# Morphological Edge Detection Algorithm of Colon Pathological Sections Based on Shearlet

Authors: Shasha Li; Caixia Deng; Tong Wang; Zhaoru Zhang
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper proposes an idea of combining the Meyer Shearlet and mathematical morphology to produce the edge detection of pathological sections of the colon. First, the method of constructing a class of sufficiently smooth sigmoid functions along with its relative scale function and Meyer wavelet function is provided in this paper. Based on those, in order to get the new Meyer wavelet function, we use the sigmoid function to construct more general scale functions. Next, taking sufficiently smooth sigmoid functions as examples, combining the relative Meyer wavelet and Shearlet to denoise some pathological sections of the colon leads a decent feedback. At last, this paper provides an improved algorithm for the edge detection of mathematical morphology with the background of multiscale and multistructure. This algorithm is used to carry out the edge detection of images after denoising yields a new edge detection algorithm that fuses the Meyer Shearlet denoising and mathematical morphology. According to the simulation results, the new algorithm is more beneficial for the observation and diagnosis of doctors since the edge noise of the colon pathological image detected by the new algorithm is smaller and provides more continuous and clear lines. Therefore, the fusion algorithm provided in this paper is an effective way to carry out the edge detection of an image.


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## Body

## 1. Introduction

Colon cancer is one of the most common cancers in clinic. Its morbidity and mortality are increasing year by year, so it is very important to diagnose it in time. The traditional diagnosis method of pathological sections is to make sections of pathological tissues [1]. Doctors grade the pathological tissues after observing them under the microscope. This method is time-consuming and subjective, so a computer-aided diagnosis system is gradually applied in medical diagnosis [2].When using a computer-aided diagnosis system to diagnose pathological sections of the colon, doctors convert traditional pathological sections into digital pathological sections through a microscope acquisition system and then preprocess the images. Finally, the diagnosis results are obtained through feature extraction and pattern recognition.During the process, digital pathological sections are easy to be interfered by equipment, environment, and other factors during the acquisition and transmission process [3], such as the influence of light intensity in the process of image acquisition, noise caused by factors such as device material properties and circuit structure, and quantization noise during digitization. These noises make the texture of the image unclear, and the edges are blurred, which is not conducive to the doctor’s observation of the image details, which seriously affects the doctor’s clinical diagnosis. Therefore, image denoising preprocessing should be performed to reduce noise interference in the image, optimize the image quality, and improve the accuracy of the doctor’s diagnosis.Medical image processing often chooses median filtering and Gaussian filtering for denoising [4, 5], but when they are used to denoise pathological sections image, the gray value of the edge of the target gland will be changed, which is not conducive to doctors' observation. In recent years, research on image denoising methods has focused on the transform domain. Because wavelet transforms have properties such as multiresolution, low entropy, and decorrelation, so wavelet transforms are often used to process noisy images [6–8]. However, the wavelet transform is isotropic, it can only reduce the noise containing horizontal, vertical, and diagonal directions, and the effect of denoising in other directions is not satisfactory. Therefore, Easley et al. described a new class of multidimensional representation systems called Shearlet [9]. Shearlet uses ordinary wavelet as the basis function, by applying shearing, translating, and scaling the basis function to generate multiresolution analysis function with multidirection. It overcomes the shortcomings of the traditional wavelet transform which lacks the ability of direction expression and has good direction selectivity. Because of its differentiability, fast attenuation speed, and limited spectrum, the Meyer wavelet is selected as the basis function of the Shearlet. The image details of denoising the Meyer Shearlet which is constructed by the traditional seventh-order polynomial sigmoid function are blurred, and the edges are not continuous enough, so the smoothness of the Meyer wavelet needs to be improved. Because the properties of Meyer wavelets are closely related to the properties of sigmoid functions in Meyer scale functions, this paper constructs a class of sufficiently smooth sigmoid functions, which can be adjustable and selected in terms of different practical problems. A good result is obtained when the Meyer Shearlet related to this sigmoid function is used to denoise the image.In the construction of a computer-aided diagnosis system for colon cancer, edge detection of the target gland is one of the key steps. Traditional edge detection operators such as Prewitt, Canny, and Sobel [10–12] can obtain clear image edges without noise, but they cannot get real edge images because of their poor antinoise ability to noisy images. With the continuous development of mathematical morphology theory, it is widely used in image edge detection [13]. The edge detection method based on mathematical morphology has a better ability to suppress noise and smoother extracted edges. Therefore, this paper proposes an improved mathematical morphological edge detection algorithm and performs edge detection on the image after denoising the Meyer Shearlet. So a new edge detection algorithm combining Meyer Shearlet and mathematical morphology is proposed. The algorithm can make the edges of the image smoother and clearer, which provides a basis for applying a computer-aided diagnosis system to help doctors make a correct diagnosis.


## 2. Shearlet Transform

Definition 1 (see [14]).
LetΩ,Λ,μ be a measure space,1≤p<+∞. We define LpΩ as a set of Lebesgue measurable functionsf in Ω, that is, ∫Ωfpdμ<∞. The Lp-norm of f∈LpΩ is defined byfp=∫Ωftpdt1/p, and LpΩ is called a normed linear space. Especially, if we haveΩ=ℝ, then when p=1 andp=2 follows that L1ℝ andL2ℝ being the most commonly used two normed linear spaces in the wavelet analysis.Definition 2 (see [15]).
Letft∈L1ℝ, then f^ω=∫−∞∞fte−iωtdt is called the Fourier transform offt.
Iff^ω∈L1ℝ, then F−1f^ω=1/2π∫−∞∞f^ωeiωtdω is the inverse Fourier transform off^ω.
Letfx,y∈L1ℝ2, then we call(1)f^ξ1,ξ2=∫−∞∞∫−∞∞fx,ye−iξ1x+ξ2ydxdy.the Fourier transform of fx,y. Also, if f^ξ1,ξ2∈L2ℝ2, then F−1f^ξ1,ξ2=1/4π2∫−∞∞∫−∞∞f^ξ1,ξ2eiξ1x+ξ2ydξ1dξ2 is the inverse Fourier transform of f^ξ1,ξ2.Definition 3 (see [15]).
Letψ∈L2ℝ2. For ∀ξ=ξ1,ξ2∈ℝ2,ξ1≠0, the Fourier transform of ψ is given by the following equation:(2)ψ^ξ=ψ^ξ1,ξ2=ψ^1ξ1ψ^2ξ2ξ1,where ψ1 andψ2 are satisfying(i)
Forψ1∈L2ℝ, ∀γ∈ℝ, we have ∫0∞ψ^1γξ12dγ/γ=1 andψ^1∈C∞ℝ, suppψ^1⊂−2,−1/2∪1/2,2(ii)
ψ2L2=1, and ψ^2∈C∞ℝ, suppψ^2⊂−1,1
Then,ψp,n,mx=p−3/4ψMp,n−1x−m is called the continuous Shearlet, whereMp,n=ppn0p=BnAp, p>0, n∈ℝ,m∈ℝ2. And where Ap=p00p is a dilation matrix,Bn=1n01 is a shear matrix. In the frequency domain, we have(3)ψ^p,n,mξ1,ξ2=p3/4e−2πiξmψ^1pξ1ψ^2p−1/2ξ2ξ1−n.
The support in the frequency domain ofψ^p,n,mξ1,ξ2 is(4)ξ1,ξ2:ξ1∈−2p,−12p∪12p,2p,ξ2ξ1+n≤p.
The support in the frequency domain of some Shearlets refers to Figure1.
Letf∈L2ℝ2, SHψfp,n,m=f,ψp,n,m is called the continuous Shearlet transform, and the inverse transform of SHψfp,n,m is(5)fx=∫−∞∞∫−∞∞∫0∞f,ψp,n,mψp,n,mdpp3dndm.Figure 1
The support in the frequency domain of Shearlets with different p and n.Definition 4 (see [15]).
Based on a binary sampling of the dilation matrixAp and integer sampling ofBn, that is, p=2j, n=lj,l∈Z. The collection of matrices M2j,l−1=A2j−1Bl−1 whereA2j=2j002j/2 and Bl=1l01. Thus, the discrete Shearlet transform is ψj,l,kx=2−3j/4ψA2j−1Bl−1x−k, where j,l∈ℤ,k∈ℤ2.
Now, the Fourier transform ofψ∈L2ℝ2 is given by “(1)” which satisfies ∑j∈ℤψ^12−jξ12=1, suppψ^1⊂−1/2,−1/4∪1/4,1/2ψ^1∈C∞ℝ, and ∑l∈ℤψ^2l+ξ2/ξ1=1suppψ^2⊂−1,1, also ψ^2∈C∞ℝ. Especially, the Meyer wavelets are commonly used as the basis functions ψ1, and its Fourier transform is(6)ψ^1x=−e−ix/2sinπ2v3x2π−1,if2π3<x<4π3,−e−ix/2cosπ2v3x4π−1,if4π3≤x<8π3,0,if2π3≥xorx≥8π3,where(7)vx=0,ifx<0,x435−84x+70x2−20x3,if0≤x≤1,1,ifx>1.
The Fourier transform ofψ2 isψ^2x=ϕx/∑k∈ℤϕx+k2 where(8)ϕx=exp−11−x2,if−1<x<1,0,ifx≤−1orx≥1.
To achieve a better effect of Shearlet in image denoising, we choose to use Meyer wavelet as one of the basis functions of Shearlet since the properties of Meyer wavelet are related to the Meyer scale function in terms of the sigmoid function.


## 3. Method of Constructing a Class of Sufficiently Smooth Sigmoid Functions

Definition 5 (see [16]).
Assumef^ω is the Fourier transform of ft, for some B∈ℝ+ if whenω≥B, we have f^ω=0, then ft has limited spectrum in B.Definition 6 (see [16]).
A column of closed subspaceVjj∈ℤ in Hilbert space L2ℝ is called an orthogonal multiple resolution analysis (OMRA) if it satisfies the following conditions:(i)
Nesting:Vj⊆Vj+1 for allj∈ℤ(ii)
Scaling:ft∈Vj⇔f2t∈Vj+1 where t∈ℝ,j∈ℤ(iii)
Separation:∩j∈ℤVj=0(iv)
Density:∪j∈ℤVj¯=L2ℝ(v)
Orthonormal basis: there existsτt∈V0, such that the set of functions τt−k,k∈ℤ is an orthonormal basis of V0, where τt is the scale function of the OMRA, and Vj is called subspace of proximity or the scale function of L2ℝDefinition 7 (see [16]).
Ifσt∈L2ℝ, and its Fourier transform σ^ω satisfies the admissibility condition∫−∞∞σ^ω2/ωdω<∞, then σt is a basic wavelet or mother wavelet function.Definition 8 (see [16]).
Letτt∈L2ℝ be a scale function of OMRA, then there exists hkk∈ℤ∈l2 such that(9)τt=∑k∈ℤhkτ2t−k.
Thus,hkk∈ℤ forms a square summable sequence and is called two-scale coefficients of the scale functionτt. Equation (9) is known as two-scale relation for the scaling function τt of the OMRA.
LetHω=1/2∑k∈ℤhke−ikω, then Hω is a periodic function with2π being its period. AndHω2+Hω+π2≡1. When gk=−1kh¯1−k, we have the wavelet function(10)σt=∑k∈ℤgkτ2t−k.
Meanwhile, the Fourier transform ofσt is(11)σ^ω=−e−iω/2Hω2+π¯τ^ω2.Lemma 1.
Letτ^ω be the bounded continuous Fourier transform of τt and satisfying(I)
τ^0=1(II)
∑k∈ℤτ^ω+2kπ2≡1(III)
τ^2ω=Hωτ^ω, where Hω is a periodic function with period 2π
Then, the closed subspace sequence generated byτt is(12)Vj=span¯2j/2τ2jt−k,k∈ℤ,∀j∈ℤ,which forms an OMRA of L2ℝ.Lemma 2.
Letτ^ω be a continuous real function, and suppτ^=a,b, thus the necessary and sufficient conditions such that τ^ω satisfies the following:(I)
τ^2ω=Hωτ^ω, where Hω is a periodic function with period2π(II)
∑k∈ℤτ^ω+2kπ2≡1
are(i)
a<0,b>0(ii)
2π<b−a≤8/3π,b/2−a≤2π,b−a/2≤2π(iii)
Letc=b−a−2π, when ω∈a+c,b−c, we have τ^ω≡1(iv)
Whenω∈a,a+c,τ^ω2+τ^ω+2π2≡1Definition 9 (see [16]).
We denote byC∞I the set of continuous real-valued functionsfx defined onII⊂ℝ, which is a real vector space with respect to ordinary addition and scalar multiplication of fx.Definition 10 (see [17]).
Ifvx is a continuous function that satisfies(13)vx=0,ifx≤0,1,ifx≥1,and vx+v1−x=10<x<1, then vx is called a sigmoid function.
Meyer has provided a class of sigmoid functions in the form of probability, and the polynomial of this class of ones is as follows.
Let the polynomialvθxθ∈ℤ+ be the cumulative distribution function (CDF) of the random variableX, where X is symmetric beta-distributed; that is,X∼βθ,θ, for some x∈0,1, we have(14)vθx=Bθ,θ−1∫0xtθ1−tθ−1dt=xθF1−θ,θ,1+θ;xBθ,θ,where Fα,β,γ,z=∑kαkβk/γkz!zk,z<1,(15)αk=αα+1…α+k+1,Bθ,θ=∫01tθ1−tθ−1dtθ>0.
Thus, we obtain the relative sigmoid function(16)v¯θx=0,ifx<0,vθx,if0≤x≤1,1,ifx>1.
According to the result of the calculation, whenθ=1,v1x=x andv¯1x is a continuous nondifferentiable function on−∞,∞; when θ=2,v2x=x23−2x andv¯2x is a first-order continuous differentiable function; when θ=3,v3x=x310−15x+6x2 andv¯3x is a second-order continuous differentiable function on −∞,∞; when θ=4,v4x=x435−84x+70x2−20x3 andv¯4x is a third-order continuous differentiable function on−∞,∞. Especially, v¯4x is a frequently used seventh-order polynomial sigmoid function when constructing Meyer Shearlets.
Since sigmoid functions have finite smoothness, this paper proposes a method of constructing a class of sufficiently smooth sigmoid functions.
Based on Definition10, a necessary and sufficient condition for sigmoid functions to have infinite derivatives is as follows.Theorem 1.
Letvx be a sigmoid function, then the necessary and sufficient conditions for vx∈C∞ℝ are(i)
vx∈C∞0,1(ii)
for∀n∈ℕ, ∃vn0=vn1=0Theorem 2.
Letpx∈C∞ℝ, and when x≤0, px=0. When x>0, px>0, if(17)v˜x=pxpx+p1−x,then v˜x is a sigmoid function andv˜x∈C∞ℝ on its domainℝ.Proof.
First, we need to show thatv˜x is a sigmoid function. When x≤0, px=0, p1−x>0, we have(18)v˜x=pxpx+p1−x=0.
Whenx≥1, px>0, p1−x=0, thus(19)v˜x=pxpx+p1−x=1,
When0<x<1,(20)v˜x+v˜1−x=pxpx+p1−x+p1−xpx+p1−x=1.
According to Definition10, v˜x is a sigmoid function on the domainℝ.
Next, it remains to showv˜x∈C∞ℝ. Since px∈C∞ℝ, p1−x∈C∞ℝ. Also, because of px+P1−x>0x∈ℝ, v˜x∈C∞ℝ.
Let(21)gix=e−1/xi,ifx>0,0,ifx≤0,where i∈ℕ+ andgin0=0, thus gix∈C∞ℝ. Therefore, according to Theorem 2, we can obtain a class of sufficiently smooth sigmoid functions.(22)v˜a1,a2,…,amx=0,ifx<0,∑i=1maigix∑i=1maigix+∑i=1maigi1−x,if0≤x≤1,1,ifx>1,where ai∈ℝi=1,2,…,m, and when x>0, we have ∑i=1maigix>0. Four cases are chosen arbitrarily from “(22)” to obtain Figure 2.
Theorem3 follows Lemma 2.Figure 2
The image of four groups of sigmoid functions.Theorem 3.
Let the Fourier transform ofτt be(23)τ^ω=1,ifω≤2π3,cosπ2v˜3ω2π−1,if2π3<ω<4π3,0,ifω≥4π3.
Thus,τ^ω satisfies(I)
τ^2ω=Hωτ^ω, where Hω is a periodic function with period 2π(II)
∑k∈ℤτ^ω+2kπ2≡1
Andτt is a Meyer scale function with a limited spectrum, and v˜x is given by “(17)”.Proof.
We need to check thatτ^ω satisfies the conditions (i)–(iv) in Lemma 2.(i)
a=−b=−4π/3 satisfies a<0,b>0(ii)
b−a=8π/3,b/2−a=2π,b−a/2=2π satisfies b/2−a≤2π, 2π<b−a≤8/3π, b−a/2≤2π(iii)
Letc=b−a−2π=2π/3, thus when ω∈−2π/3,2π/3, τ^ω≡1(iv)
Whenω∈−4π/3,−2π/3, since when 0≤x≤1, we have v˜x+v˜1−x=1, thus(24)τ^ω2+τ^ω+2π2=cos2π2v˜−3ω2π−1+cos2π2v˜3ω+6π2π−1=cos2π2v˜−3ω2π−1+sin2π2v˜−3ω2π−1=1.
Therefore,τ^ω satisfies (I) & (II) in Lemma 2.
Sincev˜x∈C∞ℝ, τ^ω is bounded and continuous, andτ^0=1. From Lemma 1, we can tell that τt is a scale function of OMRA. Hence, τt is a Meyer scale function that has a limited spectrum in 4π/3.
By plugging “(23)” in “(11),” we will obtain the following theorem.Theorem 4.
Let the Fourier transform ofσt be(25)σ^ω=−e−iω/2sinπ2v˜3ω2π−1,if2π3<ω<4π3,−e−iω/2cosπ2v˜3ω4π−1,if4π3≤ω<8π3,0,ifω≤2π3orω≥8π3.Thus,σt is a Meyer wavelet function, wherev˜x is given from “(17)”.Based on these, the more general case of scale function and Meyer wavelet function is constructed in this paper.Theorem 5.
If(26)τ^ω=sinπ2v1cω−a−c2+12cosπ2v1cω−b+c2+12,where a<0,b>0,2π<b−a≤8/3π,b/2−a≤2π,b−a/2≤2π, c=b−a−2π, and vx is a sigmoid function, then the inverse Fourier transform of τ^ω, τt, is a Meyer scale function with limited spectrum and suppτ^=a,b.Proof.
It is obvious thata,b satisfy the conditions (i) and (ii) in Lemma 2. Let θ1t=π/2v1/ct+1/2, when t≤−c/2, θ1t=0; when t>c/2, θ1t=π/2. Now, it only remains to show that τ^ω satisfies (iii) and (iv) in Lemma 2.
Ifω∈a+c,b−c, then ω−a−c/2∈c/2,2π−c/2 and ω−b+c/2∈−c/2−2π,−c/2 which leads to θ1ω−a−c/2=π/2 and θ1ω−b+c/2=0. Therefore,(27)τ^ω=sinθ1ω−a−c2⋅cosθ1ω−b+c2=1.
Ifω∈a,a+c, then ω−a−c/2±2π∈−c/2±2π,c/2±2π. Since c=b−a−2π, 2π<b−a<8/3π, we have c∈0,2/3π.
Soω−a−c/2+2π>c/2 andω−a−c/2−2π<−c/2. Therefore, θ1ω−a−c/2+2π=π/2 andθ1ω−a−c/2−2π=0. By calculation, we obtain τ^ω2+τ^ω+2π2=1.
Therefore,τ^ω satisfies (iii) and (iv) in Lemma 2.
Ifω<a, then τ^ω=sinθ1ω−a−c/2cosθ1ω−b+c/2=0.
Ifω>b, then τ^ω=sinθ1ω−a−c/2cosθ1ω−b+c/2=0.
Thus, we havesuppτ^=a,b. Overall, according to Lemma 2, τt is a Meyer scale function with a limited spectrum.
By plugging “(26)” in “(11),” we can obtain Theorem 6.Theorem 6.
Given “(11)” and “(26),” the Fourier transform of Meyer wavelet σt is(28)σ^ω=−e−iω/2τ^ω+2πτ^ω/2+πτ^ω2.For theτ^ω in “(26),” when a=−b=−4π/3, c=b−a−2π=2π/3.We can construct a common expression for the Meyer scale function.(29)τ^ω=sinπ2v32πω+2cosπ2v32πω−1=0,ifω≤2π3,cosπ2v32πω−1,if2π3<ω<4π3,1,ifω≥4π3.In this paper, we construct a general scale function “(26)” by using multiresolution analysis, and then, the corresponding Meyer wavelet function is given by “(28).” The common Meyer wavelet scale function, “(29)”, is a special case of “(26).”


## 4. Denoising Simulation Experiment by Using Meyer Shearlet

First, Gaussian noise is added with an intensity of 10 to 4 images of colon pathological sections with different gland shapes. The followings are special cases of a class of sufficiently smooth sigmoid function, that is, “(22)”.Whenm=2,a1=0.1,a2=1,(30)v˜0.1,1x=0,ifx≤0,0.1e−x−1+e−x−20.1e−x−1+e−x−2+0.1e−1−x−1+e−1−x−2,if0<x<1,1,ifx≥1.when m=2,a1=3,a2=11,(31)v˜3,11x=0,ifx≤0,3e−x−1+11e−x−23e−x−1+11e−x−2+3e−1−x−1+11e−1−x−2,if0<x<1,1,ifx≥1.To denoise images, we use the Meyer wavelet by combining Theorem3 and Theorem 4 as a basis function of Shearlet, and then using MATLAB to carry out simulation experiments, and compare the results with Gaussian filtering, wavelet transform (sym4 and db4), and the Meyer Shearlet which is constructed by the traditional seventh-order polynomial sigmoid function as a basis function for image denoising. The results are shown in Figure 3.Figure 3
Denoised images of different algorithms for pathological sections of the colon.
(a)(b)In this paper, the evaluation indicators, mean squared error (MSE) and peak signal to noise ratio (PSNR), are used to test the effectiveness of the method. For the original image I and denoised image K with the size ofM×N, the formulae of MSE and PSNR are given as follows:(32)MSE=1M×N∑i=1M∑j=1NIi,j−Ki,j2,PSNR=10⋅log102552MSE,where Ii,j andKi,j represent the pixel gray level of imagesI andK at i,j. In general, the smaller the value of MSE, the larger the value of PSNR, and the better the method in this paper. Table 1 displays the results.Table 1
Objective evaluation results.
AlgorithmMSE (dB)PSNR (dB)Section 1Section 2Section 3Section 4Section 1Section 2Section 3Section 4Denoising with Gaussian filter7.897.517.608.0139.1639.3839.3239.10Denoising with wavelet transform (db2)4.575.945.915.2741.5340.3940.4140.91Denoising with wavelet transform (sym4)4.395.605.745.0041.7140.6540.5441.14Denoising with traditional Meyer Shearlet4.265.425.554.8541.8440.7940.6941.27Denoising with Meyer Shearlet in the form of “(30)”4.105.285.474.7442.0040.9140.7541.37Denoising with Meyer Shearlet in the form of “(31)”4.125.285.484.7641.9840.9140.7441.35From the visual effects in Figure3 and evaluation indicators in Table 1, it can be seen that in denoising images of colon pathological sections, the denoising effect of the Meyer Shearlets constructed by the sigmoid functions “(30)” and “(31)” provided in this paper is better than that of the Meyer Shearlets constructed by the Gaussian filtering, wavelet transform (sym4 and db4), and the traditional seventh-order polynomial sigmoid functions. Moreover, the edge details of denoising are processed better, and the contours are clearer. Therefore, the Meyer Shearlet denoising method is a relatively effective image denoising method. In order to better extract the edge of the image, this paper proposes an improved mathematical morphological edge detection algorithm.


## 5. Improved Edge Detection Algorithm Based on Mathematical Morphology

The mathematical morphology edge detection algorithm finds the edge of an image by performing logical operations on the structuring elements with a certain shape and the image. Common basic morphological operations are erosion, dilation, opening, and closing, and other complex morphological operations can be obtained by combining them, thereby improving the accuracy of edge extraction. Letfx,y be input images, sx,y be structuring elements, andDf, Ds be domains offx,y, sx,y, respectively. The definitions of dilation, erosion, opening, and closing in morphology are given as follows:(i)
Dilation:f⊕sx,y=maxfx−x′,y−y′+sx′,y′|x−x′,y−y′∈Df,x′,y′∈Ds(ii)
Erosion:fΘsx,y=minfx+x′,y+y′+sx′,y′|x+x′,y+y′∈Df,x′,y′∈Ds(iii)
Opening:f∘s=fΘs⊕s(iv)
Closing:f•s=f⊕sΘsA mathematical morphological edge detection algorithm is proposed in [11](33)Gd=Y⊕S2−Y,Ge=Y−YΘS2,G=Gd+Ge+minGd,Ge,where Y=f∘S1∘S2•S2, S1 is a5×5 structuring element in the shape of a rectangle, andS2 is a5×5 structuring element in the shape of a disc.Gd means gray dilation edge detection operator.Ge means gray erosion edge detection operator.It can be seen from the simulation experiments that the use of two structuring elements of the same scale to extract edge details is easy to lose, and there will be small jagged or discontinuous. To overcome these shortcomings, this paper proposes a multiscale and multistructure morphological edge detection algorithm.For the extraction of edges in each direction, we choose structuring elements in the size of3×3 and5×5 as follows:(34)b1=100010001,b2=010010010,b3=001010100,b4=000111000,b5=1000001000001000001000001,b6=0010000100001000010000100,b7=0000100010001000100010000,b8=0000000000111110000000000.In order to make the detected image edges more continuous, we use “(33)” to improve it. Because the morphological closing operation can fill the small holes at the edges of the image and make up for small cracks, this paper adds the closing operation after the dilation operation to make the image edges more smooth. Also, because the morphological opening operation has the effect of removing image glitches, an opening operation is added to the erosion operation. The improved mathematical morphology edge detection algorithm in this paper is as follows.Step 1.
The following improved edge detection operators are proposed:(35)Y1i=f⊕bi•bi−f,Y2i=f−fΘbi∘bi,where i=1,2,…,8, f represents denoised images. By taking the minimum of the edge obtained above, Ymini=minY1i,Y2i, the final improved operator is(36)Yi=Y1i+Y2i+Ymini.Step 2.
The3×3 structuring elements bii=1,2,3,4 are plugged from different directions in “(33)” to perform the edge detection, and then, the test results are fused to get small scale image edges, D1=1/4∑i=14Yi.Step 3.
To obtain the big scale image edges,D2=1/4∑i=58Yi, we can use the 5×5 structuring elements from different directions bii=5,6,7,8 and follow Step 2.Step 4.
The final image edgeD fuses the two edges, D1 andD2. We have D=1/2D1+D2.


## 6. Simulation Experiment Based on Fusion Algorithm

Combining Meyer Shearlet with improved mathematical morphology edge detection algorithm to perform the edge detection of the image of colon pathological section, the following specific process is obtained in Figure4.Figure 4
The image edge detection algorithm is based on edge fusion.In order to test the effectiveness of the edge detection algorithm based on Meyer Shearlet and mathematical morphology in the image edge detection of pathological sections of the colon, we have chosen four sections with noise added images from Figure3 to produce the detection. The results are shown in Figure 5.Figure 5
Edge detection of images of colon pathological sections.
(a)(b)The objective of evaluation criteria is also required besides the subjective evaluations of images which contributes to overcoming the human visual error, mental state, etc., and improving the accuracy of judgment. In addition to the peak signal to noise ratio (PSNR) and the mean squared error (MSE), information entropy, correlation coefficient, and the root mean square error (RMSE) are also commonly used as objective evaluation criteria.Information entropy is an important index to measure the richness of image information, which can be defined as(37)H=−∑i=0L−1Pi⋅logPi,where H represents the entropy of an image, Pi is the probability that the pixel gray value in an image is i, and L is the total gray level of an image. The greater the entropy of an image, the more information the image has.The correlation coefficient is a statistical analysis index gauging how closely the two facts are correlated which provides the correlation between two images. The value of the correlation coefficient lies in the interval−1,1. We have a positive correlation, negative correlation, and no linear relationship when CA,B is greater than 0, less than 0, and equals 0, respectively. The greater the absolute value of CA,B, the stronger the correlation we have. And, CA,B is defined as follows:(38)CA,B=∑i,jAi,j−A¯×Bi,j−B¯∑i,jAi,j−A¯2×∑i,jBi,j−B¯2,A¯=1M×N∑i=0M−1∑j=0N−1Ai,j,B¯=1M×N∑i=0M−1∑j=0N−1Bi,j,where Ai,j andBi,j are the gray values of two images, A¯ andB¯ are the average values, respectively. The closer the correlation coefficient is to 1, the closer the two images are correlated.The root mean square error (RMSE) between the imageF and the standard reference image R is defined as(39)RMSE=∑i=0M−1∑j=0N−1IFi,j−Ii,j2M×N,where M and N are the numbers of rows and columns of the image, respectively. The smaller the RSME, the closer the detected image is to the standard image.In order to prove the effectiveness of the algorithm given in this paper more objectively, MSE, PSNR, information entropy, correlation coefficient, and RMSE are used as evaluation indices. The objective evaluations of Figure5 are shown in Table 2.Table 2
Objective evaluation results.
ImagesMethodsEvaluation indicatorsMSE (dB)PSNR (dB)Information entropyCorrelation coefficientRMSESection 1Algorithm in [11]4.5041.606.3448−0.45431.3725Algorithm in original3.81542.2556.1397−0.46631.5692Fusion algorithm in this paper3.7342.416.4002−0.53071.63751Section 2Algorithm in [11]6.2940.146.6383−0.39749.0812Algorithm in original5.4240.556.6396−0.426310.0898Fusion algorithm in this paper5.1141.046.6441−0.487311.9120Section 3Algorithm in [11]5.6640.606.7226−0.34061.0502Algorithm in original4.9041.126.7553−0.38451.1294Fusion algorithm in this paper4.7541.366.8441−0.39231.2922Section 4Algorithm in [11]3.5242.676.1136−0.63361.7190Algorithm in original3.4342.966.2232−0.65871.8934Fusion algorithm in this paper3.0343.326.4380−0.67091.9249In the mathematical morphological edge detection algorithm, the construction of the morphological edge detection operator and the selection of structuring elements directly affect the accuracy of image edge detection. Based on the idea of multiscale and multistructure, this paper proposes an improved morphological edge detection algorithm and uses this algorithm to extract the denoised edges of colon pathological sections images. Based on observing the images, given in Figure5, which present the edge details from the algorithm in [11], the algorithm is the original and one given in this paper. Their objective evaluation results are presented in Table 2. It follows that for noisy images, the edge of the image extracted by the fusion algorithm given in this paper is smoother, which can effectively remove the noise while keeping the image position and shape better and thus leading to the basis for the follow-up application of computer-aided diagnosis system for feature extraction and pattern recognition.


## 7. Conclusion

In this paper, we construct a class of sufficiently smooth sigmoid functions and provide a more general expression of the Meyer scale function and Meyer wavelet function, making the commonly used Meyer scale function a special case of the more general Meyer scale function constructed in this paper. Then, taking the sufficiently smooth sigmoid functions as an example, the combination of Meyer wavelet and Shearlet is applied to denoising images of colon pathological sections. The denoising effect is better than other denoising algorithms, and the shape of sigmoid functions can be adjusted to achieve different denoising effects. Finally, an improved morphological edge detection algorithm is used to detect the edge of the colon pathological sections after the Meyer Shearlet denoising. The simulation results show that combining the Meyer Shearlet denoising method with the morphological edge detection algorithm can obtain a more continuous and clear edge of the colon pathological sections image. In addition, the results of the evaluation indicators MSE and PSNR also prove its effectiveness.



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