A Multi-Feature Framework for Quantifying Information Content of Optical Remote Sensing Imagery

Abstract

Quantifying the information content of remote sensing images is considered to be a fundamental task in quantitative remote sensing. Traditionally, the grayscale entropy designed by Shannon’s information theory cannot capture the spatial structure of images, which has prompted successive proposals of a series of neighborhood-based improvement schemes. However, grayscale or neighborhood-based spatial structure is only a basic feature of the image, and the spatial structure should be divided into the overall structure and the local structure and separately characterized. For this purpose, a multi-feature quantification framework for image information content is proposed. Firstly, the information content of optical remote sensing images is measured based on grayscale, contrast, neighborhood-based topology, and spatial distribution features instead of simple grayscale or spatial structure. Secondly, the entropy metrics of the different features are designed to quantify the uncertainty of images in terms of both pixel and spatial structure. Finally, a weighted model is used to calculate the comprehensive information content of the image. The experimental results confirm that the proposed method can effectively measure the multi-feature information content, including the overall and local spatial structure. Compared with state-of-the-art entropy models, our approach is the first study to systematically consider the multiple features of image information content based on Shannon entropy. It is comparable to existing models in terms of thermodynamic consistency. This work demonstrates the effectiveness of information theory methods in measuring the information content of optical remote sensing images.

1. Introduction

With the rapid development of remote sensing image acquisition technology, these massive, multi-source, and heterogeneous image data have significantly changed the way humans use remote sensing data to perceive the world. Rapid and automatic screening of images that meet the user’s needs from massive image data is a key technology for the improvement of image data utilization and to fully exploit the value of images’ big data. The information content of images can be considered to be one of the critical reference indicators for image value assessment [1,2]. However, when considering the information content as an evaluation index for image screening, it is necessary to ensure the objectivity of the information content and the richness of the image information content covered. Therefore, fully extracting multiple features of images and designing information quantification methods can help to promote the development of automatic image big data screening technology [3].

In fact, establishing an information theory-based model for analyzing remote sensing information content is considered to be fundamental in quantitative remote sensing [2,4]. Geographic information science, including remote sensing science, as a branch of information science, has recently been the subject of calls for attention [5,6]. On the one hand, Shannon entropy [7] has been widely adopted as the most popular method in the study of measuring image information, because of its interpretability and computational simplicity. However, since Shannon entropy relies entirely on the statistics of pixel grayscale, which fails to capture the spatial structure of images, a series of improved Shannon entropies have been proposed [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. On the other hand, some scholars have reconsidered the Boltzmann entropy [23,24,25,26,27,28,29,30,31] model developed in the field of landscape ecology, which is mainly used to measure the degree of disorder in spatial structure. As can be seen, existing entropy-based information measurement models tend to assess the information content of images by utilizing a single feature. However, the information content of remote sensing images theoretically consists of multiple features, e.g., grayscale, the basic feature of images, which characterizes the degree of diversity and homogeneity of land cover. The spatial structure of the image contains both textural and geometric features, the former characterizing the heterogeneity of the same land cover type at a local scale, and the latter characterizing the structural disorder of all land cover types at an overall scale. Therefore, the grayscale Shannon entropy, improved neighborhood-based Shannon entropy, and Boltzmann entropy are measurements of information content on based only on individual features. In order to reflect the richness of features covered by image information content metrics and to satisfy the application of information metrics in, for example, image screening, the quantification of remote sensing image information requires multiple features to be considered simultaneously.

To address this issue, a multi-feature information content measurement framework for the remote sensing of images is proposed to quantify images’ comprehensive information content. A series of computational methods for quantifying the information content of each low-level feature is established based on Shannon’s information theory, while the high-level features obtained by image classification or segmentation are not considered to avoid the introduction of external uncertainties. Specifically, firstly the information content of optical remote sensing images is divided into grayscale, contrast, neighborhood-based topology, and spatial distribution features instead of simple grayscale or spatial structure. Secondly, in quantifying the information of each feature the entropy metrics based on grayscale, local binary pattern (LBP), neighborhood pixels, and intra-distance are designed to quantify the uncertainty of images in terms of pixel and spatial structure, respectively. Finally, a weighted model is used to calculate the comprehensive information content of the images.

The major contributions of this work are as follows:

• A framework for measuring the information content of remote sensing images based on Shannon’s information theory is developed. In this framework, the information of an image is defined as the uncertainty of the pixels’ properties and spatial locations. Instead of simply measuring single feature information, multiple features including grayscale, contrast, neighborhood topology, and spatial distribution are modeled to calculate the information content of images. The spatial structure information content of the image at both the overall and local scales is measured as two parts of the comprehensive information.
• Entropy metrics at each feature are designed to quantify the uncertainty of the image in terms of pixel and spatial structure. Compared with state-of-the-art entropy models, our approach is the first study to systematically consider the multiple features of image information content based on Shannon entropy. It is comparable to existing models in terms of thermodynamic consistency.

The rest of our paper is as follows: Section 2 provides an overview of related works. In Section 3, the framework for measuring the information content of remote sensing images based on multi-feature is elaborated, and details of the measurement method are described. Experiments and discussion are presented in Section 4. Finally, the conclusions and future work are summarized in Section 5.

2. Related Works

Spatial data contain spatial information, and the basis of information measurement lies in the definition of spatial information. Shannon’s information theory uses uncertainty as the definition of information and is widely used in urban sprawl analysis [32,33], landslide sensitivity analysis [34], and image processing applications such as band selection [35], image fusion [36,37], classification [38,39], and quality assessment [40]. In addition, Boltzmann entropy, which uses disorder information, has also made significant progress recently [23,24,25,26,27,28,29,30,31]. In addition to the entropy model, a few scholars take the classification accuracy of images [41], pixel difference [42], and human visual features [1] as information of the image, opening ideas in image information quantification theory. In general, entropy models are the mainstream method for information quantification. There are currently two main types: Shannon entropy, which uses uncertainty information and Boltzmann entropy, which uses spatial structure disorder information.

2.1. Shannon Entropy

Shannon entropy regards the image as a two-dimensional discrete signal, and takes the image pixels as independent random variables. The grey entropy is calculated through the statistics of the proportion of each grey pixel in the image [7,43]. Therefore, this model assumes that the greater the uncertainty of the ground object, the higher the amount of information, which is essentially the statistical information of the grayscale feature of the image. Among the wide range of applications of Shannon entropy, some key factors that may lead to changes in image information have been analyzed, such as noise [44], band-to-band relationship [45], neighborhood pixel relationship [45,46], and image scale [47], these factors are quantified to construct probabilistic statistical indicators. In addition, integrals [48], Markov chains [49], geostatistics [2], and multi-feature correlation [50,51] are used to model the correlation of image pixels and the correlation of bands to reduce redundant information content in remote sensing images. However, Shannon entropy relies entirely on the statistics of pixel grayscale and fails to capture the spatial structure of imagery. Therefore, a series of efforts have been made in academia to improve Shannon entropy into a spatial entropy that reflects spatial heterogeneity. For example, there are three main types of these improved spatial entropy models. The first type is Batty spatial entropy [8,9,10,11], which divides the unequal space into sub-regions to cause the Shannon entropy to have spatial properties. Quweider [12] modified the Batty spatial entropy and used it as a cost function for image segmentation. The second type of improved spatial entropy models are the discrimination ratio-based spatial entropy model proposed by Claramunt et al. [13] and its improvement scheme [14,15]. Gao et al. [16] simplified the algorithm of this model to improve its computational efficiency for image processing. The third type is spatial entropy based on co-occurrence patterns, which was proposed by O ’Neilld et al. [17], followed by Leibovici [18] and Leibovici et al. [19] who introduced the concepts of co-occurrence distance and co-occurrence order. Subsequently, Altieri et al. [20,21] considered spatial distance as a variable and proposed a range–occurrence approach-based spatial entropy model. Yu et al. [22] introduced the co-occurrence spatial entropy to land cover data and constructed a metric for evaluating spatial heterogeneity. Moreover, the spatial entropy based on human vision proposed by Fang et al. [3] can also be seen as a special form based on co-occurrence patterns. A thermodynamics-based evaluation of various improved Shannon entropies was performed to analyze the effectiveness for configuration capture by Gao et al. [52].

2.2. Boltzmann Entropy

Recently, the entropy model based on Shannon’s information theory has been questioned because the link between entropy and thermodynamics is only a form of parallelism in landscape ecology [53]. It indicates that even though the improved Shannon entropy can capture the difference between two different spatial structure images, its entropy value is independent of the disorder in thermodynamics. Therefore, some scholars have redirected their focus to Boltzmann entropy in classical thermodynamics, to quantify the disorder of the spatial structure of the landscape. In general, there are two main types of Boltzmann entropy models according to the application data. The first type is used for spatial heterogeneity analysis of qualitative raster data such as land cover data, for example, the proposed Boltzmann configuration entropy based on the total number of edges [23,24] and the Wasserstein metric-based Boltzmann configuration entropy [25,26,27]. The second type of Boltzmann entropy is used for numerical raster data, such as remote sensing images and digital elevation model (DEM). Gao et al. [28] designed the resampling-based method, which took the spatial uncertainty in the scale transformation as the macroscopic state. Subsequently, Gao et al. [29] and Zhang et al. [30] improved the model to improve the computational efficiency. In addition, Gao et al. [31] further proposed an improved version based on aggregation techniques in order to maintain the thermodynamic consistency of the Boltzmann entropy model. These novel Boltzmann entropy models have shown their effectiveness and potential to replace Shannon entropy in hyperspectral band selection [54,55], landscape ecology applications [56,57], and measurement of remote sensing image information content [1,58].

2.3. Imperfection of the Entropy-Based Information Model

However, it is worth noting that Shannon entropy, the improved neighborhood-based Shannon entropy, and the Boltzmann entropy all measure the information content based on a single feature. More specifically, Shannon entropy only focuses on the greyscale feature information content of images. Boltzmann entropy, developed in geography, was originally proposed to quantify the disorder of the spatial configuration of landscapes. Although it has shown the ability to quantify the spatial structure information of remote sensing images [1,58], it does not consider the multi-feature hierarchical characteristics of the image information content. Theoretically, the information content of remote sensing images consists of multiple features, including both spatial and non-spatial features of the image, such as grayscale, the basic feature of images, which characterizes the degree of diversity and homogeneity of land cover. The spatial structure of the image contains both textural and geometric features, the former characterizing the heterogeneity of the same land cover type at a local scale, and the latter characterizing the structural disorder of all land cover types at an overall scale. Previous work has lacked this consideration. Furthermore, when the classification accuracy [41] and human visual features [1] are used as the measurement solution of image information, the essence is to measure image information using several higher features. However, the segmentation and classification techniques used to evaluate the shape and spatial relationships of objects in an image inevitably introduce external uncertainty, breaking the closure of the original self-system of the image. Therefore, the results obtained from the Shannon entropy model will hardly be accurate and objective, and other information quantification models may need to be constructed instead.

3. The Multi-Feature Framework for Image Information Measurement

Firstly in this section, the measurement framework of the information content of remote sensing images is introduced. Secondly, the sources of image uncertainty and the reasons for selecting image features are discussed. Thirdly, the modelling method based on Shannon’s information theory at each feature is elaborated. Finally, a weighted entropy-based fusion scheme is used to calculate images’ comprehensive information content.

3.1. The Multi-Feature Measurement Framework

The grayscale image information content measurement framework can be divided into both non-spatial and spatial features. The non-spatial information content uses pixel grayscale entropy information and the spatial structure information content is divided into overall and local information, with each being measured separately. In the measurement of spatial structure features, the LBP operator, neighborhood pixels, and intra-distance are designated as the feature variables for calculation of the entropy metric. The overall flow chart is shown in Figure 1. For multi-spectral/multi-band remote sensing images the final information can take the sum value of the information content of each band, without considering the correlation of bands.

3.2. Feature Selection of Images

Shannon’s information theory quantifies the information content in terms of uncertainty; specifically, the amount of uncertainty eliminated equals the amount of information content obtained [7]. Accordingly, identifying the origins of image uncertainty is the key to quantifying the information content of an image. Many efforts have been made to study the uncertainty of remotely sensed images and summarize the uncertainty of images into two aspects: uncertainty of pixel’s properties and spatial location [59,60]. The uncertainty of its properties is represented by a grayscale of pixels, the pixel’s spectral response as detected by the sensor system, is a mixture of the individual responses of each component within the pixel [61]. The uncertainty brought by the spatial position comes from the difference between the attributes of the pixel itself and the spatial position. The pixels at these different positions contain the spatial position information content of the image, i.e., texture and geometric structure information.

Moreover, the information content of the spatial location at the observation scale can be further divided into local and global representation information. Therefore, for the characterization of the uncertainty of the spatial location of pixels, the information content at contrast and the neighborhood topological feature are used to quantify the local representation information. In contrast, the spatial distribution of pixels is used to quantify the global representation information.

Based on the above considerations, the following four low-level image features are selected to characterize the information content of the image in spatial and non-spatial attributes:

• Grayscale is the most basic feature of remote sensing images. It reflects the spectral characteristics of different objects in the form of digital numbers (DN). Grayscale information content can be directly quantified by statistical histograms according to Shannon entropy;
• Contrast is a common visual feature of images and is used to reflect ground objects’ texture and geometric characteristics. To keep the texture features of the image light invariant, LBP entropy is designed to quantify the contrast information;
• The complexity of the neighborhood topology relationship is measured by the neighborhood pixel differences. For a certain pixel, the more that adjacent pixels are similar, the less important the pixel is. Neighborhood topological information is measured using neighborhood topological entropy, which describes the relationship between a central pixel and its eight neighbor pixels;
• The spatial distribution of pixels reflects the configuration of the spatial structure of the image as a whole. To measure the degree of disorder in the spatial distribution of pixels, the intra-class distance entropy is used to quantify the disorder of each grayscale pixel in the spatial distribution;

Notably, when considering the use of Shannon information theory to measure the information content of remote sensing images, the closure of the image system is necessary to ensure the objectivity and accuracy of the information quantification results. Therefore, the spatial information content should derive from the spatial location of the image pixels themselves rather than based on the spatial location of the objects. This is because there are high-level feature contents from human vision, such as the shape and spatial relationships of objects; however, these high-level features require segmentation and classification to extract objects from images, which will introduce external uncertainty in the imaging system and inevitably disturb the information content of the images.

3.3. Information Quantification Model for Each Feature

In this section, we will elaborate on how the information content of each feature is measured. The final comprehensive information content of the image is the weighted sum of the four basic feature information contents mentioned above.

3.3.1. Grayscale Information

Image can be regarded as two-dimensional finite discrete signals [3,43]. A discrete random vector $\mathrm{X}=\left\{x_1,x_2,\dots ,x_m\right\}$ with finite elements and its statistical probability space is $\mathrm{P}=\left\{p_1,p_2,\dots p_m\right\}$, then Shannon entropy is given as follows [7]:

$$H_{gray}=-{{\displaystyle \sum }}_{i=1}^Q{p}_{i}lo{g}_2{p}_i$$

(1)

where $x_i$ is the pixel number of the ith grey level in the remote sensing image, $p_i$ is its probability of $x_i$, and Q is the number of classes of grayscale. When the image grayscale is uniformly distributed, $H_{gray}$ is the maximum value $lo{g}_{2}Q$. It can only measure the information of a single grayscale feature but cannot reflect the information content of other features of the image. Figure 2a,b has the same spatial distribution but different composition, while Figure 2b,c has the same composition but different spatial distribution. However, according to Equation (1), the Shannon entropy measurement results for the three simulated data are the same.

3.3.2. Contrast Information

Contrast is one of the core metrics for image texture feature representation [62] and it has been proven to be a powerful visual feature [63,64]. Contrast reflects the texture’s sharpness and heterogeneity; therefore, it can effectively reduce the uncertainty in target recognition. As a characteristic indicator, the contrast can be directly measured regarding the range and polarization degree of the grayscale values in images. However, to calculate the contrast entropy, a random feature vector related to contrast needs to be established. The LBP operator is a texture feature extraction method that is widely used in computer vision and image processing. It uses binary encoding to describe the difference between the grey value of the center pixel and the neighboring pixels to characterize the local information content of texture.

The LBP operator is derived in a 3 × 3 window, with the central pixel of the window as the threshold. The gray values of the adjacent 8 pixels are compared with it; if the surrounding pixel value is greater than the central pixel value, the pixel’s position is marked as 1; otherwise, it is marked as 0. This way, an 8-bit binary number can be generated for comparison in this window and converted into a total of 256 decimal numbers. The LBP code of the center pixel (p, q) can be expressed as:

$$LB{P}_{n_s,r}\left(p,q\right)={{\displaystyle \sum }}_{k=0}^{n_s-1}f\left(g_k-g_{\left(p,q\right)}\right)·2^k$$

(2)

where f (∙) is the threshold function, $f\left(x\right)=\{\begin{array}{c}0, x<0\\ 1, x\ge 0\end{array}$, $n_s$ represents the number of sample points with radius r and $g_k$ and $g_{\left(p,q\right)}$ are the grayscale values of the neighborhood pixel k and the center pixel $\left(p,q\right)$, respectively. Figure 3 shows a simple LBP operator calculation example and clearly reflects that the diversity of the LBP codes is huge, ranging from 0 to 255.

Obviously, the contrast is only related to the central and adjacent pixel’s relative grey values. Therefore, theoretically, using LBP as the contrast feature variable of the image can better characterize the local features of the image, because of its independence from the light intensity. Using a 3 × 3 sliding window, the LBP value of each pixel of the remote sensing image is calculated in turn to generate the LBP feature map. The formula for calculating image LBP entropy is:

$$H_{con}=-{{\displaystyle \sum }}_{i=0}^{255}\frac{N_{lp{b}_i}}{N_c}lo{g}_2\frac{N_{lp{b}_i}}{N_c}$$

(3)

where $N_{lp{b}_i}$ is the number of elements in the LBP feature map whose LBP value is equal to I and $N_c$ is the total number of LBP codes categories.

3.3.3. Neighborhood Topology Information

The pixels of a raster image are arranged in a regular grid, adjacent to each other so each pixel has a spatial neighborhood. This means that any pixel on a spatial unit will be positively or negatively affected by the surrounding units (i.e., its neighbors) to some extent [65]. According to Li and Huang [66] in the thematic information of map symbols, “if a symbol has neighbors all of the same thematic type, then the importance of this symbol is very low, in terms of thematic meaning.” Therefore, pixels are regarded as regular grid symbols in the image, a 3 × 3 neighborhood space is established pixel by pixel, and the information content of the neighborhood topology feature is calculated to characterize the difference of local spatial structure.

As shown in Figure 4, the frequencies of the same pixel values in the eight neighbors are calculated. For each central pixel p, its neighborhood topology information is defined as the entropy of its eight neighboring pixel vectors. The information content of image neighborhood topology is the average value of neighborhood topology information of all pixels, as shown in the following formula:

$$H_{nei}=-\frac{1}{N}{{\displaystyle \sum }}_{i=1}^N{{\displaystyle \sum }}_{j=1}^{}\frac{n_{ne{i}_{i,j}}}{8}lo{g}_2\frac{n_{ne{i}_{i,j}}}{8}$$

(4)

where $n_{ne{i}_{i,j}}$ indicates the number of neighboring pixels with the same grayscale value as the pixel at the jth neighboring position in the ith 3 × 3 sliding window and N is the total number of the image pixels.

It is obvious that not all pixels have a 3 × 3 neighborhood when computing an image’s contrast and neighborhood topological information content. Therefore, the edge of the image is expanded and filled in advance. In this paper, the filled pixels are the same as those of the edge row and column.

3.3.4. Spatial Distribution Information

In the measurement of spatial distribution feature information, pixels with the same grey value are regarded as elements of the same type, rather than clustering image pixels according to the shape or classification attributes of ground objects. The intra-class distance is then calculated at each pixel class level to generate a distance feature vector, as shown in Figure 5. Figure 5a–c all consist of four pixels; thus, six line segments are formed between the two points. However, due to differences in the distribution of spatial locations, the same number of line segments may have different Euclidean distances (d) and therefore generate vectors with different spatial distributions. For example, there are two cases of line distance in Figure 5a, four in Figure 5b and six in Figure 5c. It is clear that the more disordered the distribution, the more categories of elements the intra-class distance vector has.

For each grayscale q, q ∈ (0, Q), its spatial distribution information is defined as the intra-class distance entropy of each grayscale. The information content of spatial distribution is the average value of all grey-level spatial distribution information as shown in the following formula:

$$H_{spa}=-\frac{1}{N_q}{{\displaystyle \sum }}_{i=1}^{N_q}{{\displaystyle \sum }}_{j=1}^J\frac{N_{d_{i,j}}}{N_{d_i}}lo{g}_2\frac{N_{d_{i,j}}}{N_{d_i}}$$

(5)

$$N_{d_i}=C_{N_i}^2=\frac{N_i!}{2!·\left(N_i-2\right)!}$$

(6)

where $N_q$ is the total number of image gray levels; $N_{d_{i,j}}$ is the number of pixel levels with ith grayscale whose intra-class distance is equal to $d_{i,j}$; $N_{d_i}$ is the total number of intra-class distances of the ith grayscale, which can be calculated by the $N_i$ of pixel i according to Equation (6); and J is the total number of elements of the intra-class distance vector. Notably, the spatial distribution information is 0 when $N_i$ is equal to 1, since it takes at least two points to calculate the distance.

3.4. Comprehensive Information Content of Images

After the information content of individual features is calculated, the comprehensive information content of the images is calculated with a weighted model. Since the measurement of each feature was performed with a consistent form of the Shannon entropy model, they are of the same magnitude and do not need to be normalized. The calculation formula is as follows:

$$H=w_{gray}{H}_{gray}+w_{con}{H}_{con}+w_{nei}{H}_{nei}+w_{spa}{H}_{spa}$$

(7)

where w is the weight of each feature and H is the amount of information contained in each feature. The weight w can be adjusted according to different practical problems. For example, the greater the weight of the spatially distributed feature, the more suitable for evaluating the amount of image information with complex spatial structure. In this paper, conservative parameter settings are used, giving equal weights (w = 1) to calculate the comprehensive information. For multi-spectral/multi-band remote sensing images, the final comprehensive information content of the image is the sum of the information content of each band. For example, the comprehensive information content of RGB images is calculated as follows:

$$H_{RGB}=H_R+H_G+H_B$$

(8)

where $H_R, H_G$, and $H_B$ are the results of the image information calculated using Equation (7). In addition, it should be noted that the RGB images were converted to grayscale images for information measurement in our experiments.

4. Experiment and Discussion

In this section, three groups of experiments are designed to verify the validity and applicability of the proposed method. Among them, the scene images have significant surface differences, which facilitate intuitive judgments about the richness of land cover and the complexity of spatial structure. In addition, a set of remote sensing images of a large-scale range is necessary to analyze the changing pattern of the amount of information content at different features with the spatial resolution. The simulated data are used to assess the proposed method’s ability to capture spatial structural changes and the thermodynamic consistency of the disordered processes. The experimental scheme and data are as follows.

• Experiment 1: A few images are randomly selected from the AID [67] image dataset to assess consistency trends between information content and scene complexity. The AID dataset includes 30 different aviation scenes and 10,000 samples of image size 600 × 600 × 3;
• Experiment 2: The large-scale Gaofen-2 (GF-2) satellite images from the GID dataset [68] were used to analyze the changing trends in image information content with spatial resolution;
• Experiment 3: Simulated images generated from an AID original image. Specifically, a series of disordered images are generated by exchanging the local spatial structure or by simulating the random motion of gas molecules in pixel units, which have the same grayscale histogram but different spatial structures. This experiment analyzes the ability of the proposed method to capture the spatial structure and the proposed method’s thermodynamic consistency in measuring disorder is evaluated;

The proposed method is compared with six baseline methods. The classical Shannon entropy (H_gray) [7] is used to evaluate the basic performance of the proposed method. Some state-of-the-art entropy methods, geometrical mapping entropy (SE2) [3], Wang’s spatial entropy (WHs) [14], Claramunt’s spatial entropy (CHs) [13], Boltzmann relative entropy (BLHR) [28], and Wasserstein-based Boltzmann entropy [25] with von Neumann neighborhood [69] (WEH4), were performed as comparison schemes herein.

4.1. Analysis of Scene Image Information Content

To evaluate the differences in the information content of natural scene images, nine typical surface scenes were selected from the AID dataset, and two images were randomly selected in each scene, as shown in Figure 6.

The information content measurement results of various scenes are consistent with the human visual experience, as shown in Figure 7. The comprehensive information of bare land is the lowest, while that of dense residential is the highest. According to each feature’s information sequence result, if the image’s grayscale diversity is small, there will be less information at various features. Since the grayscale of bare land image pixels is the purest, the information content of only the bare land spatial distribution feature is not the highest. For other scenes where the land cover is not completely composed of a single type, the spatial distribution information is generally larger than other features.

Meanwhile, due to the difference in the richness of land cover, images of the same scene do not have the same feature information, such as pond and mountain. It can be seen that the spatial distribution information of pond_392 and mountain_18 is relatively high due to the more complex texture structure. Comprehensively, the amount of information of various features varies in different scenarios. For example, in this experiment, dense residential has the highest grayscale information and spatial distribution information, the park has the highest amount of contrast information, and forest and mountain have the highest amount of neighborhood topology information due to vegetation differences and shadows.

To compare and analyze the performance of other information measurement models, the measurement results of different scene image samples obtained by the same method are normalized, that is, $x^{\prime }=\frac{x-\min \left(x\right)}{\max \left(x\right)-\min \left(x\right)}$, as shown in Figure 8.

Intuitively, except for the WHs spatial entropy model, the trends of other types of information measurement models are roughly consistent with slight differences, as shown in Figure 8. Specifically, except for the results of WEH4, the images with the lowest amount of measured by other models are bareland_9, and the images with the highest information are denseresidential_144 and mountain_18. It can also be observed that the information distribution trend calculated according to Shannon entropy is more consistent than that calculated by Boltzmann entropy models (WEH4, BLHR). Essentially, the results of different information measurement models should not be compared because of their significantly different definitions of information. Thus, the normalized value is suitable for qualitative analysis of the relative content of each scene information. Since the comprehensive information content (H) includes both the grayscale diversity and the spatial structure information of the texture, the relative information content may be significantly different, such as the two image samples of the mountain and pond.

4.2. Analysis of Information and Resolution

This experiment analyzes the relationship between the information content and spatial resolution of remote sensing images. The experimental data derive from three large-scale images randomly selected from the GID dataset, and the size of the remote sensing image is cropped to 6000 × 6000 × 3. The 2 × 2 sliding window is used for mean filtering, which is calculated by rounding the average of four adjacent pixel values, i.e., f(x) = ROUND (($x_1+x_2+x_3+x_4$)/4). In this manner, the original image is iteratively downsampled to obtain image pyramids of different spatial resolutions, as shown in Figure 9. Theoretically, a new low-resolution image is generated after each filtering, and then the information content of each feature of this low-resolution image is calculated separately. In our experiment, in order to improve the computational efficiency, the epoch interval is set to 50, and then the various feature information content of the current resolution image is calculated.

As shown in Figure 10, as the image’s spatial resolution decreases, the information content of various features decreases significantly. Overall, during the downsampling process, the grayscale diversity of the image tends to decrease gradually, and this trend becomes stronger as image size decreases. As the basic attribute of pixels, as grayscale diversity decreases, the uncertainty of feature variables of image neighborhoods and spatial distribution will also decrease gradually. It can be observed in Figure 10 that the blurring effect on the image edge features during the downsampling process decreases the spatial distribution information, indicating that it has the highest linear correlation with the spatial resolution. However, due to the uncertainty of the downsampling, the image information may decrease rapidly. Then, the information content increases, as shown in the information change result of image (c) in Figure 10. Finally, when the spatial resolution is reduced to a certain extent, the contrast and neighborhood topological information will gradually become stable. Then, it may increase slowly as the resolution continues to decrease.

In addition, it can be observed that there is some correlation between the multiple feature information contents in Figure 10. The correlation between the various feature information content results is further analyzed using the three original images and the data generated by downsampling to verify the independence of the proposed multi-feature model. The Spearman correlation coefficients of the information measurement results from the downsampling process are shown in

Table 1. Spearman correlations of the information content of each feature in downsampled images.

Features	${\mathit{H}}_{\mathit{g}\mathit{r}\mathit{a}\mathit{y}}$	${\mathit{H}}_{\mathit{c}\mathit{o}\mathit{n}}$	${\mathit{H}}_{\mathit{n}\mathit{e}\mathit{i}}$	${\mathit{H}}_{\mathit{s}\mathit{p}\mathit{a}}$
$H_{gray}$	1.00	0.89	0.83	0.97
$H_{con}$	0.89	1.00	0.91	0.81
$H_{nei}$	0.83	0.91	1.00	0.65
$H_{spa}$	0.97	0.81	0.65	1.00

. It can be seen that there are certain correlations between the information contents of different features, among which $H_{gray}$ and $H_{spa}$ have the highest degree of correlation, ${\rho }_{gray-spa}=0.97$, followed by $H_{con}$ and $H_{nei}$, ${\rho }_{con-nei}=0.91$; the least correlated are $H_{nei}$ and $H_{spa}$, ${\rho }_{nei-spa}=0.65$. On one hand, this is because the spatial distribution information is closely related to the diversity of image gray levels. Obviously, the larger the grey level of the image pixels, the more information levels of spatial distribution that need to be calculated. The resolution will significantly reduce the number of grayscales, reducing the amount of spatially distributed information. On the other hand, both $H_{con}$ and $H_{nei}$ use a 3 × 3 sliding window. The difference between $H_{con}$ and $H_{nei}$ is that the former mainly measures the relative difference of neighboring pixels. At the same time, the latter represents the absolute difference of neighboring pixels, which causes them to have a certain correlation.

4.3. Analysis of Spatial Configuration and Disorder

The symmetry defect of the classical Shannon entropy model makes it unable to effectively capture the changes and disorder in the spatial structure of images. To verify the ability of each feature’s information content to perceive an image’s spatial structure, we exchange the local spatial structure of an image, as shown in Figure 11. The original image is divided into nine sub-image blocks. Then, the spatial position of each sub-image block is gradually exchanged to generate a series of simulated images with the same grayscale histograms but obvious different spatial structures.

As shown in Figure 12 and

Table 2. Information content results of the original and simulated image (bit/pixel).

Image	${\mathit{H}}_{\mathit{g}\mathit{r}\mathit{a}\mathit{y}}$	${\mathit{H}}_{\mathit{c}\mathit{o}\mathit{n}}$	${\mathit{H}}_{\mathit{n}\mathit{e}\mathit{i}}$	${\mathit{H}}_{\mathit{s}\mathit{p}\mathit{a}}$	$\mathit{H}$
A	6.64	5.39	1.86	11.61	25.50
C	6.64	5.39	1.87	12.47	26.37
D	6.64	5.39	1.87	15.45	29.35
E	6.64	5.39	1.87	17.42	31.32
F	6.64	5.40	1.87	21.40	35.31

, grayscale entropy does not change after the local sub-blocks are exchanged. In addition, since the scale of the exchanged sub-blocks is relatively large compared to the pixels, the pixels whose local space changes in each sub-block are mainly at the edge and the number is much smaller than the pixels inside the sub-block image. Therefore, the images’ contrast and neighborhood topological information did not change significantly. Fortunately, the spatial distribution information changes significantly as the overall spatial structure of the image changes, indicating that it captures the changes occurring in the overall spatial structure of the image. Therefore, the experiments demonstrate the rationality of splitting the spatial features of images into both local and global scales. The proposed multi-feature image information quantization framework in this paper effectively enhances the multi-scale perception of image spatial structure features with certain robustness. It can not only effectively measure the spatial structure information of the image at the local and global scales, but also meet a wider range of applications with the decomposability of comprehensive information.

To analyze whether the information at each feature satisfies thermodynamic consistency in the perception of spatial structure changes, we reduced the scale of the above experimental sub-block exchange and simulated the mixing process of gas molecules, as shown in Figure 13. The pixels of the original image are randomly swapped in each iteration. Specifically, the random swapping algorithm of pixels selects half the pixels of the image produced by the previous iteration and then swaps the position of each cell with a randomly selected neighbor cell. In this simulation, the maximum number of iterations was set to 25,000 epochs. Similarly, the information content of the simulated images was calculated and recorded every 50 epochs, resulting in 500 simulated images with disordered spatial configurations.

When the pixels of the image are regarded as gas molecules exhibiting thermodynamic motion, the spatial distribution of the image will gradually change from highly ordered to increasingly disordered, as shown in Figure 13. However, since the spatial distribution of pixels does not affect the grayscale statistical histogram, the amount of grayscale information will not change. For other feature information content, as shown in Figure 14a, it increases logarithmically with the number of iterations of mixing, then remains relatively stable. Therefore, it can be seen that in the small-scale local exchange process, consistent with the spatial distribution entropy, both the LBP entropy and the neighborhood topological entropy can perceive the local spatial distribution of the image to a certain extent through the neighborhood space and have strong thermodynamic consistency. These feature information contents also help the comprehensive information of the image to inherit thermodynamic consistency.

Compared with other entropy models, as shown in Figure 14b, the results of SE2 are not consistent with thermodynamics. Although this model can perceive the spatial structure of images, it cannot capture disorder. The thermodynamic consistency of WHs and CHs is poor because their changing trends during the iterative process are not smooth and fail to converge at the end. The thermodynamic consistency of the two Boltzmann entropy models (BLHR, WEH4) showed validity in this experiment. However, Boltzmann entropy is mainly used to quantify the disorder of landscape configuration, and it does not take into account the hierarchical information content of remote sensing images. For example, it cannot measure the most basic grayscale information of images.

4.4. Discussion

Overall, the experimental results show that the information of images are consistent with the trend of visual experience. The richer the land cover and the more complex the spatial structure of the corresponding image is, the higher the value of the comprehensive information content. In addition, the change characteristics of each feature information and spatial resolution of the image were analyzed. Specifically, all feature information decreased as the resolution decreased, but the spatial distribution information had the highest linear correlation with the degree of change in resolution. Contrast and neighborhood topological information decrease as the resolution decreases, tends to stabilize, and may even increase information. Moreover, other information measurement models, such as the Shannon entropy, which uses grayscale as the information content of the remote sensing image, and the Boltzmann entropy, which uses spatial structure disorder as information content, essentially base their information content on individual features. Since neither considers the hierarchical nature of comprehensive information of image, the measurement results of the information are biased to a certain extent.

The performance of different entropy models to characterize spatial structure and disorder was evaluated in the experiments. However, it should be noted that the nature of their information is different. As noted by Wiener [70], “Information is not material nor energy while it is the information”. There is currently no clear and accepted definition of information, and various different and implicit concepts are at play in different fields [71]. In the field of communication, Shannon’s information entropy, which uses uncertainty as the definition of information, fails to reflect the information content at the spatial structure feature in remote sensing image data. On the contrary, Boltzmann entropy, which takes the spatial disorder as image information, has no statistical information content such as image grayscale. In theory, the two forms of entropy are similar and can be converted into each other under certain conditions [72]. Therefore, when comparing the performance of existing information models of images, it is important to define the nature of information to ensure the objectivity of quantitative information model performance analysis and evaluation.

5. Conclusions

Shannon’s information theory still has advantages in image information measurement due to its simple calculation and theoretical interpretability. A multi-feature quantification framework for optical remote sensing image information content is proposed in this paper. Firstly, the information content of remote sensing images is measured based on grayscale, contrast, neighborhood-based topology, and spatial distribution features instead of simple grayscale and spatial structure. Secondly, the entropy metrics of the different features are designed to quantify the uncertainty of images in terms of pixel and spatial structure. Finally, a weighted model is used to calculate the comprehensive information content of the image. Three groups of experiments are designed to verify the validity and applicability of the proposed method. The results show that the amounts of image information are consistent with the trend of visual experience. Our model can effectively capture the spatial structure, and it is comparable to existing state-of-the-art entropy models in terms of thermodynamic consistency.

In the future, the applicability of information theory in optical remote sensing images still needs further research to better reveal the various types of image data from acquisition, and for production of subsequent applications, such as image interpretation, the changing law of surface information in the process of data conversion. First, because optical remote sensing images are the most common raster geospatial data, only optical remote sensing images are used for experiments in this paper, and the applicability of other types of image data needs to be verified. Second, in order to compare the objectivity of various information quantification models, the information content from different features of an image can be considered as a four-dimensional feature vector. Then, a machine learning approach in which the results are compared/classified over a set of relevant users labeled image classes could be more relevant. Moreover, the correlation in pixels and bands of remote sensing image and the effect of noise are also worthy of attention. In terms of applications, the performance of this framework in practical image processing and other applications, such as multi-source heterogeneous image screening and data mining, can be explored.