The Optimal Solution of Feature Decomposition Based on the Mathematical Model of Nonlinear Landscape Garden Features

The test area is flat, and the primary landscape is cultivated land and urban landscape types. Each accounted for 25% and 40% of the area. IKONOS image was 1 m full-colour data in May 2020. We use Magellan ‘Bumblebee’ GPS/GIS acquisition system to collect 32 ground control points and perform geometric correction on IKONOS full-colour data [4]. The actual point position error is 1.6 m. We segmented the corrected remote sensing data into 1024 rows and 1024 columns of the urban landscape and farmland landscape as the types and data of landscape pattern analysis (Figs. 1 and 2). At the same time, 256 rows and 256 columns of landscape areas were extracted from the upper right of the two types of landscape images as data for studying the local structure of the landscape.

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Wavelet analysis comes from Fourier analysis. This method has thrived since its establishment in the 1980s. The wavelet function can be regarded as the impulse response of a band-pass filter. The wavelet transform filters the original signal with band-pass filters of different scales [5]. This method decomposes the signal into a series of frequency bands for analysis and processing. In digital image processing, the wavelet transform is often binary discretised. Therefore, we can use discrete binary wavelets to perform multi-channel and multi-resolution analysis on images.

Assuming that {V_j} is a volume product space, then L²(R²) formed by Vj2=Vj⊗Vj is a multi-resolution analysis. If φ and ψ are the one-dimensional scaling function and wavelet function, respectively, then we can define the two-dimensional scaling function and the two-dimensional separable wavelet function as: (1)ϕ(x,y)=φ(x)φ(y)ψ1(x,y)=φ(x)φ(y)ψ2(x,y)=ψ(x)φ(y)ψ3(x,y)=ψ(x)ψ(y) Then any image f(x, y) can be decomposed and expressed as a combination of a series of sub-channel images: (2)A2−Jf,D2j1f,D2j2f,D2j3f,−J≤j≤1A2−Jf=f(x,y)×ϕ2J(−x,y)2−Jm,2−JnD2j1∼3f=f(x,y)×ψ2j1∼3(−x,−y)2−Jm,2−Jn

The image is decomposed into a pyramid structure composed of a low-frequency smooth image at 2^–J resolution and a high-frequency image composed of J layer image detail information, where D2j1∼3f represents the high-frequency information of the image in the horizontal, vertical and diagonal directions at the resolution 2j, respectively. Thus, they are orthogonal to each other.

Looking at wavelet analysis from the perspective of the structure of ground features, the wavelet coefficient measures the intensity of local variation of the signal under a particular observation scale. The greater the variation, the higher the value of the wavelet coefficient. Therefore, in a particular analysis scale, the wavelet coefficients of different pixel positions reflect the characterisation of the structure at that position [6]. It can be seen that the multi-resolution wavelet analysis decomposes the structural information on the original image. We distribute them on the D2j1∼3f image according to different directions and scales. On the 2j resolution scale, the D2j1∼3f channel indicates that the feature-length of the ground feature is between 2j – 1 and 2j. For A₂ – _J f image, it is not <2j scale feature structure information. We adopt similar G. A. The method of Bradshaw et al. calculates the sum of the squares of the wavelet coefficients of the corresponding pixels of each channel image under a particular decomposition scale, which is called the wavelet variance. Therefore, the wavelet variance becomes a function of a particular scale and the structural information under this scale. It uses scale and wavelet variance to map, reflect the structural characteristics on different scales through graphical methods, and then obtain the characteristic scale for analysing the landscape pattern. The more significant the wavelet variance on the corresponding scale, the richer the structural information. It is the main characteristic scale of the landscape structure. The more commonly used tap-4 Daubechies wavelet (D4) was selected in the experiment (Table 1). We substitute it into formula (1) to generate a two-dimensional scaling function and wavelet function. Then we substitute it into formula (2) to analyse the structure of IKONOSPAN urban landscape and farmland landscape to generate smooth and high-frequency images on the corresponding scale.

The semivariance function can be fitted with some theoretical models to obtain parameters for spatial variability analysis of geological attributes. It mainly includes base station value, nugget variance, structure variance, and range. The semivariance function is similar to spectral analysis [7]. However, what they provide is the overall average structural information in the data. The ability to express the local structural characteristics of the data is limited, which is incapable of multiscale structural analysis of the data. This article uses the following formula to calculate the semivariance in different directions.

(1)

Semivariance of landscape structure in an east-west direction (3)γe−ω(k)=1/2m∑i=1n∑j=1n[Z(i,j)−Z(i,j+k)]2

In the experiment, we used Daubechies wavelet to decompose the urban landscape and farmland landscape image from 1 to 9. Figs. 2 and 3 are the respective A₂–_J f channel wavelet variance-scale diagrams, and Figs. 4 and 5 are the respective D2j1∼2 channel wavelet variance-scale diagrams [8]. For example, in Fig. 2, the wavelet variance of the urban landscape smooth sub-graph has peaked at 16 m and 64 m, which means that the urban landscape structure information will be concentrated on two characteristic scales and appear at not <16 m and 64 m. On the other hand, in Fig. 3, the wavelet variance of the farmland landscape smoothing sub-graph has a peak at 32 m, which means that the information of the farmland landscape structure A₂–_J ƒ will be concentrated on a characteristic scale and not <32 m. Therefore, the size of the information content of the image channel is to guide whether it is necessary to continue the wavelet analysis.

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Figs. 4 and 5 show that the wavelet variance of the two diagonal directions is relatively very small. In image compression and noise removal applications, this direction is usually considered noise. In Fig.4, there are two peaks in both the horizontal and vertical structure of the urban landscape, being peak at 16 m and 128 m in the horizontal direction and 16 m and 256 m in the vertical direction. Through corresponding scale image analysis and field investigation, the urban landscape is mainly composed of buildings, streets and green spaces in buildings. The width of the buildings in the test area is about 16 m. Therefore, the horizontal and vertical directions of the building lead to the characteristic scale appearing at 16 m, which is mainly the characteristic scale of the width of the building. The rectangular area composed of 256 m and 128 m is the characteristic scale where the combined features of buildings and streets appear in the horizontal and vertical directions [9]. Through field investigation, the rectangular area is usually two buildings in the horizontal direction. The average length of the building is about 60 m and the vertical direction is usually eight rows of buildings. The width of the building plus the road between the buildings is 32 m on average. This is a typical combination in this area. This combination feature can also be seen in Fig. 1. Except for these typical characteristic scales, there is a small wave crest at 2 m horizontally and vertically. Image analysis and field investigations at the 2 m analysis scale show that the shadows of buildings are generally 2 m in size, or 2 m wide green belts and paths are reflected on this characteristic scale. It can be seen that the IKONOSPAN1 m urban landscape with a size of 1024 rows and 1024 columns shows three levels of characteristic scales.

The farmland landscape has 128 m in the horizontal direction and 256 m in the vertical direction. Through image analysis and field investigation of corresponding scales, it is found that this is the field size scale of available farmland [10]. The farmland blocks are surrounded by shelter forests and work roads, constituting a typical characteristic scale in this area. This farmland area is planted with green trees commonly used in the north. In addition, there is a small wave crest at 2 m in the horizontal direction, which is the average row spacing between planted trees. The typical characteristics of farmland and tree planting rows can also be seen in Fig. 1.

For urban landscapes and farmland landscapes of 1024 rows and 1024 columns, we use 1, 2, 4, 8, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300 and 400, respectively. The calculation interval of 500 m, 600 m is used for semivariance calculation. Figs. 6 and 7 are the semivariograms of the urban landscape and farmland landscape, respectively. We used a calculation interval of 1 m to 100 m for semivariance analysis of urban landscapes and farmland landscapes with 256 rows and 256 columns. Figs. 8 and 9 are the semivariance plots of urban and farmland sub-areas. Figs 6 and 7 show that the greyscale variables of the urban landscape and farmland landscape IKONOSPAN do not satisfy the second-order stationary hypothesis and intrinsic hypothesis in the 1024 rows and 1024 columns of the entire experimental area. Still, it meets the second-order stationary hypothesis and Eigen Order stationary hypothesis. Therefore, we use the experimental data of comprehensive average semivariance within 200 m to fit the theoretical model [11].

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Similarly, our experimental semivariance for urban landscape sub-areas is also a quasi-second-order stationary hypothesis and quasi-intrinsic. Their semivariance trends in different directions have the same shape. Therefore, we use the experimental data of comprehensive average semivariance within 30 m to fit the theoretical model. The other directions except the vertical direction and the comprehensive average semivariance show a cavitation effect for our experimental semivariance of the farmland landscape sub-areas. It is a non-stationary series. Therefore, it does not perform theoretical semivariance fitting for the sub-areas of farmland landscape. Table 2 shows that the fitting results of each landscape type using the spherical model have reached a significant level.

It can be seen from Figs. 6 and 7 that the semivariance analysis cannot reveal the structural differences in different directions of urban and farmland landscapes, nor can it effectively show the structural differences at different scales and levels. The fitting results in Table 2 reflect that the overall structure of the urban landscape has a range of 135 m. This reflects that the overall average characteristics of the combination of buildings and roads are between 128 m and 256 m in the characteristic scale of wavelet analysis [12]. The overall structure of the farmland landscape is 152 m, which is between 128 m and 256 m in the characteristic scale of wavelet analysis. It can be seen from Figs. 8 and 9 that some regional landscapes extracted from urban landscapes and farmland landscapes continue to undergo semivariance analysis. The urban landscapes show another structural feature on another scale with a range of 15 m. This is very close to the characteristic scale of 16 m in wavelet analysis. The half-length of the cycle where the cavitation effect appears in the farmland landscape sub-area except the vertical direction is about 2 m, precisely the size of the characteristic scale of the row spacing structure of the planted trees obtained by the wavelet analysis.

We used the wavelet analysis method to detect the three-level spatial pattern and combination relationship of the urban landscape in the experimental area from the IKONOSPAN1m image and the characteristic scale of each level of pattern. This provides essential information for evaluating and planning the urban landscape. In addition, from IKONOS images, it was detected that the field planting row spacing and the secondary landscape pattern of field plots in the experimental area and their sizes reflect the farming management information in this area.