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.
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]). Let be a measure space,. We define as a set of Lebesgue measurable functions in , that is, . The -norm of is defined by, and is called a normed linear space. Especially, if we have, then when and follows that and being the most commonly used two normed linear spaces in the wavelet analysis.
Definition 2 (see [15]). Let , then is called the Fourier transform of.
If , then is the inverse Fourier transform of.
Let , then we callthe Fourier transform of . Also, if , then is the inverse Fourier transform of .
Definition 3 (see [15]). Let . For , the Fourier transform of is given by the following equation:where and are satisfying(i)For , , we have and, (ii), and , Then, is called the continuous Shearlet, where, , . And where is a dilation matrix, is a shear matrix. In the frequency domain, we haveThe support in the frequency domain of isThe support in the frequency domain of some Shearlets refers to Figure 1.
Let , is called the continuous Shearlet transform, and the inverse transform of is
Definition 4 (see [15]). Based on a binary sampling of the dilation matrix and integer sampling of, that is, , . The collection of matrices where and . Thus, the discrete Shearlet transform is , where .
Now, the Fourier transform of is given by “(1)” which satisfies , , and , also . Especially, the Meyer wavelets are commonly used as the basis functions , and its Fourier transform iswhereThe Fourier transform of is whereTo 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]). Assume is the Fourier transform of , for some if when, we have , then has limited spectrum in .
Definition 6 (see [16]). A column of closed subspace in Hilbert space is called an orthogonal multiple resolution analysis (OMRA) if it satisfies the following conditions:(i)Nesting: for all(ii)Scaling: where (iii)Separation: (iv)Density: (v)Orthonormal basis: there exists , such that the set of functions is an orthonormal basis of , where is the scale function of the OMRA, and is called subspace of proximity or the scale function of
Definition 7 (see [16]). If , and its Fourier transform satisfies the admissibility condition, then is a basic wavelet or mother wavelet function.
Definition 8 (see [16]). Let be a scale function of OMRA, then there exists such thatThus, forms a square summable sequence and is called two-scale coefficients of the scale function. Equation (9) is known as two-scale relation for the scaling function of the OMRA.
Let , then is a periodic function with being its period. And. When , we have the wavelet functionMeanwhile, the Fourier transform of is
Lemma 1. Let be the bounded continuous Fourier transform of and satisfying(I)(II)(III), where is a periodic function with period Then, the closed subspace sequence generated by iswhich forms an OMRA of .
Lemma 2. Let be a continuous real function, and , thus the necessary and sufficient conditions such that satisfies the following:(I), where is a periodic function with period(II)are(i)(ii),(iii)Let , when , we have (iv)When ,
Definition 9 (see [16]). We denote by the set of continuous real-valued functions defined on, which is a real vector space with respect to ordinary addition and scalar multiplication of .
Definition 10 (see [17]). If is a continuous function that satisfiesand , then 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 polynomial be the cumulative distribution function (CDF) of the random variable, where is symmetric beta-distributed; that is,∼, for some , we havewhere ,Thus, we obtain the relative sigmoid functionAccording to the result of the calculation, when and is a continuous nondifferentiable function on; when and is a first-order continuous differentiable function; when and is a second-order continuous differentiable function on ; when and is a third-order continuous differentiable function on. Especially, 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 Definition 10, a necessary and sufficient condition for sigmoid functions to have infinite derivatives is as follows.
Theorem 1. Let be a sigmoid function, then the necessary and sufficient conditions for are(i)(ii)for ,
Theorem 2. Let , and when , . When , , ifthen is a sigmoid function and on its domain.
Proof. First, we need to show that is a sigmoid function. When , , , we haveWhen , , , thusWhen ,According to Definition 10, is a sigmoid function on the domain.
Next, it remains to show . Since , . Also, because of , .
Letwhere and, thus . Therefore, according to Theorem 2, we can obtain a class of sufficiently smooth sigmoid functions.where , and when , we have . Four cases are chosen arbitrarily from “(22)” to obtain Figure 2.
Theorem 3 follows Lemma 2.
Theorem 3. Let the Fourier transform of beThus, satisfies(I), where is a periodic function with period (II)And is a Meyer scale function with a limited spectrum, and is given by “(17)”.
Proof. We need to check that satisfies the conditions (i)–(iv) in Lemma 2.(i) satisfies (ii) satisfies , , (iii)Let , thus when , (iv)When , since when , we have , thusTherefore, satisfies (I) & (II) in Lemma 2.
Since , is bounded and continuous, and. From Lemma 1, we can tell that is a scale function of OMRA. Hence, is a Meyer scale function that has a limited spectrum in .
By plugging “(23)” in “(11),” we will obtain the following theorem.
Theorem 4. Let the Fourier transform of be
Thus, is a Meyer wavelet function, where 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. Ifwhere , , and is a sigmoid function, then the inverse Fourier transform of , , is a Meyer scale function with limited spectrum and .
Proof. It is obvious that satisfy the conditions (i) and (ii) in Lemma 2. Let , when , ; when , . Now, it only remains to show that satisfies (iii) and (iv) in Lemma 2.
If , then and which leads to and . Therefore,If , then . Since , , we have .
So and. Therefore, and. By calculation, we obtain .
Therefore, satisfies (iii) and (iv) in Lemma 2.
If , then .
If , then .
Thus, we have . Overall, according to Lemma 2, 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 is
For the in “(26),” when , .
We can construct a common expression for the Meyer scale function.
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)”.
When ,when ,
To denoise images, we use the Meyer wavelet by combining Theorem 3 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.
(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 of , the formulae of MSE and PSNR are given as follows:where and represent the pixel gray level of images and at . 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.
From the visual effects in Figure 3 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. Let be input images, be structuring elements, and, be domains of, , respectively. The definitions of dilation, erosion, opening, and closing in morphology are given as follows:(i)Dilation: (ii)Erosion: (iii)Opening: (iv)Closing:
A mathematical morphological edge detection algorithm is proposed in [11]where , is a structuring element in the shape of a rectangle, and is a structuring element in the shape of a disc. means gray dilation edge detection operator. 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 of and as follows:
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:where , represents denoised images. By taking the minimum of the edge obtained above, , the final improved operator is
Step 2. The structuring elements 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, .
Step 3. To obtain the big scale image edges, , we can use the structuring elements from different directions and follow Step 2.
Step 4. The final image edge fuses the two edges, and. We have .
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 Figure 4.
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 Figure 3 to produce the detection. The results are shown in Figure 5.
(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 aswhere represents the entropy of an image, is the probability that the pixel gray value in an image is , and 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 . We have a positive correlation, negative correlation, and no linear relationship when is greater than 0, less than 0, and equals 0, respectively. The greater the absolute value of , the stronger the correlation we have. And, is defined as follows:where and are the gray values of two images, and 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 image and the standard reference image is defined aswhere and 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 Figure 5 are shown in Table 2.
In 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 Figure 5, 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.
Data Availability
The data that support the findings of this study are available from the submitting or corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 51875142) and (Grant no. 11871181).