Research ArticleUrban Planning Image Feature Enhancement and SimulationBased on Partial Differential Equation Method

Duo Li

Department of Architecture, Nanyang Institute of Technology, Nanyang, Henan 473004, China

Correspondence should be addressed to Duo Li; 3051003@nyist.edu.cn

Received 28 July 2021; Revised 12 August 2021; Accepted 14 August 2021; Published 2 September 2021

Academic Editor: Miaochao Chen

Copyright © 2021 Duo Li. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Based on the introduction of the basic ideas and related technologies of partial differential equations, as well as the method ofpath planning, the application of partial differential equations in solving urban path planning is studied. The path planningmodel of partial differential equations and the setting of obstacle boundary conditions are introduced, and adaptive.Theoretical research and experimental results show that it is feasible and effective to solve urban path planning by partialdifferential equations, which provides a new way for urban path planning research ideas and methods. This paper proposes animage detection algorithm based on diffusion equation. According to the logarithmic transformation, the multiplicativespeckle noise in the image can be converted into additive noise. We first perform logarithmic transformation on the imageand then use the denoising model of the diffusion equation to filter out the noise in the image and then use the logarithm torecognize the image. The difference image is obtained by the domain difference method, and finally, the difference image isclassified by the clustering algorithm, and the change area is extracted. Experiments show that the algorithm can effectivelyreduce the effect of multiplicative speckle noise on the change detection results, improve the accuracy of change detection,and shorten the change detection time. This article takes the path planning problem of a two-dimensional space city as anexample to discuss the application of partial differential equations. According to the principle of energy conservation, thispaper uses the two-dimensional space radiant heat conduction equation as an example to model and illustrate the solution ofthe path planning problem.

1. Introduction

With the development of human society, the society hashigher and higher technical requirements for multimediainformation processing. Images are an important source forhumans to obtain external information. Digital image pro-cessing technology has become a research hotspot in the field[1]. Under normal circumstances, the process of digitalimage processing will be interfered by different noise signals,which will reduce the quality of the image. This not onlymakes the image blurry but also affects the subsequent pro-cessing of the image, such as image segmentation and imagecompression [2]. Therefore, finding a fast and effective imagedenoising algorithm is a big challenge facing researchers.Image restoration has always been a concern in the field ofimage processing, and it is also a preprocessing process forhigh-level image processing applications [3]. Traditional

image restoration methods often destroy image features suchas edges, lines, and textures while removing noise. Algo-rithms based on partial differential equations can maintainthe details of the image while restoring the image. Therefore,it has received more and more attention in recent years [4].Urban path planning is an important link and subject ofnavigation research. Traditional path planning methodsmainly include artificial potential field method, genetic algo-rithm, artificial neural network, simulated annealing algo-rithm, ant colony optimization algorithm, and RRT randomsearch algorithm [5]. The artificial potential field algorithmis the most typical algorithm. It has mathematical complete-ness and simplicity, but it is difficult to deal with obstaclesand narrow channel oscillations. The advantage of intelligentalgorithms such as genetic algorithm and ant colony algo-rithm is that the path can converge to the optimal or it isapproximately optimal. The disadvantage is that the evolution

HindawiAdvances in Mathematical PhysicsVolume 2021, Article ID 1700287, 10 pageshttps://doi.org/10.1155/2021/1700287

speed is difficult to control, and too many empirical parame-ters are required, which is not conducive to automatic pro-cessing [6]. The RRT algorithm is a single query pathplanning method based on sampling, because it avoids themodeling of the space. Compared with the method, it hasunique advantages, but because the random search is uniformin the global space, the algorithm is meaninglessly expensiveand the planning time is longer [7].

Partial differential equations can effectively simulatedynamic processes, such as one-dimensional string vibra-tion, two-dimensional membrane vibration and heatconduction, and three-dimensional acoustic wave oscilla-tion. Therefore, they are widely used in physics, mechanics,engineering technology, and other natural science fields[8]. The rich content and better performance of partialdifferential equations provide new enlightenment for urbanpath planning algorithms. Image restoration is also calledimage restoration; it is the process of restoring degradedimages. It can be seen that image restoration is actually theinverse process of the image degradation process [9]. Thepurpose of image restoration technology is to restore thedegraded image to its original appearance as much as possi-ble. The method is to first analyze the mechanism of imagedegradation, that is, use mathematical models to describethe process of image degradation. Then, on the basis ofthe degradation model, through the master’s thesis basedon the partial differential equations of the image restorationresearch to find the mode calculation of the inverse process,the real image can be obtained more accurately from thedegraded image, and the original information of the imagecan be restored [10]. Image restoration has always been aconcern in the field of image processing, and it is also apreprocessing process for high-level image processing applica-tions. Image restoration occupies an extremely importantposition in primary visual processing and has a wide rangeof applications in aerospace, national defense, public security,biomedicine, cultural relic restoration, and other fields [11].

This paper introduces the path planning problem of par-tial differential equation modeling and proposes a new urbanpath planning algorithm. At the same time, an image changedetection based on an adaptive total variation (TV) denois-ing algorithm is proposed. First, the image is converted tothe logarithmic transform domain, and then, the image isdenoised using the adaptive total variation denoising model.In this algorithm, in view of the shortcomings of the originalTV denoising algorithm, we use a generalized energy func-tion and introduce a difference variable to automaticallyselect the number of iterations required for denoising, avoid-ing insufficient denoising or waste caused by manual selec-tion of the number of iterations matter of time. Finally, thek-means clustering algorithm is used to cluster the differenceimages. (1) Considering the step effect of P-M in the processof restoring images, a complex diffusion model that couplesmedian filtering and the P-M model is improved. Withoutaffecting the denoising performance, the visual effect of theimage is improved. (2) Considering that the model cannoteffectively remove the shot noise points, it is improved,and a time-weighted selective diffusion model is proposed.(3) Considering that the total variation model will have a

step effect in the process of restoring the image, an improvedmodel that couples the second-order and fourth-orderpartial differential equations is proposed. The improvedalgorithm improves the visual effect of the image, and theconvergence speed is much higher than that of the fourth-order model. Experimental results show that the adaptivetotal variation denoising algorithm can effectively suppressimage noise and improve the detection accuracy andefficiency of the algorithm. This paper focuses on theresearch of image modeling and studies three differentimage restoration models from different perspectives forthe basic problems of image restoration. The first type ofmodel is a restoration model based on the scale-space axiomsystem. The main research is to analyze and design aniso-tropic diffusion equations from the perspective of direc-tional filters; the second type of model is partialdifferential equation models derived from the perspectiveof functional variation, focusing on the total variationmodels and four that are widely used today. The third typeof model is a hyperbolic partial differential equation modelcalled impulse filtering, which can be used for imageenhancement or deblurring.

2. Related Work

Torquato and Haslach Jr. [12] use hyperbolic partial differ-ential equations to simulate the path of the endoscope in thehuman body. The advantage is that it can handle the peekpath of various organ structures, and the propagation speedin one direction depends on its investigation solution. Thisis limited. The solution may be singular within time, leadingto the rupture of the solution. According to the order inwhich the images are classified in the change detection pro-cess, the image change detection method can be divided intotwo types: comparison before classification and comparisonafter classification. The method is to first classify the twoimages separately and then compare the two remote sensingimages after classification. This type of algorithm relies onthe accuracy of the classifier, and the initial error will accu-mulate in the subsequent comparison process.

The first comparison and then classification methodgenerally obtains the difference image first and then classifiesthe difference image to obtain the detection result. The accu-racy of such image change detection mainly depends on thequality of the difference image and the accuracy of the differ-ence image classification algorithm. For example, Naghaviet al. [13] proposed multiscale and multitemporal satelliteimage change detection (MCD algorithm for short), whichuses logarithmic ratio algorithm to calculate the differenceimage of SAR image, uses undecimated discrete wavelet todecompose the difference image, and finally uses k-meansclustering algorithm that clusters the decomposed differenceimages to obtain the detection results. Evans [14] and othersproposed SAR image change detection based on differenceimage fusion and k-means clustering (PPK algorithm forshort), which uses PPB Filter denoising; they use weightingmethod to combine logarithmic ratio difference images anddifference images in the time domain and then use k-means clustering algorithm to cluster the difference images,

2 Advances in Mathematical Physics

which improves the accuracy of change detection, but thedetection time is relatively long. Berger and Oliger [15]and others proposed a SAR image change detection basedon the fusion of ratio image data and fuzzy clustering (CIVBalgorithm for short). The algorithm uses a context-independent variable fusion algorithm to fuse the meanratio difference image and the log ratio difference imageand then uses fuzzy clustering algorithm to process thedifference images. This algorithm improves the effect ofchange detection, but after obtaining the fused differenceimage, it needs to use multiple scales to extract homoge-neous regions, and the total running time is very long.Others proposed that all image change detection can beclassified according to the detection purpose, detection data,detection dimension, detection time scale, and detectioncontent and divided the image change detection technologyinto direct comparison method, classification comparisonmethod, object-oriented method, and model seven methods,such as method, time trajectory method, visualizationmethod, and hybrid method.

In recent decades, image denoising processing technol-ogy based on partial differential equations has become aresearch hotspot in the field of image processing, and ithas gradually taken shape with the introduction of imagesmoothing and multiscale space theory. Later, Pai [16] firstgave a model based on anisotropic diffusion, which is calledthe Perona-Malik model (PM model), which overcomes thedisadvantages of isotropic diffusion. The biggest advantageof the PM model is to combine the smoothing process ofthe image with edge detection, use the edge spread functionto control the spread rate of the edge, and try to spread inthe nonedge area of the image, so this model can not onlyeffectively remove noise but also, moreover, the edge textureof the image can be sharpened, which lays the foundationstone for the further development of digital image process-ing technology. However, the model still has defects. Thereis uncertainty in the choice of the model’s diffusion coeffi-cient. In addition, it is easy to ignore the strong isolatednoise and edges in the image, which will cause a serious stepeffect. In order to overcome these shortcomings, laterresearchers have proposed many improved algorithms.Huo and Chai [17] proposed an image denoising modelbased on the curvature of a weighted Gaussian function.Helbing et al. [18] proposed an algorithm based on aniso-tropic diffusion, mainly for speckle noise, using the probabil-ity density function of the edge and the correlationinformation of the pixel to control the diffusion rate, whichnot only protects the edge information but also solves theimage oversmoothing defects. Some scholars have proposeda modified diffusion model based on adaptive edge thresh-old, which improves the PM model and mainly introducesa directional filter [19]. This coefficient takes into accountthe variability of local pixels in order to better remove edgenoise and selectively smooth the image. Some scholars intro-duced the directional Laplacian operator to improve the PMmodel, which diffuses the image along the edge of the origi-nal image, effectively alleviating the step effect, and protect-ing the sharp edges. This model can effectively remove noisewhile preserving edges, borders, and textures [20, 21].

3. Urban Planning Image Feature EnhancementModel Construction Based on PartialDifferential Equation Method

3.1. Method Hierarchy of Partial Differential Equations.Given an equation, generally, it can only describe the generallaw of a certain motion, and the specific state of motion can-not be determined, so the equation is called a universal equa-tion. If some conditions are attached (such as the conditionof the start of motion or the boundary after being subject toexternal constraints), the specific motion state can becompletely determined, and such conditions are called defi-nite solution conditions. The additional conditions that rep-resent the starting situation are called initial conditions, andthe conditions that are restricted on the boundary are calledboundary conditions. Figure 1 shows the partial differentialequation method hierarchical distribution structure.

Suppose that the function xðtÞ satisfies the universaldefinite equation in the region R. When the point tends togive the initial hyperplane or the boundary surface of theboundary condition from the region R, the u required inthe definite solution condition and its limit of the derivativeexists everywhere and satisfies the corresponding definitesolution condition, so x is called the solution of the definitesolution problem.

X t½ � = x ∈ R ∣ x ið Þ, t ∈ Rf g, i = 1,⋯, t: ð1Þ

The constrained least power recovery is derived based onfunction smoothing. This restoration method establishes anoptimal criterion based on a smoothness measure, such asminimizing the second derivative of the image (i.e., Lapla-cian operator). In order to be meaningful, the recovery pro-cess must be constrained with known parameters. Therefore,it is necessary to find the minimum value of the criterionfunction uðx, tÞ. For example, the function uðx, tÞ can bedefined as

u x, tð Þ =x 1ð Þ 0 00 ⋯ 00 0 x tð Þ

2664

3775

0 1 11 ⋯ 11 1 0

2664

3775: ð2Þ

If all the conditions are met (the existence of the filterfunction and the existence of the inverse Fourier transform),for the restored signal x, force is equal to the ideal signal fðx, tÞ, so there is a perfect restoration. The edge gradient ofthe image is large, the coefficient is small, the diffusion isweak, and the edge is maintained; the internal gradient ofthe image is small, the diffusion coefficient is large, the equa-tion is close to the heat conduction equation, the diffusionstrength is large, and the image is smoothed, thus realizingthe anisotropic diffusion ability. It is obtained by passingan imperfect image through a linear filter. Logically, the filterw is called an inverse filter, and its transfer function is theinverse of the transfer function of the degraded div ðuÞ:

3Advances in Mathematical Physics

∂u x, tð Þ∂t

= div c x, tð Þ × ∇uð Þ, x, t ∈ R,

f x, tð Þ = div ∇uj j∇uð Þ+∇u ∇x tð Þð Þ∇t:ð3Þ

Themotion law of many objects can be described by waveequations. For example, string vibration can be described byone-dimensional wave equation; membrane vibration can bedescribed by two-dimensional wave equation; acoustic andelectromagnetic wave oscillations can be described by three-dimensional wave equation. Therefore, the path planning ofmobile robots can be used with the two-dimensional orthree-dimensional equation description of the vibration orwave from the start point to the end point.

min u x, t ∣ t ∈ 0,∞ð Þð Þ = conv u xð Þ − u 0ð Þ, u xð Þð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivar u xð Þ − u 0ð Þð Þ ⋅ var u xð Þð Þp ,

g ∣∇u ∣ð Þ − e− ∇uj j/k = 0,conv X, Yð Þ − E X − �X

� �Y − �Y� �� �

= 0:ð4Þ

If this condition is met, the mold can be mutually usediteratively. It needs to be pointed out that because all opera-tions are a combination of image information, no otherfeatures are added in this process, so the larger the scaleparameter vðx, tÞ is, the simpler the content of the image is.

∇uj j div ∇uð Þ = u xð Þx tð Þ − 2u xð Þ2∇u xð Þ∇x tð Þ + u xð Þx tð Þ ,

v x, tð Þ − k2

k2 + ∇uj j2= 0,

v x, 0ð Þ = v tð Þx 0ð Þ:

8><>:

ð5Þ

It can be seen from the above formula that if div ðu, vÞtakes 0 or is very small on the plane, it will bring about com-putational difficulties. On the other hand, noise can cause

more serious problems. In this case, recovery can only be per-formed within a range close to the origin (close to the centerof the frequency domain).

3.2. Image Feature Enhancement Algorithm. The physicaldomain g is the passable area of uðx, yÞ with the city wherethe obstacle is removed in the working environment. If theobstacle is dynamic, the obstacle is divided according to the sizeand direction of the speed. The boundary is set to the corre-sponding multiple relationships of the Dirichlet condition.

〠n

i=1g ∣∇u ⋅ x tð Þ∣,u xð Þ, u yð Þð Þ ⋅

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

1+∇u xð Þ∇x tð Þð Þ2s

= −1,

∂u x, y, tð Þ∂x tð Þ + ∂u x, y, tð Þ

∂y tð Þ + ∂u x, y, tð Þ∂y xð Þ = 0:

ð6Þ

Figure 2 shows the flow chart of the image feature enhance-ment algorithm. Therefore, the algorithm for solving the modelequation based on the adaptive grid is as follows: (1) initializethe city’s passable area as a quasiregular triangulation grid, setu as the preset uniform grid unit side length value, and set tas the passable area of the minimum value of the energy preset;(2) use the finite element difference format to obtain theapproximate solution of yðtÞ on the triangulation grid of thecity starting area in R.

∇u x, y, tð Þ∇t

−u x + t, yð Þ − u x, yð Þ

t + 1 = 0,

x ið Þ ⋯ x 1ð Þ½ �y 1ð Þ⋯

y ið Þ

2664

3775 = ∂u x, y, tð Þ

∂u tð Þ ⋅ ∇u:ð7Þ

If the error meets the predetermined tolerance, the gridaccuracy is normal. We perform linear interpolation betweenthe triangle node and the midpoint and return the gradient of

Definite condition

A

A

A

A

A

B B

B B

C

C

C

C CC

X(A,B,C)

X(A,B,C)

X(A,B,C)

Y(A,B,C)

Y(A,B,C)

Y(A,B,C)

Conn

ectio

n

Conn

ectio

n

Conn

ectio

n

Conn

ectio

n

Partial differential equation method hierarchy

Definite condition

Definitecondition

Image feature

Image feature

Image feature

Image feature

Urban planningimage

Universal equationDefinite

condition

Figure 1: Partial differential equation method hierarchical distribution structure.

4 Advances in Mathematical Physics

the triangle center solution. If the gradient is less than the presetenergy minimum, the calculation ends; otherwise, it will enterthe next step of grid and gradient calculation.

Δu x, tð Þ = 〠n

i=1

∇u s, tð Þ∇u xð Þ = u s + t, tð Þ − u x, tð Þ

x s, tð Þ ,

Δu x, tð Þu x, tð Þ = ∂u x, y, tð Þ

∂u tð Þ ∗ div c Im u x, tð Þð ÞΔu x, tð Þð Þð Þ,

Δu x, 0ð Þu x, 0ð Þ = ∂u x, 0ð Þ:

8>>><>>>:

ð8Þ

Linear diffusion is the oldest and best way to understandscale space. From the perspective of scale space, the imageafter the continuous family of operators will get a series ofsmoother images. The original image corresponds to the scalet. For the image at =0, an image with a simpler structure canbe obtained by increasing the scale t. Because the scale-spacetheory introduces a hierarchical structure in the image charac-teristics, it constitutes an important step in the transformationfrom pixel-related image expression to semantically describedimage expression.

For a fixed node x, let y traverse all the nodes adjacent tox in the grid, and the redistributed nodes after the refine-ment form a new grid. The default is triangle division, andall nodes are the vertices of the triangle. After adding anew adaptive grid step by step, return to step to recalculate.Research shows that more than 70% of all informationreceived by humans is obtained through vision. Comparedwith voice and text information, images contain larger, moreintuitive, and more accurate information and therefore havehigher practical efficiency and wide applicability. Whenimages are processed and transmitted in digital form, thisstorage and transmission format has become the main devel-opment trend in this field due to its advantages such as goodquality, low cost, miniaturization, and easy implementation.

However, in the process of image formation, recording, pro-cessing, and transmission, due to the imperfect imagingsystem, recording equipment, transmission medium, andprocessing method, the image quality may be reduced,which cannot fully reflect the real content of the scene. Thisphenomenon is called image degradation. In this model, theimage degradation process is modeled as a system functionthat acts on the input image (usually called the point spreadfunction). It is combined with an additive noise nðxÞ, whichforces to produce a degraded image gðx, yÞ. The concept ofphysics is introduced into the image, and the whole imagecan be regarded as a flat plate with different heat. The heatconduction equation follows that time changes make thetemperature of the entire plate tend to balance. The high-temperature point delivers energy to the low-temperaturepoint to cool down, and the low-temperature point acceptsenergy to heat up until the iteration is terminated. The noisepoints in the image can be regarded as high-temperatureenergy points, which follow the equation evolution; thefrequency gradually decreases until it is smoothed out.

3.3. Optimization of Planning Evaluation Factors. In theprocess of smoothing the image with the Perona-Malikequation, sometimes “blocking effect” appears. That is, thegrayscales in some areas after the 4 image processing arethe same. The same grayscale in the area means that thefirst-order reciprocal of the gray value at any point in thearea is 0. This shows that as the number of iterationsincreases, the image transitions to the same grayscale image.Figure 3 shows the comparison of the histogram of imagefeature smoothness. This can also be seen from the assump-tion of the image model in the robustness estimationmethod. In the robustness estimation method, the assumedimage is the sum of the horizontal image of the block (thegradient of each point in the same block is zero) and thenoise with zero mean and small variance. Using an attenu-ated enhancement factor can make high-contrast pixels lessenhanced, and the enhanced image result is generally softer,and the visual effect is not too sharp, but some useful but

Input 1

Input 2

Image unit 1

Image unit 1

Image unit 2

Image unit 2

Condition 1

Condition 2

Solving algorithm

Passage area

Linear diffusion

Scale space

Semanticdescription

Scaleparameter

Scaleparameter

Semanticdescription

Output 1

Output 2

Pixel Pixel X X Node Node

Urban route planning based on partial differential equationsBordersettings

Subdividedgrid

++

Figure 2: Flow chart of image feature enhancement algorithm.

5Advances in Mathematical Physics

originally difficult to observe details are better enhanced. Inthis way, smoothing and eliminating noise multiple timeswill inevitably make the image transition to a block-levelimage. If we change u to the form of v (v is the Laplacianoperator), when v = 0, the image is only a plane, but a hori-zontal plane is not necessarily required.

In order to avoid damage to image features and achievegood results, smoothing should follow the following twoprinciples: (1) the areas with strong image features are lesssmooth; the areas with weak image features are moresmooth; (2) the vertical smoothness of the image featureis small; the smoothness along the image feature is large.In the direction of image features which are mentioned,the method based on partial differential equations judgesthe flatness of the image according to the size of the gradi-ent during processing. This can be seen from the pixelpoint gradient in the coefficient distribution functionduring iteration. On the other hand, it also detects thedirection of the image feature and reduces the smoothnessin the direction perpendicular to the feature. Accuratelyextracting the image feature direction is critical for imagesmoothing and other image processing tasks. For someimage feature directions, gradient-based extraction methodsare prone to errors. Partial differential equations began tobe used in physics to describe the objective world. Later,after continuous in-depth research by researchers, partialdifferential equations have gradually been applied to thefield of digital image processing, and good results have beenachieved in this field. The image denoising method of dif-ferential equation not only effectively filters out the noisein the image but also the important structural feature infor-mation of the image is basically not lost. In addition, theimage denoising algorithm based on partial differentialequations is a nonlinear method and has a relatively com-plete mathematical theory system and design structure. Inaddition, the calculation is fast, simple, accurate, and stable.More importantly, the partial differential equations areclosely related to the physical characteristics of the imagesuch as gradient and variance.

4. Application and Analysis of Urban PlanningImage Feature Enhancement Model Based onPartial Differential Equation Method

4.1. Numerical Processing of Partial Differential Equations.The image itself has a complicated structure, the pixel grayvalue of the noise and texture detail area of the image istoo large, and the gray value of the flat area is relativelysmall. Taking into account this characteristic, the frequencydomain low-pass filter is used to filter out the noise signal,and then, the image information of the low-frequency partis retained, so as to achieve the purpose of removing noise.Partial differential equations use the same diffusion strategyin different regions, that is, regardless of the direction, thediffusion capacity and speed of the equation are the same.The ideal conductivity coefficient should be more spread inthe smooth area, less spread around the grayscale changes,so as to protect the edge while smoothing out some smallchanges such as noise and some unnecessary textures. How-ever, the image itself has complex structural information,and the use of the same diffusion coefficient will inevitablyignore the importance of the image’s structural feature infor-mation, which in turn leads to image distortion. The modeluses the local gradient characteristics of the image to adap-tively determine the diffusion coefficient. Different regionsand moments have different diffusion coefficients, whichcan effectively retain the edge information and texturedetails of the image. Figure 4 shows the denoising effect curvedistribution at different noise points in the image. Specifi-cally, in areas where noise is densely distributed, the diffusionability is relatively large, which can filter out the noise pointsof the image; in the edge area of the image, the spread func-tion is almost no longer diffused, thereby protecting the edgeof the image from being affected by the filter.

In order to verify the effectiveness of the method in thispaper, the article selects a 128 × 128 image as the test imageand adds various mixed noises to it (this article takes one ofthe mixed noises as an example, the Gaussian noise and den-sity with a variance of 0.01%, 0.05% of the salt and peppernoise is mixed), and then, the second-order PDE denoisingmodel, the fourth-order YK model, the LC model, and themodel in this paper are used for denoising comparison.According to the classification accuracy effect on the data-base, the results are divided into two groups for analysis, asshown in the paper.

It can be seen from the classification results that theclassification accuracy of the method is greatly improvedcompared to other methods. Figure 5 shows a schematic dia-gram of the numerical processing effect of partial differentialequations. The scene contains obvious foreground targetsand background with similar local structure. The classifica-tion effect of low-rank coding methods such as LrrSPM(FastLow-rank Representation based Spatial Pyramid Matching)is significantly higher than that of sparse coding ScSPM,because these images have similar or uniform structural fea-tures. The low-rank coding adopts a joint coding method,and the minimum rank regularization can capture these fea-tures. Commonality at the same time, it can also be seen thatthe Local Coding Method (LLC) is better than the Sparse

Dat

a gro

up

1

2

3

4

Image Feature smoothness

0 20 40 60 80 100

Image 1Image 2

Image 3Image 4

Figure 3: Comparison of smoothness histograms of image features.

6 Advances in Mathematical Physics

Coding (ScSPM). The reason for the analysis is that the localfeatures of each type of image contained in the scene areextremely similar; the local features of the same snowboard-ing are very similar. It is further verified that local con-straints can effectively capture the similarities and commonfeatures between local feature descriptors.

4.2. Image Feature Model Simulation. The experimentalenvironment is developed with Matlab, running on a PC,CPU frequency is 512MB, and collision radius is 0.4m.The environment is a rectangular area under 20m × 20m,obstacles are randomly set, and the size is arbitrary.Dynamic environment parameters are random settings.The starting point is set to (0,100), and the end point is setto (100,0). When there are dynamic obstacles in the experi-mental environment, we use the heat conduction partial dif-ferential equation to solve the urban path planning. In theobstacle, point 1 is the starting point of movement, point 2initially meets the city, point 3 meets the boundary of thesolid wall, and point 4 is the path point the ball has traveledthrough. It can be seen that the adiabatic condition of theobstacle boundary affects the heat conduction in the plan-ning process, making the original heat conduction directionchange. By predicting the movement speed of obstacles, theobstacle avoidance and detour can be successfully achieved.Because the movement speed of obstacle is too large, it doesnot affect the original conduction planning path, so there isno need to detour.

In order to analyze the effectiveness of the proposedadaptive fractional denoising algorithm based on theimproved PM model, four grayscale images with a size of512 × 512 are selected for comparative analysis. Secondly,combined with the experimental results, we use Matlab tosimulate the experiment. Solving the lowest matrix represen-tation of a 128-dimensional image gets a rank of 59, whichshows that the rank of natural images is much smaller thanthe dimensionality, it is divided into several small blocks

according to the local similarity of the image, and the mini-mum matrix representation of each local small block isobtained. It can be seen that local constraint coding cancapture the relationship of similar features. Figure 6 showsthe image threshold parameter gradient denoising effectdistribution. There is also an important parameter K in theproposed model. The value of the threshold parameter Kdetermines the gradient value that starts to weaken the diffu-sion ability. If its value is unreasonable or inaccurate, therewill be incomplete noise filtering or excessively smooth flatareas and the loss of edge information.

Figure 7 shows the before and after comparison of theconvergence speed of the image algorithm. In order to seethe influence of K value on the diffusion rate more intui-tively, here, we take K as 20 and 100, respectively, to drawa graph of the relationship between diffusion coefficientsand gradient modulus. The image signal-to-noise ratio afterdenoising is 11.83, and the calculation time is 21.44 seconds.The noise in the image has been removed to a certain extent,but the visual effect is still not satisfactory, and the conver-gence speed of the algorithm is too slow, resulting in themodel’s denoising running time is too long. For the param-eters, we improved their values. In the paper, we set theparameter s = 70, the quantity = 20 (the value of the harmfulvalue in Matlab is the minimum real number), the visualeffect of the image after denoising is better, and the visualquality of the image has been improved to a large extent.The image SNR after denoising is 14.37, and the calculationtime is greatly shortened to 19.66 seconds.

4.3. Example Application and Analysis. It is the planned pathobtained after solving the elliptic partial differential equationmobile robot path planning model established in this paper.Due to the adaptive mesh refinement of the experimentalenvironment, the planned path remains in a narrow area.It can be passed safely, and the planned path is relativelysmooth, avoiding sudden changes in city-related operatingparameters, so it can be applied to path planning in incom-plete cities. All experiments in Matlab R2012a for Windows10 are used for simulation, and computer configuration is anIntel Core i5 2.40GHz and a 4G operating core. From thepaper, the setting of obstacle boundary conditions and thethree-dimensional results of model calculation can be seenmore intuitively with the results of smoothing experimentson images with fluctuation characteristics.

Figure 8 shows the box plot of the denoising image rec-ognition degree deviation. It can be seen that when imagepixels are used as the diffusion function, the denoised imagehas a better recognition effect. Therefore, it shows that thetwo diffusion coefficients can effectively remove the noise.However, according to the data, the PSNR (peak signal-to-noise ratio), SSIM (Structural Similarity), and informationentropy of the denoised image when image index 1 is usedas the spread function are slightly higher than the corre-sponding values of the denoised image when image index 2is used as the spread function. In addition, when imageindex 1 is used as the spread function, the running time ofthe denoising image is significantly shortened. Therefore,

OriginalFitting

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Imag

e noi

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Figure 4: Denoising effect curve distribution at different noisepoints of the image.

7Advances in Mathematical Physics

the spread function image index 1 is selected to analyze theimage in the following models.

Figure 9 shows a three-dimensional histogram of thepeak signal-to-noise ratio versus pixel diffusion time. It canbe seen that the peak signal-to-noise ratio value graduallydecreases as the number of diffusion increases, because thelarger the peak signal-to-noise ratio value is, the longer thediffusion time and the loss of image texture details are, sothe peak signal-to-noise ratio value should theoreticallyconform to the decreasing function. The basic idea is togradually increase the number of diffusions until the peaksignal-to-noise ratio reaches the maximum after the diffu-sion starts. After each iteration of the noise image, the noise

image will become smoother. The smoother the image is, thesmaller the gradient of the image is. It can also be seen fromthe results that the effect of sparse coding is better than thatof low-rank coding, which shows that sparse coding adoptsan independent coding method for each image to preservethe individual characteristics of the image, but this codingmethod has differences in characteristics. If a slightly chan-ged image is added to a similar scene class, the coding coef-ficient will change greatly. Therefore, as the number ofdiffusion increases, the value of peak signal-to-noise ratio

Sparse coding

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Diffusion coefficientDiffusion coefficient

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Edge information

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solv

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Filter out the noise signal

Texture detail Texture detail

Numerical processing of partialdifferential equation

SPM

SPM SPM SPM

SPM CLR

CLR

CLR

CLRCLR

Figure 5: Schematic diagram of the numerical processing effect of partial differential equations.

Imag

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ld g

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ent r

atio

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Figure 6: Image threshold parameter gradient denoising effectdistribution.

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erge

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Figure 7: Convergence speed of image algorithm before and aftercomparison line chart.

8 Advances in Mathematical Physics

must be reduced. Only in this way can the details and textureinformation be better preserved, so the peak signal-to-noiseratio depends on pixel diffusion time to decrease.

In order to verify the effect of the partial differentialequation algorithm, in the environment shown above, thegenetic algorithm, the Voronoi diagram search, the artificialpotential field algorithm, the RRT algorithm, and the partialdifferential equation algorithm in this paper are compared,respectively, to carry out urban route planning. Figure 10shows a fan chart of comparison of the effects of urban plan-ning image enhancement algorithms based on partial differ-ential equations. Compared with other planning algorithms,the planning algorithm in this paper reduces the plannedpath length by 33.45%, 27.23%, 18.21%, 15.24%, and5.87%, respectively. It can be clearly seen that the denoisingimage of the model proposed in this chapter is clearer,

namely, it has a good visual effect, indicating that the algo-rithm can effectively remove noise. In addition, the edgeand texture details of the denoised image are also clearer,indicating that the algorithm proposed in this chapter canbetter preserve the edge and texture information to a certainextent without being destroyed, and more importantly, itreduces the “staircase effect” to a certain extent.

5. Conclusion

This paper analyzes the feasibility of the partial differentialequation algorithm in urban path planning and proposes anew method of urban path planning. We use partial differ-ential equations to model urban path planning, throughthe adaptive mesh refinement of the experimental environ-ment; it realizes the application of partial differential equa-tions in urban path planning. Theoretical research showsthat it is feasible to solve the path planning of mobile robotsby partial differential equation algorithms, which providesnew ideas and methods for urban path planning researchfor static and dynamic. The experimental results prove theeffectiveness of the partial differential equation algorithmto solve the urban path planning problem.

The main improvements in this paper are as follows: (1)considering that PM will have a step effect in the process ofrestoring images, the improved coupling median filter andPM model are proposed with the complex diffusion model.The improved algorithm effectively combines the advantagesof various models and improves the visual effect of the imagewithout affecting the denoising performance. (2) The modelis unable to effectively remove the shot noise points, so it isimproved. A time-weighted selective diffusion model isproposed. The improved algorithm is relative in terms ofrecovery effect and convergence speed. The model has beenimproved to some extent. (3) Considering that the total var-iation model will have a step effect in the process of restoringthe image, an improved model that couples the second-orderand fourth-order partial differential equations is proposed.The improved algorithm improves the visual effect of theimage, and the convergence speed is much higher than thatof the fourth-order model.

Test number

1 2 3 4 5 6 7 8

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oisin

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Figure 8: Box diagram of denoising image recognition degreedeviation.

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0Peak

sign

al-to

-noi

se ra

tio/%

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34

Image index10

2030

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Pixel diffusion time/s

23Image ind 30

Figure 9: Three-dimensional histogram of peak signal-to-noiseratio with pixel diffusion time.

15.24%

18.21%27.23%

33.45%5.87%

Algorithm 1 Algorithm 2 Algorithm 3 Algorithm 4 Algorithm 5

Figure 10: Comparison of the effects of urban planning imageenhancement algorithms based on partial differential equations.

9Advances in Mathematical Physics

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The author declares no known competing financial interestsor personal relationships that could have appeared to influ-ence the work reported in this paper.

Acknowledgments

This study was supported by the Department of Scienceand Technology Research Project: Theory and Practice ofSeismic Optimal Design of Urban and Rural Water SupplyNetwork (Grant No. 202102310246), Henan ProvincialKey Scientific Research Projects of Colleges and Universi-ties: Research on Safety Performance and OptimizationDesign of Urban and Rural Water Supply Network (GrantNo. 20A560017), and Henan Province Project Plan ofArchival Science and Technology: Study of EarthquakeDisaster Emergency Management Mechanism of ArchivesDepartment in Henan Province (Grant No. 2021-R-20).

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