Spatial Geometric Transformations and Shortest Path Planning in Plantscape Design

As people’s demands for the natural environment and for spatial comfort continue to increase, research on the use of plantscaping is gradually increasing. The plant landscape is very diverse, both potted plants and landscape greenery, which can be conveniently planted in a variety of indoor as well as outdoor spaces, such as parks, interiors and rooftops, and these diverse plant landscapes not only decorate the surrounding environment, but also let people feel the atmosphere of nature [1-2]. The functionality of plantscapes directly affects the comfort and applicability of the environment, and its functionality is mainly reflected in space utilization, environment improvement, and comfort improvement [3-5]. In space utilization, bright and spacious space to place tall green plants can highlight the sense of openness of the space, while closed and small space should be configured with smaller green plants in order to reduce the sense of congestion of the space, and space ventilation and lighting is also an important influence on plant landscaping [6-9]. In terms of environmental improvement, plants can respirate to absorb carbon dioxide and formaldehyde, benzene and other harmful substances commonly found in indoor air, and release the oxygen necessary for human beings, thus playing the role of air purification [10-13]. In terms of spatial comfort improvement, plantscape can optimize some details in the space, thus improving the comfort of people’s life [14-17].

In the actual plant landscape design process, it is necessary to fully consider the color, form and survival conditions of different plant landscapes, and effectively integrate the plant landscape with the environmental space by using appropriate spatial geometric transformation methods [18-20]. Plant landscape has become one of the indispensable materials in spatial design, therefore, it is necessary to optimize the application method of spatial design plant landscape to maximize the important role of plant landscape in spatial design.

In this paper, Harris corner point detection algorithm and two-way matching algorithm are first used to extract and match the feature points of spatial geometric transformation process in plant landscape design. After that, the plant landscape design planning map is constructed according to the matching results of the extracted data. Finally, Multi-factor Evolutionary Algorithm (MFEA) is used to solve the weights of influencing factors and the shortest paths of the carrier network topology map in the plant landscape design, so as to get a more suitable plant landscape design planning map and the shortest route of the plan.

2

Modeling of shortest path planning algorithms in plantscape design

2.1

Alignment technique for feature points of landscape spatial geometry transformation design map

2.1.1

Extraction of feature points

Spatial Geometric Transformation of Plant Landscape Design: The point feature of an image is an important local feature of an image, which is important information in image understanding and pattern recognition. Image feature point extraction technique is also one of the powerful tools for image processing and is a key technology for image matching, pattern recognition, target tracking, and other related applications.

Harris corner point detection algorithm [21-22] is a gray scale based autocorrelation function. The autocorrelation function of Harris algorithm is computed by calculating the gray scale change of the window when it makes a small translation along any direction, so Harris algorithm has rotational invariance.

The gray scale autocorrelation function of an image at point (x,y) is: (1) $r (x, y) = \sum_{w} w (x_{i}, y_{i}) {[I (x_{i} + Δ x, y_{i} + Δ y) - I (x_{i}, y_{i})]}^{2}$ Where w is the window function centered on (x,y) and w(x_i,y_i) represents the weights of the points, it has two forms: binary and Gaussian.

The formula for the two-dimensional Gaussian window function is w(x,y) = exp[–(x² + y²)/2σ²], (x_i,y_i) is the pixel coordinates within the window, Δx and Δy are the translations in the x and y directions, respectively, and I(·,·) is the gray scale function of the image. When the translation (Δx,Δy) is very small, I(x_i + Δx,y_i + Δy) can be approximated by a first order Taylor series: (2) $I (x_{i} + Δ x, y_{i} + Δ y) \approx I (x_{i}, y_{i}) + [I_{x} (x_{i}, y_{i}) I_{y} (x_{i}, y_{i})] [\begin{matrix} Δ x \\ Δ y \end{matrix}]$ where I_x and I_y are the derivatives in the x and y directions, respectively. There: (3) $\begin{matrix} r (x, y) = \sum_{w} w (x_{i}, y_{i}) {[I (x_{i} + ∆ x, y_{i} + ∆ y) - I (x_{i,} y_{i})]}^{2} \\ \approx \sum_{w} w (x_{i}, y_{i}) {(I (x_{i}, y_{i}) + [I_{x} (x_{i}, y_{i}) I_{y} (x_{i}, y_{i})] [\begin{matrix} Δ x \\ Δ y \end{matrix}] - I (x_{i}, y_{i}))}^{2} \\ = \sum_{w} w (x_{i}, y_{i}) {([I_{x} (x_{i}, y_{i}) I_{y} (x_{i}, y_{i})] [\begin{matrix} Δ x \\ Δ y \end{matrix}])}^{2} \\ = [\begin{matrix} Δ x & Δ y \end{matrix}] [\begin{matrix} \sum_{w} w (x_{i}, y_{i}) I_{x}^{2} (x_{i}, y_{i}) & \sum_{w} w (x_{i}, y_{i}) I_{x} (x_{i}, y_{i}) I_{y} (x_{i}, y_{i}) \\ \sum_{w} w (x_{i}, y_{i}) I_{x} (x_{i}, y_{i}) I_{y} (x_{i}, y_{i}) & \sum_{w} w (x_{i}, y_{i}) I_{y}^{2} (x_{i}, y_{i}) \end{matrix}] [\begin{matrix} Δ x \\ Δ y \end{matrix}] \\ = [\begin{matrix} Δ x & Δ y \end{matrix}] M (x, y) [\begin{matrix} Δ x \\ Δ y \end{matrix}] \\ = [\begin{matrix} Δ x & Δ y \end{matrix}] [\begin{matrix} A & C \\ C & B \end{matrix}] [\begin{matrix} Δ x \\ Δ y \end{matrix}] \end{matrix}$

Let λ₁ and λ₂ be the two eigenvalues of matrix M(x,y), respectively, then the following relations exist between them and the planes, edges and corner points in the image: 1)

Plane: λ₁ and λ₂ are both smaller, and the autocorrelation function r(x,y) does not vary much in all directions:

2)

Edges: λ₁ ≫ λ₂ or λ₁ ≪ λ₂, the autocorrelation function r(x,y) is large in one direction and large in its

3)

Corner points: λ₁ and λ₂ are both larger, and autocorrelation function r(x,y) is large in all directions.

In practice, it is not necessary to specifically compute λ₁ and λ₂. Harris describes a simple criterion for detecting corner points in his article: (4) $H = \det M - k \cdot {(t r a c e M)}^{2}$ where detM = λ₁λ₂ = AB–C² is the determinant of matrix M(x,y), traceM = λ₁ + λ₂ = A + B is the trace of matrix M(x,y), and k is generally taken as 0.04-0.06. At this point, the complete steps of Harris’ algorithm can be summarized as follows: 1)

Calculate the gradients of the spatial geometric transformation image I(x,y) of the plantscape design in the x-direction and y-direction: I_x and I_y, respectively;

2)

Calculate $I_{x}^{2}$ , $I_{y}^{2}$ and I_y;

3)

Gaussian weighting of $I_{x}^{2}$ , $I_{y}^{2}$ and I_xI_y to obtain matrix M(x,y);

4)

Setting the read value T and calculating H, the points of H > T are then candidate corner points;

5)

Perform local non-maximum suppression on the candidate corner points, and the result is the final corner point.

2.1.2

Representation of feature points

1)

Determination of the main direction of feature points

In order to make the feature description operator of the spatial geometric transformation image of the plant landscape design rotationally invariant, a principal direction is to be assigned to each feature point by utilizing the distribution of the gradient direction of the pixel points in the circular area of a certain size around the feature point. Assuming that the scale where a feature point is located is σ, the magnitude and angle of the Gaussian image are computed within a circular neighborhood centered on the feature point and with a radius of 4.5 σ, as shown in the following two equations: (5) $m (x, y) = \sqrt{{(L (x + 1, y) - L (x - 1, y))}^{2} + {(L (x, y + 1) - L (x, y - 1))}^{2}}$ (6) $θ (x, y) = \tan^{- 1} (\frac{L (x, y + 1) - L (x, y - 1)}{L (x + 1, y) - L (x - 1, y)})$ where m(x,y) is the magnitude of the image gradient, θ(x,y) is the magnitude angle of the image gradient, and the scale of L is the scale where the keypoints are located.

2)

Generation of feature point description vector

Before generating the feature point description vector, it is necessary to rotate the feature point neighborhood by an angle centered on the feature point so that the main direction falls at the 0° position, and the result of the assignment of the main direction of the feature point is shown in Fig. 1, which can make the feature description vector rotationally invariant.

2.1.3

Matching of feature points

Two-way matching method:

The nearest/next nearest ratio method described earlier is a one-way matching strategy for finding the corresponding points of feature points of the other shortest path planning image in one image of the spatial geometric transformation of plant landscape design. Using the principle of uniqueness of matching pairs of feature points, the matching point pairs in the two to-be-aligned images should be one-to-one, i.e., for matching point pairs p and q, there should exist a matching mapping [23] relation between these 2 points {p → q} and {p ← q}. In this way, the final matching result should be the set of matching point pairs obtained by matching the spatially geometrically transformed image of the plant landscape design of the reference image to the to-be-aligned shortest path image and the set of matching point pairs obtained by matching the to-be-aligned image to the reference image matching. Since the whole process is to carry out the matching process in two directions, it is called the two-way matching method.

Applied to the actual algorithm, the specific steps are as follows: 1)

Using some feature point extraction method, extract the feature points in the two images to be aligned respectively:

2)

According to some feature point matching method, calculate the set of matching point pairs in one direction;

3)

Within the set of matching point pairs obtained in the previous step, calculate the matching mapping in the opposite direction according to the same method.

2.2

Plant landscape path planning topology map construction

A topology map is a diagram that shows the topology of a network. It shows how the components (e.g., nodes) in a system are connected and interact with each other. In a topology map, each component is usually represented as a node, and the connections between them are represented as edges. These nodes and edges can be marked and distinguished by different colors, shapes, line types, etc. Topological maps have the advantage of being able to clearly present complex structures and interrelationships, which allows users to better understand and analyze them.

In this paper, topological structure [24-25] is used to construct the map needed for path planning because in this paper, only the length of the path is considered in the process of path planning task optimization, and does not care about the positional relationship between the nodes, so the topological map is a kind of map construction method suitable for this paper. The flowchart of path planning topological map construction is shown in Fig. 2.

2.3

Shortest path solving in landscape design planning

The Multifactor Evolutionary Algorithm [26-27] (MFEA) aims to solve multiple optimization tasks simultaneously using a single population. The MFEA algorithm employs an implicit parallelism algorithm based on a population search that attempts to uncover implicit associations between different tasks, and in this way accelerates the rate of convergence of the respective problems. The MFEA algorithm aims to find a set of solutions such as Eq. (7): (7) ${x_{1}, x_{2}, \dots, x_{k - 1}, x_{k}} = a r g m i n {f_{1} (x), f_{2} (x), \dots, f_{k - 1} (x), f_{k} (x)}$ where x_j is the feasible solution on the search space X_j and K is the number of tasks in the multitasking.

The MFEA algorithm has the following four definitions for each individual p_i:

Factor cost: for a given task T_j, the factor cost of individual p_i on task T_j is: (8) $φ_{j}^{i} = λ \cdot δ_{j}^{i} + f_{j}^{i}$ where λ is a larger penalty multiplier, and $f_{j}^{i}$ and $δ_{j}^{i} δ_{j}^{i}$ are the objective value of p_i with respect to T_j and the total constraint violation, respectively. Thus, e.g., p_i is feasible for T_j (zero constraint violation).

Factor Ranking: assuming that Individual p_i is in Task k, the factor ranking of an individual is defined as r_i,k means that Individual p_i evaluates the factor cost of the individual for Task k and ranks the index positions in ascending order of factor cost.

Skill factor: skill factor τ_i indicates which task individual p_i corresponds to, and skill factor τ_i for each individual p_i is shown in equation (9): (9) $τ_{i} = a r g m i n {r_{i}, k}, k = 1, 2, \dots, K$

Scalar fitness: the ranked list of factorials is: (10) ${r_{1}^{i}, r_{2}^{i}, \dots, r_{k}^{i}}$

Reduction to scalar fitness φ_i based on the best position of a single p_i in all tasks: (11) $φ_{i} = 1 / m i n_{j ò {1, 2, \dots, K}} {r_{j}^{i}}$ $${\varphi _i} = 1\,/\,mi{n_{j\`o \left\{ {1,2, \cdots ,K} \right\}}}\left\{ {r_j^i} \right\}$$

The MFEA algorithm solves multiple cross-domain optimization tasks with a single population, and different tasks provide different factors that affect the construction of shortest paths, and the information of shortest path planning is affected by the terrain, landforms, and vegetation types of multiple different tasks. Therefore, this paper relies on this method to obtain the optimal shortest path after comprehensively analyzing the factors of the shortest path planning and construction process in the following section.

3

Plant landscaping shortest path planning results and analysis

3.1

Comparison of Characteristic Top Alignment Data in Plant Landscaping

In this paper, the extracted data points are divided into three groups, and the running time and distance sum of squares for data alignment are investigated using the Harris corner point detection algorithm of this paper and the traditional corner point detection algorithm (TAMM) with 25 and 50 iterations. The results of the alignment data comparison are shown in Table 1. The results show that the Harris corner point detection algorithm with 50 iterations has more iterations and larger alignment errors. In this paper, the improved Harris corner point detection algorithm uses the improved feature point corresponding to the point pair search method while adding the super-linear distance constraint method to remove the wrong point pair, so that the point cloud data in the alignment process of the number of alignment iterations and alignment error is effectively reduced, and the convergence condition is reached in 25 iterations. In terms of the alignment time, in the three sets of data, the running time of the Harris corner detection algorithm (163.59 s, 177.32 s, and 47.32 s) is smaller than that of the traditional alignment algorithm (207.41 s, 203.55 s, and 79.48 s), and it can be concluded that the improved Harris corner detection algorithm in this paper can align the point cloud data faster and more efficiently to finally meet the accuracy requirements. The accuracy requirement has finally been achieved.

Table 1.

Comparison results with criterion

Dot cloud data	Group 1		Group 2		Group 3
Data point	229673	203974	248696	247885	325589	314977
Matching method	Harris	TAMM	Harris	TAMM	Harris	TAMM
Iteration number	25	50	25	50	25	50
Run time (s)	163.59	207.41	177.32	203.55	47.32	79.48
Distance of the sum of squares	0.0001	0.0032	0.0002	0.0093	0.0006	0.0093

3.2

Multi-factorial comprehensive analysis of plant landscape design Shortest Path

3.2.1

Determination of shortest path selection factor weights

Since there are differences in topography and geomorphology in the process of plant landscape design, these differences will have an impact on the process of plant landscape design, therefore, in order to determine the spatial geometric transformation of plant landscape design species and the shortest path planning, it is necessary to carry out a study on the weights of the four influencing factors, namely “geology, geomorphology, elevation and slope”. The results of the relative importance of the four factors are shown in Figure 3. The investigation results of the influence factors of path planning in plant landscape design show that: the average value of the relative importance weights of geology, features, elevation and slope = 0.2: 0.5: 0.2: 0.1. In order to facilitate the calculation, when choosing the comprehensive modeling of plant landscape factors, the weight ratio of the weighted superposition of each raster layer is chosen as geology: features: elevation: slope = 2:5:2:1.

3.2.2

Results of Shortest Path Analysis of Plant Landscaping Designs

In this paper, the MFEA algorithm is used to determine the shortest path for landscape design planning. The plant landscape design shortest path trend direction diagram is shown in Figure 4. From the figure, it can be more intuitive to see the important factors in the path selection of feature points. And under the condition of topological map construction, the three optimized routes obtained from the results of feature point matching are found to be suitable as optimized routes for plant landscape design. However, the distances of the three routes are not equally short, and after the investigation of the routes, it is found that the distance of route No. 1 is the shortest.

The shortest path is determined on the basis of economic values such as cost, and its influencing factors are explored in the previous section, followed by the estimation of its cost in order to better determine the reasonableness of its shortest path. The cumulative trend of the minimum cost of the shortest path for plant landscape design is shown in Figure 5. It was discovered that the shortest path is almost entirely confined to the minimum cumulative cost cost interval. This indicates that the minimum cumulative cost from any point in the routing area to the starting point is the lowest, and the shortest route has the lowest cost (0-40,000 yuan). The planned path’s cost is expected to be between 350,000 yuan and beyond the minimum cost region, which is why the shortest route chosen in this paper is very much as expected.

3.3

Evaluation of the design effect of spatial geometric transformation of plantscape

The standard deviation, entropy, and peak signal-to-noise ratio of the spatial geometric transformation design map of plant landscape design are used to test the visual communication effect of this paper’s method on plant landscape design. The standard deviation can express the contrast of the color of the local area of the plant landscape design image after the application of the method in this paper, and the larger the standard deviation is, the more significant the contrast is. The entropy can express the amount of color information in the local area of the plant landscape design image after the application of the method of this paper, and the larger this value is, the more significant the amount of color information exists in the local area. The peak signal-to-noise ratio can express the noise resistance of the optimized local color features of the plant landscape design image after the application of the method in this paper, and this value is larger, which indicates that the method in this paper has a significant suppression performance of the image noise points. In this paper, 100, 200, 300, and 400 image frames were selected to evaluate the results of the design map and design route.

The object-based 3D image color transfer and parallax optimization method (3Dict-POM), and Chaotic Particle Swarm Optimization Support Vector Machine for Image Editing (CPGOSVM) are used as the comparison methods to test the visual communication effect of the plant landscape design after the application of the three methods. The evaluation results of the design effect of spatial geometric transformation of plantscape are shown in Table 2. The experimental results show that, based on the comparison of this paper’s method (MFEA) algorithm, 3Dict-POM method, and CPGOSVM method, the MFEA method has the best visual communication effect on plant landscape design. After the application of this paper’s method, the standard deviation of the plant landscape design image is the largest, and the maximum value is 0.9851, while the maximum value of the standard deviation of the 3Dict-POM method and the CPGOSVM method is 0.7306 and 0.6631, respectively, which is 0.2545 and 0.332 less than that of this paper’s method. The MFEA, 3Dict-POM, and CPGOSVM methods determine the The image information entropy becomes larger along with the increase of the number of image frames, and the maximum values are 0.9659 bit, 0.4839 bit and 0.4977 bit, respectively; while the peak signal-to-noise ratios are not affected by the number of image frames, and their mean values are stabilized at 1.6903 dB, 1.3051 dB and 1.2870 dB, respectively (with four bits of valid data retained). The MFEA method has a higher standard deviation, entropy, and peak signal-to-noise ratio than the comparison method, according to a comprehensive analysis. It can be seen that after the application of the method in this paper, the amount of color information information in the local area of the plant landscape design image is the most significant, the contrast is the most significant, and the suppression performance of the image noise points is the most significant, and the overall design effect is better.

Table 2.

The design effect of the spatial geometry of plant landscape is evaluated

Evaluation index	Method	Frame number
Evaluation index	Method	100	200	300	400
Standard deviation	MFEA	0.9308	0.9463	0.9658	0.9851
	3Dict-POM	0.7306	0.6809	0.6647	0.5339
	CPGOSVM	0.6631	0.6396	0.6268	0.6262
Entropy/bit	MFEA	0.6186	0.7399	0.8424	0.9659
	3Dict-POM	0.4367	0.4805	0.4936	0.4839
	CPGOSVM	0.4356	0.4647	0.4855	0.4977
Peak signal-to-noise ratio/dB	MFEA	1.6812	1.6919	1.6814	1.7067
	3Dict-POM	1.4786	1.3335	1.2169	1.1912
	CPGOSVM	1.4533	1.3263	1.2009	1.1674

4

Conclusion

In this paper, the plant landscape design map under spatial geometric transformation is obtained by aligning the feature points of the landscape spatial geometric transformation design map and constructing the path planning topology map, and the shortest path in the design map is planned. The primary conclusions are as follows: 1)

The alignment error is larger when the Harris corner point detection algorithm has more iterations (50 times). The alignment time on the Harris corner point detection algorithm for the three sets of data (163.59 s, 177.32 s and 47.32 s) was significantly smaller than that of the traditional alignment algorithm (207.41 s, 203.55 s and 79.48 s).

2)

In the process of plant landscape design, this paper determines the relative importance of four factors affecting the design process: geology, features, elevation and slope (0.2:0.5:0.2:0.1), and then determines the weight selection ratio of each raster layer weighted superposition (2:5:2:1). Afterwards, three optimized routes were obtained according to the results of feature point matching, and after comparison, it was found that the distance of route No. 1 was the shortest, and the shortest path was included in the interval of the minimum cumulative cost cost, and the shortest route was determined very much in line with the expectations.

3)

Under the comparison of MFEA, 3Dict-POM, and CPGOSVM methods, the MFEA method has the best visual communication effect on the plant landscape design, with the maximum value of standard deviation of 0.9851, the maximum value of entropy of 0.9659 bit, and the mean value of peak signal-to-noise ratio of 1.6903 dB. Obviously, after the application of the MFEA method, under the spatial geometrical transformation of the plant landscape design image local area has the largest color difference, the most significant contrast, and the most significant suppression performance of image noise points, and the overall design effect is better.

Funding:

This research was supported by the 2024 Scientific Research Basic Ability Improvement Project of Young and Middle-aged Teachers in Guangxi Universities: “The Association Study of Garden Plant Biodiversity and Landscape Design” (2024KY0258).

Language:: English

Publication timeframe:: 1 times per year
Journal Subjects:: Life Sciences, Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics, Physics, other

Journal RSS Feed

Spatial Geometric Transformations and Shortest Path Planning in Plantscape Design

Hanhan Zhou

Peiyun Bie

Published Online: Mar 17, 2025

Received: Oct 18, 2024

Accepted: Jan 29, 2025

DOI: https://doi.org/10.2478/amns-2025-0323

KeywordsHarris corner point detection algorithm, Bi-directional matching method, Topological map, Multi-factor evolutionary algorithm, Plantscape design, Path planning

© 2025 Hanhan Zhou et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Harris corner point detection algorithm, Bi-directional matching method, Topological map, Multi-factor evolutionary algorithm, Plantscape design, Path planning