The proposed method of this paper consists of two main sections as follow: 1) calculating the classes’ proportions, 2) implementing Cellular Automata model to map sub-pixels.
2.1 Calculation of the classes’ proportions
The initial input of the proposed CA based SRM approach is classes’ proportions. Regarding to the soft classification uncertainties in classes’ memberships within mixed pixels (K C Mertens et al. 2003), a new absolute classes’ proportions calculation techniques is proposed. This approach provides ability to systematically compare the performances of existing SRM algorithms and also provides increased confidence of interpreting the results of different SRM’s results. Before calculating the classes’ proportions a pre-determined parameter named Scale Factor (SF) must be determined, in this step assume\(SF=3\). The total number of sub-pixels in each given course pixel is equal to \(SFSF\) (K C Mertens et al. 2003).
In order to explain this method in detail, an example is provided in Fig. 1a that illustrates a binary image consists of two classes in different colours (Yellow and Blue) with the size of \(15\times 15\) pixels. At the first step, it is assumed that Fig. 1a is a fine resolution map that produced from a course resolution map (Fig. 1b) using a particular SF value (\(SF=3\)). Thus, Fig. 1b can be considered as a synthetic coarse resolution image where each given pixel of that consists of an exact number of classes. This map is then employed to produce two abundance maps which illustrated as Figs. 1(c-1) and 1(c-2). Considering a given coarse pixel (row= 2 and column= 3) in Fig. 1b composed of 7 Yellow sub-pixels and 2 Blue sub-pixels, a set of the abundance values is assigned to the associated classes of that pixels in Figs. 1(c-1) and 1(c-2), respectively. This example will being used all over this paper.
2.2 Using cellular automata in sub-pixel mapping
Figure 2 illustrates the flowchart of the proposed CA based SRM approach. Utilizing Cellular Automata in sub-pixel mapping has six steps as follow:
In an overview, this method consists of five main steps and one iterative step that repeats from step 3 to 5. Firstly, the Scale Factor (SF) value and neighbourhood kernel size must be determined. Let \(SF\) be the scale factor, that is, each coarse pixel is divided into \(SFSF\) equal size fine sub-pixels. Then, the pure coarse pixels should be identified and their class label must simply assign to their corresponding subpixels. In the third step, the number of neighbours for unmapped sub-pixels are counted and a sub-pixel with the highest number of neighbours is selected. In the fourth step, Eq. 1 is used to calculate the membership value for all existing classes. In the next step, the sub-pixel is assigned to the class with the maximum suitability, and finally, step 3 and 5 are repeated to the point that all sub-pixels are mapped. These steps are explained in detail below.
2.2.1 First step
The original course spatial resolution map directly provides the location and class labels of pure pixels; therefore, in this step of the methodology, we employ that information to generate the partially completed SR maps which represents the pure pixels (Fig. 3). The number of generated partially completed SR maps is equal to the total number of classes in the original course spatial resolution map.
In this section an example is given in Fig. 3 to illustrate the first step of the proposed approach. Aa Fig. 3 represents, the Blue proportion matrix and Yellow proportion matrix are illustrated in Fig. 3(a-1) and Fig. 3(b-1), respectively. Regarding the \(SF\) value (\(SFSF=9\)), if the value of a given cell in the proportion matrix is 9, then that pixel could be considered as a pure pixel for that class. For every pure pixel location, their corresponding subpixels are assigned to appropriate class on the basis of their pure class labels, namely Blue and Yellow (Fig. 3(a-2) and Fig. 3(b-2)).
2.2.2 Second step
In this step, regarding the same size and resolution of both Fig. 3(a-2) and Fig. 3(b-2), the summation operation is applied between these two rasters to produce a combined pure sub-pixel maps (Fig. 4). As can be seen, there are three classes of pixels in Fig. 4: Blue colour, Yellow colour, and White colour; where, as mentioned before, Yellow and Blue colours represent the pure pixels, and White colour shows the location of the mixed pixels in the proportion matrices (Fig. 3(a-1) and Fig. 3(b-1)). Then, each White sub-pixel was considered as the centre pixel of a square kernel and the number of neighbourhood classified sub-pixels (Yellow and Blue colours) around the centre pixel was counted. In this example, a square kernel with size \(3\times 3\) is employed. Then, White sub-pixels with the maximum number of certain neighbours are identified (see sub-pixels in Red colour in Figs. 4(a) and 4(b)).
2.2.3 Third step
As mentioned before, the sub-pixels in red colour (Fig. 4(a) and (b)) denote the unlabelled sub-pixels with the maximum number of immediate neighbouring sub-pixels (highlighted in green colour) assigned to different labels. In this example, for each given red colour pixel, two integer values, which represents the number of neighbouring subpixels in different labels, are computed. For example, the Red sub-pixel in row 4 and column 7 has 5 Yellow neighbours and 0 Blue neighbour and sub-pixel in row 12 and column 9 has 0 Yellow neighbour and 5 Blue neighbours. Regarding to these data, an energy functions is proposed, as follow:
$${ef}_{ic}=\sum _{k=1}^{N}{\text{d} }_{k}/{d}_{ik}$$
1
Where:
\({ef}_{ic}\) Is the amount of energy for sub-pixel i in c class.
\({d}_{ik}\) Represents the Euclidean distance between sub-pixels i and k.
N is the number of neighbours.
If sub-pixel k belongs to an original pure pixel then \({\text{d} }_{k}=1\), otherwise \({\text{d} }_{k}=0\) .
For example, the energy function (Eq. 1) for the sub-pixel in red colour located at row 4 and column 7 is calculated as follows:
$${ef}_{\left(\text{4,7}\right)B}=\frac{\sum _{k=1}^{9}{\text{d} }_{k}}{{d}_{ik}}=\frac{0}{\sqrt{2}}+\frac{0}{1}+\frac{0}{\sqrt{2}}+\frac{0}{1}+\frac{0}{0}+\frac{0}{1}+\frac{0}{\sqrt{2}}+\frac{0}{1}+\frac{0}{\sqrt{2}}=0$$
2
$${ef}_{\left(\text{4,7}\right)Y}=\frac{\sum _{k=1}^{9}{\text{d} }_{k}}{{d}_{ik}}=\frac{1}{\sqrt{2}}+\frac{1}{1}+\frac{1}{\sqrt{2}}+\frac{1}{1}+\frac{0}{0}+\frac{0}{1}+\frac{1}{\sqrt{2}}+\frac{0}{1}+\frac{0}{\sqrt{2}}=4.12$$
3
In this examples, \({ef}_{\left(\text{4,7}\right)Y}=4.12\) represents the computed energy of a given sub-pixel (row 4 and column 7) belonging to Yellow class and \({ef}_{\left(\text{4,7}\right)B}=0\) shows the calculated energy of belonging to Blue class for that given sub-pixel.
2.2.4 Forth step
The aim of this step is assign an appropriate label to each given unlabelled sub-pixel (red colour) which was identified in the second step. The criterion to select a certain class is based on two conditions: (1) the corresponding portion value of that class in the proportion matrix must be higher than 0; (2) the computed energy value (Eq. 1) of that class must be higher than the energy value of other classes. With respect to these criterion, as Fig. 5 illustrates, a certain label is assigned to each given unlabelled sub-pixel (red colour). Then, for each given sub-pixel, its corresponding portion value in the proportion image must be updated by subtracting 1 from it. Therefore, through these processes, the initial sub-pixel map is constructed and the impacts of the changes in initial SRM on other sub-pixels labelling is iteratively considered.
2.2.5 Fifth step
In this section step 2 to step 4 are repeated until no more unlabelled sub-pixel can be found. Therefore, each sub-pixel will only have assigned to one class. Finally, all pure images are overlaid and final image is created. It is worth to mentioning that, any one sub-pixel only has one class therefore, there are no interferences between two or more classes for sub-pixels through overlaying.
2.3 Accuracy Assessment
In this paper two indexes are used for accuracy assessment: Percent Correctly Classify (PCC) and PCC’. They both look same; however, PCC considers the pure and mixed pixels and PCC’ just evaluate the mixed pixels (Wu, Du, and Member 2017; Zhang et al. 2008).
In this article, a widely used benchmark data set provided by ISPRS are utilized. This data set includes Lidar point cloud, aerial images, Digital Surface Model (DSM) and mosaic orthophoto map from the city centre of Vaihingen and Potsdam of Germany. The Vaihingen Potsdam dataset have the ground sampling distance (GSD) of 9 cm and 5cm, respectively. Each orthophoto image has Green (G), Red (R) and Near Infrared (NIR) channels (Cramer, 2010). In this research, area 1 of Vaihingen dataset is selected which is the city centre of that region and includes historical buildings with sophisticated shapes and some trees (Fig. 6), also four areas, which includes different geometries such as circular, triangular, rectangular and linear forms, of Potsdam data set are considered (Fig. 7).
In addition, the aerial images were visually interpreted by ISPRS to produce the land use map with GSD of 9cm. It includes six classes, as Road, Building, Low height vegetation\Grass, Tree, Car and Back ground. The ISPRS benchmark has been widely utilized as the training data for different models and also to assess the performance of different algorithms and methodologies. This dataset is freely available through https://www.isprs.org/education/benchmarks.aspx.
Regarding to comparison, two synthetic images (Concentric circles and circle images) were produced using Matlab software. These images have an equal size of \(300\times 300\) pixels. The radious of circle image considered 100 pixels, and concentric circles provided in 50 pixels intervals which resulted in 3 circles (Fig. 8).