Research ArticleImage Texture Analysis and Edge Detection Algorithm Based onAnisotropic Diffusion Equation

Xiaoqin Li

College of General Education, Guangdong University of Science and Technology, Dongguan Guangdong 523000, China

Correspondence should be addressed to Xiaoqin Li; lixiaoqin@gdust.edu.cn

Received 28 August 2021; Accepted 22 September 2021; Published 4 October 2021

Academic Editor: Miaochao Chen

Copyright © 2021 Xiaoqin 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.

This paper uses partial differential equation image processing techniques to establish image texture analysis models based onnonlinear anisotropic diffusion equations for image denoising, image segmentation, and image decomposition. This paperproposes a class of denoising models based on the hybrid anisotropic diffusion equation from the characteristics of differentnoise types. The model exhibits anisotropic diffusion near the image boundary, which can protect the boundary well, andisotropic diffusion inside the image; so, it can remove the noise effectively. We use the immovable point theory to prove theuniqueness of the model solution and further discuss other properties such as asymptotics of the solution. We propose a classof image texture analysis algorithms based on anisotropic diffusion equations and discrete gray level sets. First, a class ofnonconvex generalized functions is proposed to remove the noise from the original image to obtain a smooth image whilesharpening the edges. Then, an energy generalization function based on the gray level set is proposed, and the existence of theglobal minimum of this energy generalization function is discussed. Finally, an equivalent form of this energy generalization isgiven in the discrete case, and an image texture analysis algorithm is designed based on the equivalent form. The algorithm isimproved by initial position optimization, dynamic adjustment of parameters, and adaptive selection of thresholds so that theants can search along the real edges. Experiments show that the improved algorithm for image edge detection can obtain morecomplete edges and better detection results. The energy generalization function is calculated directly on the discrete gray levelset instead of solving the corresponding partial differential equation, which can avoid the selection of the initial level set andthe reinitialization of the level set, thus greatly improving the segmentation efficiency. The new algorithm has a highimprovement in segmentation efficiency and can efficiently handle large size complex images.

1. Introduction

In the process of processing digital images, the imagesobtained in practice are subject to many contaminants andimage noise, due to the current state of technology. The noisemakes the image blurred and thus loses some extremelyimportant information, which makes the actual analysis dif-ficult. Therefore, only effective preprocessing of noisy imagescan more accurately obtain the information we need from theimages. Nonlinear diffusion filtering is often utilized in thepost-processing of fluctuating data to visualize quality-related features in computer-aided quality control, as wellas for texture enhancement [1]. Nonlinear diffusion is veryuseful for improvement, secondary sampling, line detectionin blind image recovery, scale space-based segmentation

algorithms, texture segmentation, telemetry data segmenta-tion, and for target tracking in infrared images [2]. Withthe widespread development of image processing technology,people hope to give a clear mathematical explanation forimage processing and then pay more attention to the mathe-matical nature of image processing. Mathematicians try toremodel existing image processing problems with strictmathematical theories, classify existing image processingalgorithms, and improve image processing models at thelevel of mathematical theory. Among these mathematicaltheories, partial differential equations with mature systemshave become a natural choice [3].

The anisotropic diffusion equation is introduced intotexture analysis to extract more discriminative image fea-tures. However, the anisotropic diffusion equation is an

HindawiAdvances in Mathematical PhysicsVolume 2021, Article ID 9910882, 11 pageshttps://doi.org/10.1155/2021/9910882

algorithm that requires a triple generalized cascade for imageprocessing, and more generalized functions are required toextract features for a more accurate representation of imagefeatures. The 0-dimensional real features obtained from theanisotropic diffusion equation after the triple generalizedcascade operation lose more image information, which alsolimits its application prospects [4]. In addition, in the pro-cess of image feature extraction, the features of a single algo-rithm are slightly inferior in characterizing the diversity ofimage information, the anisotropic diffusion equation is noexception, and the combination of more generalized func-tions to extract the triple features of an image is not effectivein terms of computational efficiency. Because of the aboveproblems, this project starts from the basis of an anisotropicdiffusion equation, conducts an indepth study of it, com-bines its advantages and disadvantages, and proposes anew image feature extraction algorithm based on multireso-lution anisotropic diffusion equation and multi-featurefusion for texture analysis [5]. In the framework of partialdifferential equations, it is easy to introduce and integratemethods from other fields to achieve model improvementand innovation. The theory of partial differential equationalgorithm is mature, and it is easy to carry out the numericalrealization of the model. Based on the many advantages ofpartial differential equations for image processing, this paperuses anisotropic diffusion equations (sets) to establish aseries of image denoising, image segmentation, and imagedecomposition models [6].

In image processing, the partial differential equationmethod has the following significant theoretical and compu-tational features compared with other methods. Partial dif-ferential equations have a deep background and foundationin mathematical theory, which allows models to be adaptedand stable algorithms to be obtained. Partial differentialequations can integrate the established classical methodsinto a new framework system and reinterpret them in thatsystem. In the framework of partial differential equations,it is easy to introduce and integrate methods from otherfields to achieve model improvements and innovations.The theory of partial differential equation algorithms ismature and easy to carry out the numerical implementationof the model. In the first chapter, we start from the researchbackground and significance of image literality analysis andedge detection and analyze the research difficulties and theresearch content of this topic. In the second chapter, we ana-lyze the current research and provide the relevant theoreticalresearch basis for this paper. In the third chapter, the studyof image texture analysis and edge detection based on ananisotropic diffusion equation is presented. In this paper,the energy generalization is calculated directly on the dis-crete gray level set instead of solving the partial differentialequation, thus avoiding the selection of the initial level setand the re-initialization of the level set, which can greatlyimprove the segmentation efficiency. The detailed stepsand principles of the ant colony algorithm in edge detectionare introduced, and the algorithm is improved and opti-mized for its problems and verified in natural images. Chap-ter four analyzes the image texture analysis and the imageedge detection algorithm and proves that the research

method in this paper has better performance and effective-ness. Chapter five summarizes and reviews the research con-tents of this paper, explains the areas that still needimprovement in the research process, and points out thedirection for future research work.

2. Related Work

From segmentation methods, image segmentation is mainlydivided into edge-based detection methods and region-basedsegmentation methods. Since this method can achievecompletely unsupervised segmentation, it is widely used inthe field of image segmentation. Compared with traditionalimage segmentation methods, the active contour modelcan accurately achieve segmentation accuracy and can alsobe more easily expressed in terms of energy generalizationand ensure the clarity of the image shape, intensity distribu-tion, and other attributes [7]. Jubairahmed et al. first pro-posed the famous snake model, which is a segmentationmodel based on boundary detection, and the idea is to obtainthe final segmentation curve by solving for the minimum ofthe geometric measure [8]. The main idea is to transform theboundary problem into a zero-level set of higher one-dimensional functions, which finally reduces to solving par-tial differential equations. Under the framework of the levelset method, the image segmentation model based on partialdifferential equations has been developed greatly [9]. Basedon the work of Mafi et al., a series of image segmentationmodels based on the set of reaction dilation equations wereproposed. One equation in the set of equations is used toobtain the structural part of the image, the other equationis used to obtain the structural part of the image, the otherequation is used to obtain the texture part of the image,and the two equations are coupled by the source term [10].The solution of one equation will affect the source term ofthe other equation, and finally, obtain uniform and smoothimage structure information and oscillating texture informa-tion [11].

Sun et al. examine several numerical schemes to creatediscrete nonlinear diffusion scale-spaces, for dimensionalvector models, proving that the explicit scheme is appropri-ate. They show that the diffusion overshoot process is suit-able, as well as satisfying the mean gray invariance andextremum principles, making the sequence smooth and con-vergent to a constant steady-state [12]. He et al. establish rel-evant one-dimensional semiimplicit and Crank-Nicolsoncomputational formats and compare their results. In addi-tion, four authors also examine the m-dimensional diffusion,for which they also make use of the related one-dimensionalsemi-implicit and Crank-Nicolson computational formats,under which conditions the matrix to be computed is not tri-diagonal and is a general matrix, and therefore computation-ally highland. Therefore, the authors suggest using splitting,alternating direction implicit, local one-dimensional, oradaptive operator splitting computational formats [13]. Baiet al. explain how to use anisotropic diffusion models forcolor images so that the image is defined on a multidimen-sional surface. Good results were obtained with boundaryenhancement and enhancement of coherent diffusion

2 Advances in Mathematical Physics

model, using it with grayscale images, proposing a multidi-mensional color model [14]. To deal with the vector imageapproach, the simplest one is to treat each channel of themultichannel data separately and independently by diffusionfiltering [15]. This approach causes undesired effects as if theboundaries of each channel are formed at different locationson the image. To avoid this effect, an integrated diffusionprocess must be used so that it summarizes the data of allchannels [16].

The existing research on the system of reaction-diffusionequations is getting deeper and deeper, especially the theoret-ical research on the minimum variational flow has beengreatly developed in the last decade, and the study of the sys-tem of reaction-diffusion equations using these methods willbe an important topic [17]. We will develop a series of imagedecomposition models based on the system of reaction-diffusion equations and guide the selection of model parame-ters and the design of algorithms by studying the suitability oftheir solutions [18]. A zero-crossing detector is used to sepa-rate edge regions from nonedge regions, and then an adaptiveweight function is used to select the edge detection method,which can be adaptively adjusted to use different methodsaccording to the information in the region range, and a fullvariational model is used to detect regions containing doubleedges and the presence of false edges. Meanwhile, this methodis also helpful in removing image noise [19, 20]. By splittingthe edge regions and nonedge regions, the smoothing opera-tion can be reduced in the part containing edges to retainmoreedge detail information; in the nonedge part, the smoothingoperation is used to remove noise.

3. Research on Image Texture Analysis andEdge Detection Based on the AnisotropicDiffusion Equation

3.1. Image Texture Feature Model. The texture image fea-tures extracted by this image texture representation methodcontain the following three components: segmentationstructure pattern features, fine-grained local binary patternfeatures, and neighborhood difference pattern features. Thesplit structure pattern feature contains the complete contourstructure of the image texture and is used to represent theoverall contour information of the image texture. The subdi-vided local binary pattern feature is a subdivision of the tra-ditional complete local binary pattern so that thenonuniform patterns in the traditional complete local binarypattern method are fully utilized for obtaining the localinformation of the image texture. Using the difference infor-mation between the neighbor pixels of the segmented struc-tural pattern and the subdivided local binary pattern, weobtain the neighborhood difference pattern feature. Amongthem, the neighborhood difference pattern features acquireboth the local texture information of the image texture andthe overall contour information of the image texture. Thesethree partial features are normalized to form the structuraldifference histogram representation features used to describethe texture image [21, 22]. It can be seen that this image tex-ture representation effectively combines both local and

macrotexture information of image textures. The structuraldifferential histogram representation features can obtaingood classification performance for different imaging condi-tions, such as illumination changes, image rotation, anddeflation. Moreover, the feature extraction method is com-putationally fast, and the dimensionality of the features isrelatively small, which can achieve fast texture image classi-fication while ensuring the classification performance.

The anisotropy equation can be expressed as a prob-lem consisting of a diffusive partial differential equationplus an initial condition and a mean Neumann boundarycondition, as shown in Equation (1), where n denotesthe outward normal.

d v! = d Dð K ∀vφ!� �

∀v!� �

+ ϖ v! − f!� �

,

v!x, 1ð Þ = f

!xð Þ,d v!

dm! = 0:

8>>>>>>>><>>>>>>>>:

ð1Þ

The multiscale construction tensor is a matrix derivedfrom the gradient of a function. It summarizes the maindirections of the gradient in the specified domain of pointswith powers that are coherent with these directions [23].The diffusion process with the steering of the structuretensor instead of the regularized gradient steering allowsus to adapt it to more refined goals, such as the enhance-ment of coherent flow-like structural elements. I mustreconsider the vector-valued construction descriptors ∀vφ

!in relation to the matrix basis. The matrix Kð∀vφ!Þ arisesfrom the tensor product as shown in Equation (2), where⊗ is the tensor product with a standard orthogonal basisconstructed from the eigenvectors v1

!, v2!, such that v1!⊕ ∀

vφ! and v2

!⊙ ∀vφ!. The associated eigenvalues are j∀vφ!j2,

and 0 denotes the directional contrast of the eigenvectors.

K ∀vφ!� �

≜ ∀vφ!⊗ ∀v!: ð2Þ

If we consider a color image, for example, a three-dimensional red, green, and blue image, the arithmeticvalue is taken to construct a three-dimensional construc-tion tensor, as shown in Equation (3).

Π v!� �

= ∑3i=1K við Þ3 : ð3Þ

To make the directional information complete, it isnecessary to Kð∀vφ!Þ follow the convolution of the heatkernel by the component Hφ. Therefore, the constructiontensor is obtained, as shown in Equation (4).

Kφ ∀vφ!� �

=Hφ ∗ K ∀vφ!� �

: ð4Þ

3Advances in Mathematical Physics

For a central pixel in a local neighborhood, the rela-tionship between its neighboring pixels is an importantclue for a complete representation of these local neigh-borhoods. In this section, we will obtain the rotation-invariant neighbor difference pattern features of a textureimage and its segmentation structure. For a central pixelin a local neighborhood of texture image I, its rotation-invariant NDP can be expressed as Equation (5). Here,C is the number of neighbor pixels of the central pixel,f ðm, nÞmod ði,CÞ is the ith neighbor pixel, and fðm, nÞmod ði−1,CÞ is the (i − 1)-th neighbor pixel.

NDP m, nð Þ = limC⟶∞

〠C−1

i=0h f m, nð Þmod i,Cð Þ − f m, nð Þmod i−1,Cð Þ� �

: ð5Þ

The NDP feature histogram is built from the con-structed NDP to represent the difference informationbetween neighboring pixels in the local area of the textureimage I. The square window of the NDP feature histogramcan be calculated by a mathematical expression, asexpressed in Equation (6).

gi = limM,N⟶∞

〠M

m=1〠N

n=1f NDP m, nð Þ, ið Þ, i ⊆ 1, C½ �: ð6Þ

To extract SDHR features, a given texture image is firsttransformed into a structurally encoded map; then, threetypes of patterns are computed for constructing SSP, RLBP,and NDP features, respectively; finally, we use SDHR torepresent the connection of these three parts of pattern his-togram features [24]. To have a clear and complete under-standing of our proposed method, the complete flowchartof constructing SDHR features is given in Figure 1. Fromthe computation process of SDHR features, we can easilysee that our proposed texture feature representation not

only contains microtexture information, but also the overallcontour information of the image texture is included.

Let the difference between the values on the two points(m0, n0) and (m1, n1) of the image be Equation (7).

∀v! = v! m0, n0ð Þ ⊙ v! m1, n1ð Þ: ð7Þ

When the Euclidean distance of (m0, n0) from (m1, n1) isnear zero, the arc length is equivalent to Equation (8).

d v! = 〠3

i=1

dvidm

ðdm + dvi

dn

ðdn

� �: ð8Þ

Consider a curve c in a manifold such that it is parame-trized by the arc length. If cð0Þ = p and its first-order deriva-tive are contained in the tangent plane, thenðað0Þ, vðcÞÞ = ΓðX, XÞ, the second fundamental form ofwhich has the following expression ΓðX, XÞ = Kμ ∗ sin μ,Kμ is the curvature of the curve cφ which is cut from themanifold by the plane containing X such that the anglebetween this plane, and the normal plane is equivalent to μ. Thus, due to the proven positivity, we can define theparametrization of Equation (9).

f v!� �

=ffiffiffiffiffiffiffiffiffiffiðd v!

s=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi〠3

i=1

dvidm

ðdm + dvi

dn

ðdn

� �s: ð9Þ

The features extracted from the spatial, wavelet, andspectral domains are reasonable. Specifically, the pixel valuesin the color space are extracted from the spatial domain torepresent the color information of the image. Since thewavelet transform captures the directional and multiresolu-tion information of the image, the directional and multireso-lution features are extracted from the wavelet domain. Inaddition, spectral features are also extracted from the

Texture picture

Fine local binary modefeatures

Segmentation structurepattern features

Neighborhood differencemode features

NN classifier

Figure 1: Flow chart of texture classification method.

4 Advances in Mathematical Physics

spectral domain. Finally, the features of the above threedomains are converted into an NS feature matrix using NStransform. This paper proposes an efficient image denoisingmodel based on nonlinear diffusion equation. Aiming at theshortcomings of the traditional model that the forward andbackward diffusion occurs and the step effect is generated,a new type of diffusion coefficient is first constructed, andits theoretical properties are analyzed. Then, a new diffusionequation model was constructed based on the new diffusioncoefficient. This model combines median curvature diffusionand Gaussian heat equation diffusion, and forward andbackward diffusion will occur, but the backward diffusionis controllable, so that the restored image will not producenew features.

3.2. Image Edge Detection Algorithm. Edges play an impor-tant role in image analysis as the most basic feature of animage. Discontinuities in local features, abrupt changes ingrayscale or structural information may manifest as edgeson an image. At the same time, edges are transitions thatconnect regions, and usually, this feature can be used to seg-ment an image. All edges have both direction and amplitudecharacteristics, with a flatter grayscale change following theedge, while the one perpendicular to the edge, generallyreferred to as the normal direction, and usually has anabrupt grayscale change. This change in magnitude withdirection can be measured by a differential operator, mathe-matically derived as a first- or second-order derivative.

The first-order derivative considers the edge as the loca-tion where the pixel undergoes a sharp change, and thischange is generally described by the gradient of the grayscaledistribution, with the maximum value obtained from thefirst-order derivative corresponding to the edge location.Finding the gradient amplitude is the most commonly useddifferentiation method, given a continuous image f (m, n)whose directional derivative obtains a local maximum inthe direction normal to the edge. The second-order deriva-tive can be obtained using the Laplace operator [25]. Thecalculated derivatives over the zero point correspond to theedge positions in the image. For regions with uniform graychanges, the first-order derivative will not find the boundary,and the second-order derivative can not only identify thetype of gray abrupt changes but also provide useful bound-ary information. Although in principle the higher order ofthe derivative means the better image segmentation, thederivative information beyond the second-order lacks appli-cation value. At present, the second-order derivative is rarelyused for edge detection because it is very sensitive to noise,and if the noise is not successfully removed, small noise par-ticles will be amplified into fine lines; so, a part of the noisemust be removed by smoothing filtering before edgedetection.

The second-order differential operator can locate theedge position accurately but ignores the grayscale differenceof the surrounding pixels. In particular, isolated noise pointsnear the edges that are not filtered out, the response to thesepoints can cause high bright spots or thin lines close to theedges, thus affecting the boundary recognition. This is wherelow-pass filtering becomes necessary. Therefore, it is neces-

sary to combine the windowed adaptive median filteringalgorithm proposed in this paper with the Laplace operatorto form a new template. The results of the operations underthe Laplacian differential operator are isotropic, and theform of the template illustrates its unique rotational invari-ance, for a two-dimensional image, defined here as in Equa-tion (10).

Yf m, nð Þ = f m + 1, n + 1ð Þ + f m + 1, nð Þ + f m, n + 1ð Þ½ �

− f m − 1, n − 1ð Þ + f m − 1, nð Þ + f m, n − 1ð Þ½ �+ f m, nð Þ:

ð10Þ

The pull operator is expressed as a discrete form, as inEquation (11). The gray level of 255 is represented ascompletely white, and the gray level of 0 is represented ascompletely black, a gradual process from black to white. Iftwo adjacent gray levels are very close, it is not easy for thehuman eye to distinguish clearly. This explains why thelarger the gray level span, the clearer the image resolution,and the more comfortable it is to look at. Of course, a largerpixel gray level of an image does not all mean the better itscontrast, and a certain number of pixels across these graylevels are required.

Yf =

ðdf 2

dm2 +ðdf 2

dn2: ð11Þ

The role of detection processing after image enhance-ment is to enhance the contrast between pixel gray levels.Because the image becomes blurred by averaging and inte-gration operations, details can be highlighted using imageinverse operations. The Laplacian based on differential oper-ation enhances the image region where the abrupt graychange occurs while the effect of smoothing the gray changein the region is diminished. Therefore, the original image isprocessed using the Laplacian operator to generate an imagethat reflects the grayscale abrupt change, and the imagesharpening effect is obtained by overlaying this image withthe original image. This image sharpening method, becauseit is obtained by superimposing the original image with theresult of the operator processing, can fully retain the back-ground information of the image and the gray value in dif-ferent image areas and enhance the contrast and smalldetail information of the gray mutation area, but the disad-vantage is that it is easy to produce a double response tothe edge of some areas.

The direction of all edges is quantified by delineating arange of directions, again using the horizontal edge as anexample, and defining the direction of the edge by thedirection of the normal. As shown in Figure 2(a), if thecoordinates of the edge normal direction range from-24.5° to +24.5, or from -158.5° to +158.5°, then all edgeswithin this range are horizontal edges. Figure 2(b) is usedto represent the angle ranges corresponding to the fourdirections.

5Advances in Mathematical Physics

There are two basic operators in digital morphologywhich are expansion and erosion. The principle is to obtaininformation about the image morphology and structure in aset image region using structured units of a specific shape,followed by detection and recognition of image details. Forexample, the structured element B corrodes the digital imageand is represented by Equation (12), where Ga is the transla-tion of vector a to N . Setting structural element N as a diskwhen the center of the image is on P, erosion can be consid-ered as the trajectory of the center point of element N whenstructural element B moves in image M.

M ÷N = s ⊆ P,Ga ⊆Mf g: ð12Þ

The expansion operation of structured element B forimage A can be expressed by equation (13), setting the centerof structured element N at the origin, and the expansionoperation can be understood as the whole image area cov-ered by the structured element N scrolling over the image.This operation is the one that causes the expansion of theimage target area and the narrowing of the holes present inthe image to be connected.

M ×N =ℝn⊆N ∗Mn: ð13Þ

Using a combination of these two methods, the image isfirst inflated with structural elements, followed by an erosionprocess, so that any noise that is not removed cleanly is elim-inated to the maximum. This combination of operations,also known as the closed operation, is capable of smoothingthe true contours of the image so that small faults are fusedand small voids are eliminated. This operation is representedby Equation (14).

T1 = M ⊙Nð Þ ∗N ,T2 = M ∗Nð Þ ⊙N:

(ð14Þ

Using the structure element corrosion map, and thencalculating the Boolean difference between the image aftercorrosion and the original image, assuming that G is theimage edge information, Equation (15) can be obtained.Edge detection can be performed on the inner boundary ofthe image to obtain the inner boundary information of theimage.

G =M − M ⊗Nð Þ: ð15Þ

We chose the structural elements based on the size anddimensions of the image itself, which should be as small aspossible, and approximate the shape of the noise distribu-tion, considering the image measurement used. Since theadded pretzel noise is represented as black and white noise,the structural elements are chosen as small dots for theexperiment. This composite approach of morphological fil-tering operator combines two operations. In the paper, botherosion and open operations are combined to form a peaknoise filter, and then by combining expansion and closureoperations, a low-pass noise filter is obtained. The combina-tion of two filters can get the effect of suppressing the noiseof both peaks and troughs in the image. Combined with themorphological operation to strengthen the edges, the linesense of the detected target is stronger after edge recognition.After the image edges are initially extracted with theimproved operator, the multistructure morphological filter-ing is performed to achieve further boundary enhancementof the image and retain more perfect contour edgeinformation.

To address the problems that traditional image edgecontour extraction algorithms often have discontinuousedges and broken edge segments when extracting edges,the algorithm in this paper incorporates the fuzzy triangularaffiliation function into the ant colony algorithm and com-bines it with the edge detection Canny algorithm. Firstly,the Canny algorithm is used to detect the edges of the target

+24.5°−24.5°

−155.5° +155.5°

Positive edgenormal

Negativeedge normal

Edge normal direction coordinate range

(a)

+24.5°−24.5°

+69.5°−69.5°

+113.5°−113.5°

+158.5°−158.5°

0°Horizontal edge

Vertical edge

Edge normal direction angle

(b)

Figure 2: The angle of direction of edge normal.

6 Advances in Mathematical Physics

image, then the ant colony algorithm fused with FTM isused to detect the broken edges twice, and finally, the idealcontinuous edge contour is extracted. The edges areextracted from the original image using the canny opera-tor, setting the gray value of the edge points in the edgeimage to 255 and the gray value of the nonedge part to0. The number of connections between each nonedge pixelpoint and edge pixel point in the figure is calculated asshown in Equation (16).

g xið Þ = f xið Þ∑255

i=1xi,

H = limn⟶∞

〠n

i=11 − g xið Þð Þ ∗ 1 − g xi+1ð Þð Þ ∗ 1 − g xi+2ð Þð Þ:

8>>>><>>>>:

ð16Þ

By setting the ant stop search condition, it can effec-tively reduce the invalid search of ants on edge pointsand nonedge regions, saving the search time and improv-ing the efficiency of the algorithm. After all K ants placedare iterated, by setting the pheromone concentrationthreshold, the pixel points larger than the threshold areselected and marked as edge points, thus completing thesecondary detection of the area between the edge end-points in the edge image and obtaining better continuityand clearer edge information.

4. Analysis of Results

4.1. Image Texture Analysis. To test the recognition ability ofthe features extracted by our method, we conducted testexperiments on set 1, set 2, set 3, set 4, set 5, and set 6. Mean-while, we used LRS-MD, PMC-BC, CLBP, MLEP, Cov-LBPD, and scLBP to compare with our anisotropic diffusionequation features. The ACAR of the results of the experi-

ment is shown in Figure 3. From Figure 3, it can be seen thatour method slightly outperforms the other comparisonmethods when the images are not rotated, and the standarddeviation of our method is lower than that of the other com-parison methods. In addition, we also give the ACARS of theSDMV feature and the FV feature of the texture image,where these two features are represented by the anisotropicdiffusion equation-SDMV and the anisotropic diffusionequation-FV. The ACARS of these two independent featuresillustrates that the combination of both can achieve betterclassification results.

To further test the performance of our proposed aniso-tropic diffusion equation feature in processing rotatedimages, we conducted test experiments on set 1, set 2, set3, set 4, set 5, and set 6, respectively. The ACARS of theexperimental results of our method and the comparisonmethod is shown in Figure 4. It is clear from Figure 4 thatthe ACAR of our method outperforms the six comparisonmethods. Our method obtains the best experimental resultson set 1, set 2, set 3, set 4, set 5, and set 6. Moreover, thestandard deviation of our method is smaller than the othersix comparison methods, except that the standard deviationof the CLBP method is lower than our method on set 4 andset 5. The results of image texture classification experimentsdemonstrate that the rotation-invariant property has beenincluded in our extracted anisotropic diffusion equationimage texture features, and such a combination of featuresimproves the recognition ability based on transform domainfeatures.

Combining the results in Figures 3 and 4, we can see thatour method yields good classification results with and with-out image rotation. This further proves that rotation invari-ance is included in our extracted features.

To further extensively test the performance of ourextracted image texture features for rotated image classifica-tion, we further conduct test experiments for rotation invari-ance on two more complex image texture datasets, TIPS2

1 2 3 4 5 6

80

85

90

95

100

Reco

gniti

on re

sult

Data set

LRS-MDPCM-BCCLBP

MLEPCov-LBPScLBP

MRIR-FVMRIR-SDMVMRIR

Figure 3: Comparison results of recognition experiments.

1 2 3 4 5 6

40

50

60

70

80

90

Perfo

rman

ce re

sult

Data set

LRS-MDPCM-BCCLBP

MLEPCov-LBPScLBP

MRIR-FVMRIR-SDMVMRIR

Figure 4: Performance experimental comparison results.

7Advances in Mathematical Physics

and OUTEX. The experimental results of our method andthe comparison method are shown in Figure 5. As can beseen from Figure 5, our method slightly outperforms theother comparison methods on both challenging texture sets.Thus, the results of these experiments further demonstratethe robustness of our texture feature construction methodin processing rotated images. The results of the whole imagetexture classification comparison experiments on the TIPS2and OUTEX databases show that our method can handlerotated images well and can obtain satisfactory classificationperformance.

Based on the introduction of the anisotropic diffusionequation image texture feature representation method, itcan be seen that this feature representation method effec-tively combines the advantages of the rotational invarianceof the null domain method and the advantages of thetransform domain method that can perform directionalmultiscale analysis, and it can be said that this method isan effective combination of the null domain method andthe transform domain method. At the same time, theadvantages of the two methods complement each other,thus improving the classification performance of the con-structed features. The value of this texture feature con-struction method is that it introduces the idea ofrotation invariance into the transform domain-basedmethod. However, the dimensionality of the features con-structed by this method is high, and therefore, featuredimensionality reduction is required.

4.2. Edge Detection Analysis. The quantitative analysis of theexperiments in this section still utilizes the mean squareerror MSE and peak signal-to-noise ratio PSNR as evalua-tion metrics. Since noise as blood vessels around the tissueis the main factor affecting the detection results in the massimages, the superiority of the algorithm can be judged by theamount of noise content in the results. The results are shownin Figure 6. From Figure 6, it can be seen that the meansquare error of the method in this paper is smaller comparedto the other two algorithms, indicating that the noise contentis less in the detection results of the method in this paper,and the corresponding value of the peak signal-to-noise ratiois larger. Combining the results of both subjective judgmentsand quantitative analyses illustrates the effectiveness of thealgorithm in this paper. Using the edge detection methodis one of the main methods for image segmentation, andthe lumps can be filled next to complete the segmentation.

As can be seen in Figure 7, compared with the lump seg-mentation results obtained by the other two methods, thelump region extracted using the algorithm proposed in thispaper has significantly better results and higher segmenta-tion accuracy after a comprehensive analysis of the threeindicators. The algorithm uses the Canny edge breakpointas the initial position of the ant, changes the traditional antcolony algorithm to directly use the gradient informationas the heuristic information, but uses the fuzzy triangularaffiliation function to calculate the fuzzy affiliation degreeof the pixel points in the image as the heuristic information

LRS-MD PCM-BC CLBP MLEP Cov-LBP ScLBP MRIR-FV MRIR-SDMV

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Figure 5: Results of rotational invariance tests.

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of the ant colony, and sets the search stopping condition,which makes the continuity of the image edge better andthe lumps obtained after filling more complete. From theextraction results of relevant comparison experiments, itcan be seen that compared with other algorithms, theextracted image edge contour information and the filledobtained lump segmentation image by the method in thispaper are clearer, more complete, and more accurate thanthe adaptive Canny algorithm and the comparison algo-rithm methods. The accuracy of the algorithm in this paperis more than 92%, which is 0.3% higher than other algo-

rithms. The recall rate and F1 value reached 68.9% and78.9%, respectively, which were about 10% higher than otheralgorithms.

The experimental results were statistically analyzedusing the methods in this paper. Figure 8 shows the statisti-cal results of the edge images (T is the threshold consideredappropriate) in which A denotes the number of edge points,B denotes the number of quad-connected pixels, and Cdenotes the number of pixels put into eight connections.

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Figure 6: Quantitative analysis results of the three algorithms.

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Figure 7: Analysis of segmentation results of three algorithms.

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Figure 8: Statistical results of edge images.

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The results of the numerical analysis were compared by cal-culating the ratios C/A and C/B. For any edge image, theratio of C/A to C/B is smaller when its edge lines are betterconnected. In other words, the smaller the value of A, thestronger the continuity of the binary edge image obtainedby the algorithm, and the smaller the value of B, the betterthe one-sided response of the algorithm. According to thestatistical data, the improvement of the algorithm in thispaper is obvious, the continuity is strong, and the unilateralresponse is better.

Based on the traditional median filtering, the image issubdivided into edge detail points, isolated noise points,and flat areas based on the human eye’s visual perceptionof different areas of the image, using the correlation betweenthe pixels around the noise points to extract determinationrules and filtering the image separately for different partsof the noise characteristics combined with visual require-ments. This approach avoids the bias in the detection of sig-nal points on the one hand and improves the efficiency ofthe algorithm detection compared with the algorithm ofglobal processing on the other hand. The proposed smooth-ing noise is applied to Laplace sharpening as a basis to forma new edge detection algorithm, the algorithm is applied to awide range of segmentation datasets, and the detectionresults show good performance in both visual observationand evaluation of objective metrics.

5. Conclusion

In this paper, a class of hybrid diffusion equation modelsbased on thermal equation diffusion and median curvaturediffusion is proposed by combining the characteristics of dif-ferent noises and the features of noisy images. The equationbehaves as anisotropic diffusion at the near boundary of theimage; thus, the equation is close to the median curvaturediffusion equation, which can protect the boundary well,while it behaves as isotropic diffusion in the smooth regionof the image; thus, the equation is close to the heat conduc-tion equation, which can denoise effectively. To define thediffusion tensor, the image is viewed as a three-dimensional Riemann manifold, a view that, although itrequires a fresh look at various aspects of the concept, thegeometric properties of the tensor and the final computa-tional results obtained concretely show that using aniso-tropic diffusion works much better than other classicalmethods. Defining the anisotropic diffusion tensor in termsof the above concepts satisfies all the necessary precondi-tions and next provides its eigenvalues and eigenvector cal-culations to construct the basis for numerical experiments.The next term proves the existence of a unique solution tothe diffusion problem, which results in a generalization ofthe one-dimensional similarity problem. In addition, usingpartial differential analysis, the smoothness and continuousdependence of the solution on the additive noise imageproperty are proved. The filtering convergence to the solu-tion property is also shown not only experimentally but alsotheoretically. The last item proves that the weak comparisonprinciple holds. The theory of this item is more complicated,and it is the basis for protecting the convergence of the solu-

tion and also the uniqueness of the weak solution, but thereader interested in the application does not need to lookat it carefully. For the phenomenon that the ant colony inthe algorithm is affected by pheromone and heuristic infor-mation at different stages to different degrees, the two sensi-tive factors are dynamically adjusted. For the ant colony inthe search is susceptible to the influence of the local optimalsolution, the maximum and minimum pheromone valuesare set to suppress the continuous accumulation of phero-mone of noisy pixel points; in selecting the threshold valueto determine the edge point according to the amount ofpheromone of pixel points, the maximum variance methodbetween classes is used to calculate the thresholds. By apply-ing the improved algorithm to natural images and combin-ing subjective judgment with quantitative analysis, theeffectiveness of the improved method in this paper can beproved.

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 authors declare that they have no known competingfinancial interests or personal relationships that could haveappeared to influence the work reported in this paper.

Acknowledgments

The study was supported by 2018 Guangdong ScientificResearch Project: two scale finite algorithm analysis anderror analysis for a class of piezoelectric problems with qua-siperiodic structures, 2018KQNCX300; 2019 ScientificResearch Project of Guangdong University of Science andTechnology: discreteness of 2n order J-SELF adjoint vectordifferential operator spectrum, GKY-2019KYYB-23.

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