fuzzy geospatial object semivariograms can differ significantly from the global one. The semivariogram values for different fuzzy geospatial objects are higher than the semivariogram of the global image.

Section 3.4 and Section 3.5 presented the experimental conclusions and discussed the difference between the semivariogram of geographical objects and global semivariogram in this chapter. Local modeling using semivariograms of fuzzy geospatial objects improve the accuracy of predicted images compared to ATPK downscaling using the global semivariogram.

5. Conclusions

In this paper, we proposed a new method for downscaling MF images based on the semivariograms of fuzzy geospatial objects. This method was mainly divided into three parts: the extraction of fuzzy geospatial objects, calculating semivariogram for each fuzzy geospatial object, and ATPK downscaling. Two kinds of semivariogram calculation methods were proposed for different scales of fuzzy geospatial objects: (i) the semivariogram construction method based on the boundary of fuzzy geospatial objects; and (ii) the semivariogram construction method for small-scale fuzzy geospatial objects.

This method improved the kriging interpolation method to estimate the weight of each observed data point based on the global spatial covariance. Fuzzy membership functions (MFs) were used for compounding multiple variance functions in the ATPK downscaling method.

The experiments demonstrated that local modeling using the semivariograms of fuzzy geospatial objects improve the accuracy of the predicted images relative to ATPK downscaling using global semivariograms. A simulation experiment based on the Worldview-2 dataset and a real experiment based on the Sentinel-2 dataset were used to validate the performance of the method proposed in this paper against other six classic downscale algorithms. This method achieved downscaled MF images with consistently the greatest accuracies compared to other methods, where the results contained more small-scale fuzzy geospatial object information and fitted better to real geographic boundaries.

Fuzziness is an inherent uncertainty that exists in the geographic world, as almost all natural geospatial objects have vague boundaries. The value of medium resolution geospatial data, such as digital ground models and medium resolution remote sensing images, can be enhanced by studying the downscaling method of fuzzy membership function at the geospatial object level. By using the downscaling method for geospatial objects, the ability of the fuzzy membership function to express local details can be effectively improved. This, in turn, improves the performance of fuzzy geospatial object representation, spatial analysis, and remote sensing image fuzzy analysis [42,65].